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epfl-collab | In terms of the \textbf{bias-variance} decomposition, a 1-nearest neighbor classifier has \rule{2cm}{0.15mm} than a 3-nearest neighbor classifier. | ['lower bias', 'higher variance', 'lower variance', 'higher bias'] | B | null | Document 1:::
K-nearest neighbors algorithm
This value is the average of the values of k nearest neighbors. If k = 1, then the output is simply assigned to the value of that single nearest neighbor.k-NN is a type of classification where the function is only approximated locally and all computation is deferred until function evaluation. Since this algorithm relies on distance for classification, if the features represent different physical units or come in vastly different scales then normalizing the training data can improve its accuracy dramatically.Both for classification and regression, a useful technique can be to assign weights to the contributions of the neighbors, so that the nearer neighbors contribute more to the average than the more distant ones.
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Structured kNN
Structured k-Nearest Neighbours is a machine learning algorithm that generalizes the k-Nearest Neighbors (kNN) classifier. Whereas the kNN classifier supports binary classification, multiclass classification and regression, the Structured kNN (SkNN) allows training of a classifier for general structured output labels. As an example, a sample instance might be a natural language sentence, and the output label is an annotated parse tree. Training a classifier consists of showing pairs of correct sample and output label pairs. After training, the structured kNN model allows one to predict for new sample instances the corresponding output label; that is, given a natural language sentence, the classifier can produce the most likely parse tree.
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Nearest centroid classifier
In machine learning, a nearest centroid classifier or nearest prototype classifier is a classification model that assigns to observations the label of the class of training samples whose mean (centroid) is closest to the observation. When applied to text classification using word vectors containing tf*idf weights to represent documents, the nearest centroid classifier is known as the Rocchio classifier because of its similarity to the Rocchio algorithm for relevance feedback.An extended version of the nearest centroid classifier has found applications in the medical domain, specifically classification of tumors.
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K-nearest neighbors algorithm
For example, a common weighting scheme consists in giving each neighbor a weight of 1/d, where d is the distance to the neighbor.The neighbors are taken from a set of objects for which the class (for k-NN classification) or the object property value (for k-NN regression) is known. This can be thought of as the training set for the algorithm, though no explicit training step is required. A peculiarity of the k-NN algorithm is that it is sensitive to the local structure of the data.
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K-nearest neighbors algorithm
An object is classified by a plurality vote of its neighbors, with the object being assigned to the class most common among its k nearest neighbors (k is a positive integer, typically small). If k = 1, then the object is simply assigned to the class of that single nearest neighbor. In k-NN regression, the output is the property value for the object. |
epfl-collab | In deep learning, which of these are hyper-parameters? | ['The learning rate', 'The weights $\\mathbf{W}^{[l]}$ and biases $\\mathbf{b}^{[l]}$', 'The number of layers', 'The type of weight initialization'] | A | null | Document 1:::
Hyperparameter (machine learning)
In machine learning, a hyperparameter is a parameter whose value is used to control the learning process. By contrast, the values of other parameters (typically node weights) are derived via training. Hyperparameters can be classified as model hyperparameters, that cannot be inferred while fitting the machine to the training set because they refer to the model selection task, or algorithm hyperparameters, that in principle have no influence on the performance of the model but affect the speed and quality of the learning process. An example of a model hyperparameter is the topology and size of a neural network.
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Hyperparameter (machine learning)
In machine learning, a hyperparameter is a parameter whose value is used to control the learning process. By contrast, the values of other parameters (typically node weights) are derived via training. Hyperparameters can be classified as model hyperparameters, that cannot be inferred while fitting the machine to the training set because they refer to the model selection task, or algorithm hyperparameters, that in principle have no influence on the performance of the model but affect the speed and quality of the learning process. An example of a model hyperparameter is the topology and size of a neural network.
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Grid search
In machine learning, hyperparameter optimization or tuning is the problem of choosing a set of optimal hyperparameters for a learning algorithm. A hyperparameter is a parameter whose value is used to control the learning process. By contrast, the values of other parameters (typically node weights) are learned.
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Grid search
The same kind of machine learning model can require different constraints, weights or learning rates to generalize different data patterns. These measures are called hyperparameters, and have to be tuned so that the model can optimally solve the machine learning problem. Hyperparameter optimization finds a tuple of hyperparameters that yields an optimal model which minimizes a predefined loss function on given independent data. The objective function takes a tuple of hyperparameters and returns the associated loss. Cross-validation is often used to estimate this generalization performance, and therefore choose the set of values for hyperparameters that maximize it.
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Hyperparameter (machine learning)
Examples of algorithm hyperparameters are learning rate and batch size as well as mini-batch size. Batch size can refer to the full data sample where mini-batch size would be a smaller sample set. Different model training algorithms require different hyperparameters, some simple algorithms (such as ordinary least squares regression) require none. Given these hyperparameters, the training algorithm learns the parameters from the data. For instance, LASSO is an algorithm that adds a regularization hyperparameter to ordinary least squares regression, which has to be set before estimating the parameters through the training algorithm. |
epfl-collab | What is the mean squared error of $f$ for a sample, where $\textbf{x}$ is an input, $y$ a target and $f(\textbf{x},W)$ the mapping function ?
(One answer) | [' $P(y=i |\\textbf{x}) = \\frac{e^{\\textbf{f}_i(\\textbf{x},W)}}{\\sum_j e^{\\textbf{f}_j(\\textbf{x},W)}}$ ', ' $||y - f(\\textbf{x},W)||^2 $ ', ' $-\\log(P(y=i | \\textbf{x})) = -\\log(\\frac{e^{\\textbf{f}_i(\\textbf{x},W)}}{\\sum_j e^{\\textbf{f}_j(\\textbf{x},W)}})$ ', ' $||y - f(\\textbf{x},W)|| $'] | B | null | Document 1:::
Minimum mean-square error
In statistics and signal processing, a minimum mean square error (MMSE) estimator is an estimation method which minimizes the mean square error (MSE), which is a common measure of estimator quality, of the fitted values of a dependent variable. In the Bayesian setting, the term MMSE more specifically refers to estimation with quadratic loss function. In such case, the MMSE estimator is given by the posterior mean of the parameter to be estimated.
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Mean squared deviation
In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between the estimated values and the actual value. MSE is a risk function, corresponding to the expected value of the squared error loss. The fact that MSE is almost always strictly positive (and not zero) is because of randomness or because the estimator does not account for information that could produce a more accurate estimate.
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Minimum mean-square error
Let x {\displaystyle x} be a n × 1 {\displaystyle n\times 1} hidden random vector variable, and let y {\displaystyle y} be a m × 1 {\displaystyle m\times 1} known random vector variable (the measurement or observation), both of them not necessarily of the same dimension. An estimator x ^ ( y ) {\displaystyle {\hat {x}}(y)} of x {\displaystyle x} is any function of the measurement y {\displaystyle y} . The estimation error vector is given by e = x ^ − x {\displaystyle e={\hat {x}}-x} and its mean squared error (MSE) is given by the trace of error covariance matrix MSE = tr { E { ( x ^ − x ) ( x ^ − x ) T } } = E { ( x ^ − x ) T ( x ^ − x ) } , {\displaystyle \operatorname {MSE} =\operatorname {tr} \left\{\operatorname {E} \{({\hat {x}}-x)({\hat {x}}-x)^{T}\}\right\}=\operatorname {E} \{({\hat {x}}-x)^{T}({\hat {x}}-x)\},} where the expectation E {\displaystyle \operatorname {E} } is taken over x {\displaystyle x} conditioned on y {\displaystyle y} . When x {\displaystyle x} is a scalar variable, the MSE expression simplifies to E { ( x ^ − x ) 2 } {\displaystyle \operatorname {E} \left\{({\hat {x}}-x)^{2}\right\}} .
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Mean square deviation
If a vector of n {\displaystyle n} predictions is generated from a sample of n {\displaystyle n} data points on all variables, and Y {\displaystyle Y} is the vector of observed values of the variable being predicted, with Y ^ {\displaystyle {\hat {Y}}} being the predicted values (e.g. as from a least-squares fit), then the within-sample MSE of the predictor is computed as MSE = 1 n ∑ i = 1 n ( Y i − Y i ^ ) 2 . {\displaystyle \operatorname {MSE} ={\frac {1}{n}}\sum _{i=1}^{n}\left(Y_{i}-{\hat {Y_{i}}}\right)^{2}.} In other words, the MSE is the mean ( 1 n ∑ i = 1 n ) {\textstyle \left({\frac {1}{n}}\sum _{i=1}^{n}\right)} of the squares of the errors ( Y i − Y i ^ ) 2 {\textstyle \left(Y_{i}-{\hat {Y_{i}}}\right)^{2}} . This is an easily computable quantity for a particular sample (and hence is sample-dependent).
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Mean square deviation
The MSE either assesses the quality of a predictor (i.e., a function mapping arbitrary inputs to a sample of values of some random variable), or of an estimator (i.e., a mathematical function mapping a sample of data to an estimate of a parameter of the population from which the data is sampled). The definition of an MSE differs according to whether one is describing a predictor or an estimator. |
epfl-collab | When using linear regression, how do you help prevent numerical instabilities? (One or multiple answers) | ['reduce learning rate', 'remove degenerate features', 'add a regularization term', 'add more features'] | C | null | Document 1:::
Linear Regression
In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of one explanatory variable is called simple linear regression; for more than one, the process is called multiple linear regression. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable.In linear regression, the relationships are modeled using linear predictor functions whose unknown model parameters are estimated from the data. Such models are called linear models.
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Linear Regression
This is because models which depend linearly on their unknown parameters are easier to fit than models which are non-linearly related to their parameters and because the statistical properties of the resulting estimators are easier to determine. Linear regression has many practical uses. Most applications fall into one of the following two broad categories: If the goal is error reduction in prediction or forecasting, linear regression can be used to fit a predictive model to an observed data set of values of the response and explanatory variables.
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Polynomial regression
The goal of regression analysis is to model the expected value of a dependent variable y in terms of the value of an independent variable (or vector of independent variables) x. In simple linear regression, the model y = β 0 + β 1 x + ε , {\displaystyle y=\beta _{0}+\beta _{1}x+\varepsilon ,\,} is used, where ε is an unobserved random error with mean zero conditioned on a scalar variable x. In this model, for each unit increase in the value of x, the conditional expectation of y increases by β1 units. In many settings, such a linear relationship may not hold. For example, if we are modeling the yield of a chemical synthesis in terms of the temperature at which the synthesis takes place, we may find that the yield improves by increasing amounts for each unit increase in temperature. In this case, we might propose a quadratic model of the form y = β 0 + β 1 x + β 2 x 2 + ε .
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Polynomial regression
In general, we can model the expected value of y as an nth degree polynomial, yielding the general polynomial regression model y = β 0 + β 1 x + β 2 x 2 + β 3 x 3 + ⋯ + β n x n + ε . {\displaystyle y=\beta _{0}+\beta _{1}x+\beta _{2}x^{2}+\beta _{3}x^{3}+\cdots +\beta _{n}x^{n}+\varepsilon .\,} Conveniently, these models are all linear from the point of view of estimation, since the regression function is linear in terms of the unknown parameters β0, β1, .... Therefore, for least squares analysis, the computational and inferential problems of polynomial regression can be completely addressed using the techniques of multiple regression. This is done by treating x, x2, ... as being distinct independent variables in a multiple regression model.
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Proper linear model
In statistics, a proper linear model is a linear regression model in which the weights given to the predictor variables are chosen in such a way as to optimize the relationship between the prediction and the criterion. Simple regression analysis is the most common example of a proper linear model. Unit-weighted regression is the most common example of an improper linear model. |
epfl-collab | You write a Python code to optimize the weights of your linear regression with 10 features \textbf{using gradient descent} for 500 epochs. What is the minimum number of for-loops you need to perform your optimization? | ['No for-loop is really necessary. Everything can be vectorized', 'Only one for-loop to iterate over the weights.', 'Two for-loops, one to iterate over the weights and the other to iterate over the epochs', 'Only one for-loop to iterate over the epochs.'] | D | null | Document 1:::
Gradient descent
In mathematics, gradient descent (also often called steepest descent) is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the gradient (or approximate gradient) of the function at the current point, because this is the direction of steepest descent. Conversely, stepping in the direction of the gradient will lead to a local maximum of that function; the procedure is then known as gradient ascent. It is particularly useful in machine learning for minimizing the cost or loss function.
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Coordinate descent
Coordinate descent is an optimization algorithm that successively minimizes along coordinate directions to find the minimum of a function. At each iteration, the algorithm determines a coordinate or coordinate block via a coordinate selection rule, then exactly or inexactly minimizes over the corresponding coordinate hyperplane while fixing all other coordinates or coordinate blocks. A line search along the coordinate direction can be performed at the current iterate to determine the appropriate step size. Coordinate descent is applicable in both differentiable and derivative-free contexts.
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Gradient descent
Gradient descent should not be confused with local search algorithms, although both are iterative methods for optimization. Gradient descent is generally attributed to Augustin-Louis Cauchy, who first suggested it in 1847. Jacques Hadamard independently proposed a similar method in 1907. Its convergence properties for non-linear optimization problems were first studied by Haskell Curry in 1944, with the method becoming increasingly well-studied and used in the following decades.A simple extension of gradient descent, stochastic gradient descent, serves as the most basic algorithm used for training most deep networks today.
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Learning rate
While the descent direction is usually determined from the gradient of the loss function, the learning rate determines how big a step is taken in that direction. A too high learning rate will make the learning jump over minima but a too low learning rate will either take too long to converge or get stuck in an undesirable local minimum.In order to achieve faster convergence, prevent oscillations and getting stuck in undesirable local minima the learning rate is often varied during training either in accordance to a learning rate schedule or by using an adaptive learning rate. The learning rate and its adjustments may also differ per parameter, in which case it is a diagonal matrix that can be interpreted as an approximation to the inverse of the Hessian matrix in Newton's method. The learning rate is related to the step length determined by inexact line search in quasi-Newton methods and related optimization algorithms.
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Learning rate
While the descent direction is usually determined from the gradient of the loss function, the learning rate determines how big a step is taken in that direction. A too high learning rate will make the learning jump over minima but a too low learning rate will either take too long to converge or get stuck in an undesirable local minimum.In order to achieve faster convergence, prevent oscillations and getting stuck in undesirable local minima the learning rate is often varied during training either in accordance to a learning rate schedule or by using an adaptive learning rate. The learning rate and its adjustments may also differ per parameter, in which case it is a diagonal matrix that can be interpreted as an approximation to the inverse of the Hessian matrix in Newton's method. The learning rate is related to the step length determined by inexact line search in quasi-Newton methods and related optimization algorithms. |
epfl-collab | Which loss function(s) should you use? (One or multiple answers) | ['hinge loss', 'L1 loss', 'mean square error (MSE) loss', 'cross entropy loss'] | D | null | Document 1:::
Loss functions for classification
These are called margin-based loss functions. Choosing a margin-based loss function amounts to choosing ϕ {\displaystyle \phi } . Selection of a loss function within this framework impacts the optimal f ϕ ∗ {\displaystyle f_{\phi }^{*}} which minimizes the expected risk.
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Loss functions for classification
As a result, it is better to substitute loss function surrogates which are tractable for commonly used learning algorithms, as they have convenient properties such as being convex and smooth. In addition to their computational tractability, one can show that the solutions to the learning problem using these loss surrogates allow for the recovery of the actual solution to the original classification problem.
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Loss functions for classification
In machine learning and mathematical optimization, loss functions for classification are computationally feasible loss functions representing the price paid for inaccuracy of predictions in classification problems (problems of identifying which category a particular observation belongs to). Given X {\displaystyle {\mathcal {X}}} as the space of all possible inputs (usually X ⊂ R d {\displaystyle {\mathcal {X}}\subset \mathbb {R} ^{d}} ), and Y = { − 1 , 1 } {\displaystyle {\mathcal {Y}}=\{-1,1\}} as the set of labels (possible outputs), a typical goal of classification algorithms is to find a function f: X → Y {\displaystyle f:{\mathcal {X}}\to {\mathcal {Y}}} which best predicts a label y {\displaystyle y} for a given input x → {\displaystyle {\vec {x}}} . However, because of incomplete information, noise in the measurement, or probabilistic components in the underlying process, it is possible for the same x → {\displaystyle {\vec {x}}} to generate different y {\displaystyle y} . As a result, the goal of the learning problem is to minimize expected loss (also known as the risk), defined as I = ∫ X × Y V ( f ( x → ) , y ) p ( x → , y ) d x → d y {\displaystyle I=\displaystyle \int _{{\mathcal {X}}\times {\mathcal {Y}}}V(f({\vec {x}}),y)\,p({\vec {x}},y)\,d{\vec {x}}\,dy} where V ( f ( x → ) , y ) {\displaystyle V(f({\vec {x}}),y)} is a given loss function, and p ( x → , y ) {\displaystyle p({\vec {x}},y)} is the probability density function of the process that generated the data, which can equivalently be written as p ( x → , y ) = p ( y ∣ x → ) p ( x → ) .
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Loss functions for classification
{\displaystyle p({\vec {x}},y)=p(y\mid {\vec {x}})p({\vec {x}}).} Within classification, several commonly used loss functions are written solely in terms of the product of the true label y {\displaystyle y} and the predicted label f ( x → ) {\displaystyle f({\vec {x}})} . Therefore, they can be defined as functions of only one variable υ = y f ( x → ) {\displaystyle \upsilon =yf({\vec {x}})} , so that V ( f ( x → ) , y ) = ϕ ( y f ( x → ) ) = ϕ ( υ ) {\displaystyle V(f({\vec {x}}),y)=\phi (yf({\vec {x}}))=\phi (\upsilon )} with a suitably chosen function ϕ: R → R {\displaystyle \phi :\mathbb {R} \to \mathbb {R} } .
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Loss functions for classification
Given the binary nature of classification, a natural selection for a loss function (assuming equal cost for false positives and false negatives) would be the 0-1 loss function (0–1 indicator function), which takes the value of 0 if the predicted classification equals that of the true class or a 1 if the predicted classification does not match the true class. This selection is modeled by V ( f ( x → ) , y ) = H ( − y f ( x → ) ) {\displaystyle V(f({\vec {x}}),y)=H(-yf({\vec {x}}))} where H {\displaystyle H} indicates the Heaviside step function. However, this loss function is non-convex and non-smooth, and solving for the optimal solution is an NP-hard combinatorial optimization problem. |
epfl-collab | Fill the missing line of code: (one answer)\\
\hspace*{.5cm} \#code missing\\
\hspace*{.5cm} np.mean(np.random.randn(1000))\\ | ['import numpy', 'import np.mean\\\\\n\t\timport np.random', 'import numpy as np', 'import np'] | C | null | Document 1:::
Marsaglia polar method
The Marsaglia polar method is a pseudo-random number sampling method for generating a pair of independent standard normal random variables.Standard normal random variables are frequently used in computer science, computational statistics, and in particular, in applications of the Monte Carlo method. The polar method works by choosing random points (x, y) in the square −1 < x < 1, −1 < y < 1 until 0 < s = x 2 + y 2 < 1 , {\displaystyle 0
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Mean line segment length
In geometry, the mean line segment length is the average length of a line segment connecting two points chosen uniformly at random in a given shape. In other words, it is the expected Euclidean distance between two random points, where each point in the shape is equally likely to be chosen. Even for simple shapes such as a square or a triangle, solving for the exact value of their mean line segment lengths can be difficult because their closed-form expressions can get quite complicated. As an example, consider the following question: What is the average distance between two randomly chosen points inside a square with side length 1?While the question may seem simple, it has a fairly complicated answer; the exact value for this is 2 + 2 + 5 ln ( 1 + 2 ) 15 {\displaystyle {\frac {2+{\sqrt {2}}+5\ln(1+{\sqrt {2}})}{15}}} .
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Systematic sampling
In this case, rather than simply counting through elements of the population and selecting every kth unit, we allocate each element a space along a number line according to its selection probability. We then generate a random start from a uniform distribution between 0 and 1, and move along the number line in steps of 1. Example: We have a population of 5 units (A to E).
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Random number sampling
Non-uniform random variate generation or pseudo-random number sampling is the numerical practice of generating pseudo-random numbers (PRN) that follow a given probability distribution. Methods are typically based on the availability of a uniformly distributed PRN generator. Computational algorithms are then used to manipulate a single random variate, X, or often several such variates, into a new random variate Y such that these values have the required distribution. The first methods were developed for Monte-Carlo simulations in the Manhattan project, published by John von Neumann in the early 1950s.
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Sparse Distributed Memory
The mean of the binomial distribution is n/2, and the variance is n/4. This distribution function will be denoted by N(d). The normal distribution F with mean n/2 and standard deviation n / 2 {\displaystyle {\sqrt {n}}/2} is a good approximation to it: N(d) = Pr{d(x, y) ≤ d} ≅ F{(d − n / 2)/ n / 4 {\displaystyle {\sqrt {n/4}}} } Tendency to orthogonalityAn outstanding property of N is that most of it lies at approximately the mean (indifference) distance n/2 from a point (and its complement). In other words, most of the space is nearly orthogonal to any given point, and the larger n is, the more pronounced is this effect. |
epfl-collab | What is the output of the following block of Python code? (one answer) \\
\verb|my_string = `computational'| \\
\verb|print(my_string[1])|\\
\verb|print(my_string[3:5])|
\vspace{0.25cm} | ['o\\\\put', 'o\\\\pu', 'c\\\\mp', 'c\\\\mpu'] | B | null | Document 1:::
String (computer science)
String may also denote more general arrays or other sequence (or list) data types and structures. Depending on the programming language and precise data type used, a variable declared to be a string may either cause storage in memory to be statically allocated for a predetermined maximum length or employ dynamic allocation to allow it to hold a variable number of elements. When a string appears literally in source code, it is known as a string literal or an anonymous string.In formal languages, which are used in mathematical logic and theoretical computer science, a string is a finite sequence of symbols that are chosen from a set called an alphabet.
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String functions
The most basic example of a string function is the length(string) function. This function returns the length of a string literal. e.g. length("hello world") would return 11.Other languages may have string functions with similar or exactly the same syntax or parameters or outcomes. For example, in many languages the length function is usually represented as len(string). The below list of common functions aims to help limit this confusion.
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String (computer science)
In computer programming, a string is traditionally a sequence of characters, either as a literal constant or as some kind of variable. The latter may allow its elements to be mutated and the length changed, or it may be fixed (after creation). A string is generally considered as a data type and is often implemented as an array data structure of bytes (or words) that stores a sequence of elements, typically characters, using some character encoding.
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Multi-string Turing machine with input and output
In computational complexity theory, DSPACE or SPACE is the computational resource describing the resource of memory space for a deterministic Turing machine. It represents the total amount of memory space that a "normal" physical computer would need to solve a given computational problem with a given algorithm.
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Kolmogorov Complexity
If P is a program which outputs a string x, then P is a description of x. The length of the description is just the length of P as a character string, multiplied by the number of bits in a character (e.g., 7 for ASCII). We could, alternatively, choose an encoding for Turing machines, where an encoding is a function which associates to each Turing Machine M a bitstring . If M is a Turing Machine which, on input w, outputs string x, then the concatenated string w is a description of x. For theoretical analysis, this approach is more suited for constructing detailed formal proofs and is generally preferred in the research literature. |
epfl-collab | In Machine Learning, we want to learn the \textbf{parameters W} for the mapping function f: $y=f(x,W) +\epsilon$ where x is the input, y the output, and $\epsilon$ the error term.\\
(One or multiple answers) | ['When f: $R \\rightarrow \\{1,..N\\}$, it is a classification task', 'When f: $R^M \\rightarrow R$, it is a classification task ', 'When f: $R^M \\rightarrow R$, it is a regression task', 'When f: $R^M \\rightarrow \\{1,..N\\}$, it is a classification task'] | A | null | Document 1:::
Learning with errors
There exists a certain unknown linear function f: Z q n → Z q {\displaystyle f:\mathbb {Z} _{q}^{n}\rightarrow \mathbb {Z} _{q}} , and the input to the LWE problem is a sample of pairs ( x , y ) {\displaystyle (\mathbf {x} ,y)} , where x ∈ Z q n {\displaystyle \mathbf {x} \in \mathbb {Z} _{q}^{n}} and y ∈ Z q {\displaystyle y\in \mathbb {Z} _{q}} , so that with high probability y = f ( x ) {\displaystyle y=f(\mathbf {x} )} . Furthermore, the deviation from the equality is according to some known noise model. The problem calls for finding the function f {\displaystyle f} , or some close approximation thereof, with high probability.
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Linear classification
If the input feature vector to the classifier is a real vector x → {\displaystyle {\vec {x}}} , then the output score is y = f ( w → ⋅ x → ) = f ( ∑ j w j x j ) , {\displaystyle y=f({\vec {w}}\cdot {\vec {x}})=f\left(\sum _{j}w_{j}x_{j}\right),} where w → {\displaystyle {\vec {w}}} is a real vector of weights and f is a function that converts the dot product of the two vectors into the desired output. (In other words, w → {\displaystyle {\vec {w}}} is a one-form or linear functional mapping x → {\displaystyle {\vec {x}}} onto R.) The weight vector w → {\displaystyle {\vec {w}}} is learned from a set of labeled training samples.
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Loss functions for classification
In machine learning and mathematical optimization, loss functions for classification are computationally feasible loss functions representing the price paid for inaccuracy of predictions in classification problems (problems of identifying which category a particular observation belongs to). Given X {\displaystyle {\mathcal {X}}} as the space of all possible inputs (usually X ⊂ R d {\displaystyle {\mathcal {X}}\subset \mathbb {R} ^{d}} ), and Y = { − 1 , 1 } {\displaystyle {\mathcal {Y}}=\{-1,1\}} as the set of labels (possible outputs), a typical goal of classification algorithms is to find a function f: X → Y {\displaystyle f:{\mathcal {X}}\to {\mathcal {Y}}} which best predicts a label y {\displaystyle y} for a given input x → {\displaystyle {\vec {x}}} . However, because of incomplete information, noise in the measurement, or probabilistic components in the underlying process, it is possible for the same x → {\displaystyle {\vec {x}}} to generate different y {\displaystyle y} . As a result, the goal of the learning problem is to minimize expected loss (also known as the risk), defined as I = ∫ X × Y V ( f ( x → ) , y ) p ( x → , y ) d x → d y {\displaystyle I=\displaystyle \int _{{\mathcal {X}}\times {\mathcal {Y}}}V(f({\vec {x}}),y)\,p({\vec {x}},y)\,d{\vec {x}}\,dy} where V ( f ( x → ) , y ) {\displaystyle V(f({\vec {x}}),y)} is a given loss function, and p ( x → , y ) {\displaystyle p({\vec {x}},y)} is the probability density function of the process that generated the data, which can equivalently be written as p ( x → , y ) = p ( y ∣ x → ) p ( x → ) .
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Learning rule
Depending on the complexity of actual model being simulated, the learning rule of the network can be as simple as an XOR gate or mean squared error, or as complex as the result of a system of differential equations. The learning rule is one of the factors which decides how fast or how accurately the artificial network can be developed. Depending upon the process to develop the network there are three main models of machine learning: Unsupervised learning Supervised learning Reinforcement learning
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Learning rule
Depending on the complexity of actual model being simulated, the learning rule of the network can be as simple as an XOR gate or mean squared error, or as complex as the result of a system of differential equations. The learning rule is one of the factors which decides how fast or how accurately the artificial network can be developed. Depending upon the process to develop the network there are three main models of machine learning: Unsupervised learning Supervised learning Reinforcement learning |
epfl-collab | Principle Component Analysis (PCA) is a technique for... | ['variance normalization', 'data augmentation', 'feature extraction', 'dimensionality reduction'] | D | null | Document 1:::
Principal components
Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and enabling the visualization of multidimensional data. Formally, PCA is a statistical technique for reducing the dimensionality of a dataset. This is accomplished by linearly transforming the data into a new coordinate system where (most of) the variation in the data can be described with fewer dimensions than the initial data. Many studies use the first two principal components in order to plot the data in two dimensions and to visually identify clusters of closely related data points.
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Principal components
PCA is most commonly used when many of the variables are highly correlated with each other and it is desirable to reduce their number to an independent set. PCA is used in exploratory data analysis and for making predictive models. It is commonly used for dimensionality reduction by projecting each data point onto only the first few principal components to obtain lower-dimensional data while preserving as much of the data's variation as possible.
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Principal components
Principal component analysis is the process of computing the principal components and using them to perform a change of basis on the data, sometimes using only the first few principal components and ignoring the rest. In data analysis, the first principal component of a set of p {\displaystyle p} variables, presumed to be jointly normally distributed, is the derived variable formed as a linear combination of the original variables that explains the most variance. The second principal component explains the most variance in what is left once the effect of the first component is removed, and we may proceed through p {\displaystyle p} iterations until all the variance is explained.
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Principal components
Thus, the principal components are often computed by eigendecomposition of the data covariance matrix or singular value decomposition of the data matrix. PCA is the simplest of the true eigenvector-based multivariate analyses and is closely related to factor analysis.
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Sparse PCA
Sparse principal component analysis (SPCA or sparse PCA) is a specialised technique used in statistical analysis and, in particular, in the analysis of multivariate data sets. It extends the classic method of principal component analysis (PCA) for the reduction of dimensionality of data by introducing sparsity structures to the input variables. A particular disadvantage of ordinary PCA is that the principal components are usually linear combinations of all input variables. SPCA overcomes this disadvantage by finding components that are linear combinations of just a few input variables (SPCs). |
epfl-collab | You are using a 3-layer fully-connected neural net with \textbf{ReLU activations}. Your input data has components in [0, 1]. \textbf{You initialize all your weights to -10}, and set all the bias terms to 0. You start optimizing using SGD. What will likely happen? | ['The gradient is 0 so nothing happens', 'Everything is fine', "The gradient is very large so the model can't converge", 'Training is fine, but our neural net does only as well as a linear model'] | A | null | Document 1:::
Stochastic gradient descent
Stochastic gradient descent (often abbreviated SGD) is an iterative method for optimizing an objective function with suitable smoothness properties (e.g. differentiable or subdifferentiable). It can be regarded as a stochastic approximation of gradient descent optimization, since it replaces the actual gradient (calculated from the entire data set) by an estimate thereof (calculated from a randomly selected subset of the data). Especially in high-dimensional optimization problems this reduces the very high computational burden, achieving faster iterations in exchange for a lower convergence rate.While the basic idea behind stochastic approximation can be traced back to the Robbins–Monro algorithm of the 1950s, stochastic gradient descent has become an important optimization method in machine learning.
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Pooling (neural networks)
They have applications in: image and video recognition, recommender systems, image classification, image segmentation, medical image analysis, natural language processing, brain–computer interfaces, and financial time series.CNNs are also known as Shift Invariant or Space Invariant Artificial Neural Networks (SIANN), based on the shared-weight architecture of the convolution kernels or filters that slide along input features and provide translation-equivariant responses known as feature maps. Counter-intuitively, most convolutional neural networks are not invariant to translation, due to the downsampling operation they apply to the input.Feed-forward neural networks are usually fully connected networks, that is, each neuron in one layer is connected to all neurons in the next layer. The "full connectivity" of these networks make them prone to overfitting data.
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Tensor (machine learning)
Tensors may also be used to compute the layers of a fully connected neural network, where the tensor is applied to the entire layer instead of individual unit values. The output value of single layer unit y m {\displaystyle y_{m}} is the sum-product of its input units and the connection weights filtered through the activation function f {\displaystyle f}: y m = f ( ∑ n x n u m , n ) . {\displaystyle y_{m}=f\left(\sum _{n}x_{n}u_{m,n}\right).} The vectors x {\displaystyle x} and y {\displaystyle y} of output values can be expressed as a mode-1 tensors, while the hidden weights can be expressed as a mode-2 tensor.
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Fine-tuning (deep learning)
In deep learning, fine-tuning is an approach to transfer learning in which the weights of a pre-trained model are trained on new data. Fine-tuning can be done on the entire neural network, or on only a subset of its layers, in which case the layers that are not being fine-tuned are "frozen" (not updated during the backpropagation step). A model may also be augmented with "adapters" that consist of far fewer parameters than the original model, and fine-tuned in a parameter-efficient way by tuning the weights of the adapters and leaving the rest of the model's weights frozen.For some architectures, such as convolutional neural networks, it is common to keep the earlier layers (those closest to the input layer) frozen because they capture lower-level features, while later layers often discern high-level features that can be more related to the task that the model is trained on.Models that are pre-trained on large and general corpora are usually fine-tuned by reusing the model's parameters as a starting point and adding a task-specific layer trained from scratch. Fine-tuning the full model is common as well and often yields better results, but it is more computationally expensive.Fine-tuning is typically accomplished with supervised learning, but there are also techniques to fine-tune a model using weak supervision. Fine-tuning can be combined with a reinforcement learning from human feedback-based objective to produce language models like ChatGPT (a fine-tuned version of GPT-3) and Sparrow.
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Fine-tuning (deep learning)
In deep learning, fine-tuning is an approach to transfer learning in which the weights of a pre-trained model are trained on new data. Fine-tuning can be done on the entire neural network, or on only a subset of its layers, in which case the layers that are not being fine-tuned are "frozen" (not updated during the backpropagation step). A model may also be augmented with "adapters" that consist of far fewer parameters than the original model, and fine-tuned in a parameter-efficient way by tuning the weights of the adapters and leaving the rest of the model's weights frozen.For some architectures, such as convolutional neural networks, it is common to keep the earlier layers (those closest to the input layer) frozen because they capture lower-level features, while later layers often discern high-level features that can be more related to the task that the model is trained on.Models that are pre-trained on large and general corpora are usually fine-tuned by reusing the model's parameters as a starting point and adding a task-specific layer trained from scratch. Fine-tuning the full model is common as well and often yields better results, but it is more computationally expensive.Fine-tuning is typically accomplished with supervised learning, but there are also techniques to fine-tune a model using weak supervision. Fine-tuning can be combined with a reinforcement learning from human feedback-based objective to produce language models like ChatGPT (a fine-tuned version of GPT-3) and Sparrow. |
epfl-collab | You are using a 3-layer fully-connected neural, and you are using \textbf{$f(x) = 2x$ as your activation function} . Your input data has components in [0, 1]. \textbf{You initialize your weights using Kaiming (He) initialization}, and set all the bias terms to 0. You start optimizing using SGD. What will likely happen? | ['The gradient is 0 so nothing happens', 'Training is fine, but our neural net does only as well as a linear model', "The gradient is very large so the model can't converge", 'Everything is fine'] | B | null | Document 1:::
Stochastic gradient descent
Stochastic gradient descent (often abbreviated SGD) is an iterative method for optimizing an objective function with suitable smoothness properties (e.g. differentiable or subdifferentiable). It can be regarded as a stochastic approximation of gradient descent optimization, since it replaces the actual gradient (calculated from the entire data set) by an estimate thereof (calculated from a randomly selected subset of the data). Especially in high-dimensional optimization problems this reduces the very high computational burden, achieving faster iterations in exchange for a lower convergence rate.While the basic idea behind stochastic approximation can be traced back to the Robbins–Monro algorithm of the 1950s, stochastic gradient descent has become an important optimization method in machine learning.
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Tensor (machine learning)
Tensors may also be used to compute the layers of a fully connected neural network, where the tensor is applied to the entire layer instead of individual unit values. The output value of single layer unit y m {\displaystyle y_{m}} is the sum-product of its input units and the connection weights filtered through the activation function f {\displaystyle f}: y m = f ( ∑ n x n u m , n ) . {\displaystyle y_{m}=f\left(\sum _{n}x_{n}u_{m,n}\right).} The vectors x {\displaystyle x} and y {\displaystyle y} of output values can be expressed as a mode-1 tensors, while the hidden weights can be expressed as a mode-2 tensor.
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Softmax activation function
The normalization ensures that the sum of the components of the output vector σ ( z ) {\displaystyle \sigma (\mathbf {z} )} is 1. The term "softmax" derives from the amplifying effects of the exponential on any maxima in the input vector.
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Softmax activation function
The softmax function takes as input a vector z of K real numbers, and normalizes it into a probability distribution consisting of K probabilities proportional to the exponentials of the input numbers. That is, prior to applying softmax, some vector components could be negative, or greater than one; and might not sum to 1; but after applying softmax, each component will be in the interval ( 0 , 1 ) {\displaystyle (0,1)} , and the components will add up to 1, so that they can be interpreted as probabilities. Furthermore, the larger input components will correspond to larger probabilities. The standard (unit) softmax function σ: R K ↦ ( 0 , 1 ) K {\displaystyle \sigma :\mathbb {R} ^{K}\mapsto (0,1)^{K}} where K ≥ 1 {\displaystyle K\geq 1} is defined by the formula In words, it applies the standard exponential function to each element z i {\displaystyle z_{i}} of the input vector z {\displaystyle \mathbf {z} } and normalizes these values by dividing by the sum of all these exponentials.
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Activation function
Activation function of a node in an artificial neural network is a function that calculates the output of the node (based on its inputs and the weights on individual inputs). Nontrivial problems can be solved only using a nonlinear activation function. Modern activation functions include the smooth version of the ReLU, the GELU, which was used in the 2018 BERT model, the logistic (sigmoid) function used in the 2012 speech recognition model developed by Hinton et al, the ReLU used in the 2012 AlexNet computer vision model and in the 2015 ResNet model. |
epfl-collab | What is a good representation for scores when classifying these three target classes: Car, Bike and Bus, in the context of logistic regression. (One or multiple answers) | ['{Car: $1$,} {Bike: $2$,} {Bus: $3$}', '{Car: $(0,1)$,} {Bike: $(1,0)$,} {Bus: $(1,1)$}', '{Car: $(0,1)$,} {Bike: $(1,0)$,} {Bus: $(0.5,0.5)$}', '{Car: $(0,1,0)$,} {Bike: $(1,0,0)$,} {Bus: $(0,0,1)$}'] | D | null | Document 1:::
Multiclass classifier
In machine learning and statistical classification, multiclass classification or multinomial classification is the problem of classifying instances into one of three or more classes (classifying instances into one of two classes is called binary classification). While many classification algorithms (notably multinomial logistic regression) naturally permit the use of more than two classes, some are by nature binary algorithms; these can, however, be turned into multinomial classifiers by a variety of strategies. Multiclass classification should not be confused with multi-label classification, where multiple labels are to be predicted for each instance.
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Linear classification
In the field of machine learning, the goal of statistical classification is to use an object's characteristics to identify which class (or group) it belongs to. A linear classifier achieves this by making a classification decision based on the value of a linear combination of the characteristics. An object's characteristics are also known as feature values and are typically presented to the machine in a vector called a feature vector. Such classifiers work well for practical problems such as document classification, and more generally for problems with many variables (features), reaching accuracy levels comparable to non-linear classifiers while taking less time to train and use.
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Car classification
The following table summarises the commonly used terms of market segments and legal classifications.
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Statistical classifier
Terminology across fields is quite varied. In statistics, where classification is often done with logistic regression or a similar procedure, the properties of observations are termed explanatory variables (or independent variables, regressors, etc.), and the categories to be predicted are known as outcomes, which are considered to be possible values of the dependent variable. In machine learning, the observations are often known as instances, the explanatory variables are termed features (grouped into a feature vector), and the possible categories to be predicted are classes. Other fields may use different terminology: e.g. in community ecology, the term "classification" normally refers to cluster analysis.
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Statistical classifier
In statistics, classification is the problem of identifying which of a set of categories (sub-populations) an observation (or observations) belongs to. Examples are assigning a given email to the "spam" or "non-spam" class, and assigning a diagnosis to a given patient based on observed characteristics of the patient (sex, blood pressure, presence or absence of certain symptoms, etc.). Often, the individual observations are analyzed into a set of quantifiable properties, known variously as explanatory variables or features. These properties may variously be categorical (e.g. "A", "B", "AB" or "O", for blood type), ordinal (e.g. "large", "medium" or "small"), integer-valued (e.g. the number of occurrences of a particular word in an email) or real-valued (e.g. a measurement of blood pressure). |
epfl-collab | Decision trees... | ['... have several different roots.', '... need water and sunlight to grow.', '... can be used for both classification and regression.', '... can be easily explained.'] | C | null | Document 1:::
Classification and regression tree
Decision tree learning is a supervised learning approach used in statistics, data mining and machine learning. In this formalism, a classification or regression decision tree is used as a predictive model to draw conclusions about a set of observations. Tree models where the target variable can take a discrete set of values are called classification trees; in these tree structures, leaves represent class labels and branches represent conjunctions of features that lead to those class labels. Decision trees where the target variable can take continuous values (typically real numbers) are called regression trees. More generally, the concept of regression tree can be extended to any kind of object equipped with pairwise dissimilarities such as categorical sequences.Decision trees are among the most popular machine learning algorithms given their intelligibility and simplicity.In decision analysis, a decision tree can be used to visually and explicitly represent decisions and decision making. In data mining, a decision tree describes data (but the resulting classification tree can be an input for decision making).
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Decision rules
A decision tree is a decision support hierarchical model that uses a tree-like model of decisions and their possible consequences, including chance event outcomes, resource costs, and utility. It is one way to display an algorithm that only contains conditional control statements. Decision trees are commonly used in operations research, specifically in decision analysis, to help identify a strategy most likely to reach a goal, but are also a popular tool in machine learning.
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Decision tree complexity
In computational complexity the decision tree model is the model of computation in which an algorithm is considered to be basically a decision tree, i.e., a sequence of queries or tests that are done adaptively, so the outcome of previous tests can influence the tests performed next. Typically, these tests have a small number of outcomes (such as a yes–no question) and can be performed quickly (say, with unit computational cost), so the worst-case time complexity of an algorithm in the decision tree model corresponds to the depth of the corresponding decision tree. This notion of computational complexity of a problem or an algorithm in the decision tree model is called its decision tree complexity or query complexity. Decision trees models are instrumental in establishing lower bounds for complexity theory for certain classes of computational problems and algorithms.
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Sequential game
Decision trees are the extensive form of dynamic games that provide information on the possible ways that a given game can be played. They show the sequence in which players act and the number of times that they can each make a decision. Decision trees also provide information on what each player knows or does not know at the point in time they decide on an action to take.
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C4.5 algorithm
C4.5 is an algorithm used to generate a decision tree developed by Ross Quinlan. C4.5 is an extension of Quinlan's earlier ID3 algorithm. The decision trees generated by C4.5 can be used for classification, and for this reason, C4.5 is often referred to as a statistical classifier. In 2011, authors of the Weka machine learning software described the C4.5 algorithm as "a landmark decision tree program that is probably the machine learning workhorse most widely used in practice to date".It became quite popular after ranking #1 in the Top 10 Algorithms in Data Mining pre-eminent paper published by Springer LNCS in 2008. |
epfl-collab | Which method can be used for dimensionality reduction ? | ['T-distributed Stochastic Neighbor Embedding (t-SNE)', 'Autoencoders', 'SVM', 'PCA'] | D | null | Document 1:::
Dimensionality reduction
Dimensionality reduction, or dimension reduction, is the transformation of data from a high-dimensional space into a low-dimensional space so that the low-dimensional representation retains some meaningful properties of the original data, ideally close to its intrinsic dimension. Working in high-dimensional spaces can be undesirable for many reasons; raw data are often sparse as a consequence of the curse of dimensionality, and analyzing the data is usually computationally intractable (hard to control or deal with). Dimensionality reduction is common in fields that deal with large numbers of observations and/or large numbers of variables, such as signal processing, speech recognition, neuroinformatics, and bioinformatics.Methods are commonly divided into linear and nonlinear approaches. Approaches can also be divided into feature selection and feature extraction. Dimensionality reduction can be used for noise reduction, data visualization, cluster analysis, or as an intermediate step to facilitate other analyses.
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Multifactor dimensionality reduction
Multifactor dimensionality reduction (MDR) is a statistical approach, also used in machine learning automatic approaches, for detecting and characterizing combinations of attributes or independent variables that interact to influence a dependent or class variable. MDR was designed specifically to identify nonadditive interactions among discrete variables that influence a binary outcome and is considered a nonparametric and model-free alternative to traditional statistical methods such as logistic regression. The basis of the MDR method is a constructive induction or feature engineering algorithm that converts two or more variables or attributes to a single attribute. This process of constructing a new attribute changes the representation space of the data. The end goal is to create or discover a representation that facilitates the detection of nonlinear or nonadditive interactions among the attributes such that prediction of the class variable is improved over that of the original representation of the data.
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Multifactor dimensionality reduction
Multifactor dimensionality reduction (MDR) is a statistical approach, also used in machine learning automatic approaches, for detecting and characterizing combinations of attributes or independent variables that interact to influence a dependent or class variable. MDR was designed specifically to identify nonadditive interactions among discrete variables that influence a binary outcome and is considered a nonparametric and model-free alternative to traditional statistical methods such as logistic regression. The basis of the MDR method is a constructive induction or feature engineering algorithm that converts two or more variables or attributes to a single attribute. This process of constructing a new attribute changes the representation space of the data. The end goal is to create or discover a representation that facilitates the detection of nonlinear or nonadditive interactions among the attributes such that prediction of the class variable is improved over that of the original representation of the data.
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Sufficient dimension reduction
In statistics, sufficient dimension reduction (SDR) is a paradigm for analyzing data that combines the ideas of dimension reduction with the concept of sufficiency. Dimension reduction has long been a primary goal of regression analysis. Given a response variable y and a p-dimensional predictor vector x {\displaystyle {\textbf {x}}} , regression analysis aims to study the distribution of y ∣ x {\displaystyle y\mid {\textbf {x}}} , the conditional distribution of y {\displaystyle y} given x {\displaystyle {\textbf {x}}} . A dimension reduction is a function R ( x ) {\displaystyle R({\textbf {x}})} that maps x {\displaystyle {\textbf {x}}} to a subset of R k {\displaystyle \mathbb {R} ^{k}} , k < p, thereby reducing the dimension of x {\displaystyle {\textbf {x}}} .
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Spectral clustering
In multivariate statistics, spectral clustering techniques make use of the spectrum (eigenvalues) of the similarity matrix of the data to perform dimensionality reduction before clustering in fewer dimensions. The similarity matrix is provided as an input and consists of a quantitative assessment of the relative similarity of each pair of points in the dataset. In application to image segmentation, spectral clustering is known as segmentation-based object categorization. |
epfl-collab | Mean Square Error loss: | ['Maximizing the accuracy', 'Minimizing the distance between the predicted point and the true point', 'Minimizing the score of false classes when they are close, or bigger than, the score of the true class', 'Maximizing the probability of the correct class'] | B | null | Document 1:::
Minimum mean-square error
In statistics and signal processing, a minimum mean square error (MMSE) estimator is an estimation method which minimizes the mean square error (MSE), which is a common measure of estimator quality, of the fitted values of a dependent variable. In the Bayesian setting, the term MMSE more specifically refers to estimation with quadratic loss function. In such case, the MMSE estimator is given by the posterior mean of the parameter to be estimated.
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Mean squared deviation
In machine learning, specifically empirical risk minimization, MSE may refer to the empirical risk (the average loss on an observed data set), as an estimate of the true MSE (the true risk: the average loss on the actual population distribution). The MSE is a measure of the quality of an estimator. As it is derived from the square of Euclidean distance, it is always a positive value that decreases as the error approaches zero.
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Minimum mean-square error
Let x {\displaystyle x} be a n × 1 {\displaystyle n\times 1} hidden random vector variable, and let y {\displaystyle y} be a m × 1 {\displaystyle m\times 1} known random vector variable (the measurement or observation), both of them not necessarily of the same dimension. An estimator x ^ ( y ) {\displaystyle {\hat {x}}(y)} of x {\displaystyle x} is any function of the measurement y {\displaystyle y} . The estimation error vector is given by e = x ^ − x {\displaystyle e={\hat {x}}-x} and its mean squared error (MSE) is given by the trace of error covariance matrix MSE = tr { E { ( x ^ − x ) ( x ^ − x ) T } } = E { ( x ^ − x ) T ( x ^ − x ) } , {\displaystyle \operatorname {MSE} =\operatorname {tr} \left\{\operatorname {E} \{({\hat {x}}-x)({\hat {x}}-x)^{T}\}\right\}=\operatorname {E} \{({\hat {x}}-x)^{T}({\hat {x}}-x)\},} where the expectation E {\displaystyle \operatorname {E} } is taken over x {\displaystyle x} conditioned on y {\displaystyle y} . When x {\displaystyle x} is a scalar variable, the MSE expression simplifies to E { ( x ^ − x ) 2 } {\displaystyle \operatorname {E} \left\{({\hat {x}}-x)^{2}\right\}} .
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Mean squared deviation
In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between the estimated values and the actual value. MSE is a risk function, corresponding to the expected value of the squared error loss. The fact that MSE is almost always strictly positive (and not zero) is because of randomness or because the estimator does not account for information that could produce a more accurate estimate.
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Mean square deviation
In matrix notation, MSE = 1 n ∑ i = 1 n ( e i ) 2 = 1 n e T e {\displaystyle \operatorname {MSE} ={\frac {1}{n}}\sum _{i=1}^{n}(e_{i})^{2}={\frac {1}{n}}\mathbf {e} ^{\mathsf {T}}\mathbf {e} } where e i {\displaystyle e_{i}} is ( Y i − Y i ^ ) {\displaystyle (Y_{i}-{\hat {Y_{i}}})} and e {\displaystyle \mathbf {e} } is the n × 1 {\displaystyle n\times 1} column vector. The MSE can also be computed on q data points that were not used in estimating the model, either because they were held back for this purpose, or because these data have been newly obtained. Within this process, known as statistical learning, the MSE is often called the test MSE, and is computed as MSE = 1 q ∑ i = n + 1 n + q ( Y i − Y i ^ ) 2 . {\displaystyle \operatorname {MSE} ={\frac {1}{q}}\sum _{i=n+1}^{n+q}\left(Y_{i}-{\hat {Y_{i}}}\right)^{2}.} |
epfl-collab | You need to debug your Stochastic Gradient Descent update for a classification of three bridge types.
Manually compute the model output for the feature vector $x=(1, 0, 0, 0, 0)$ and $W$ contains only zeros. The model is logistic regression, \textit{i.e.}, $\textrm{softmax}(Wx)$.
Remember:
\begin{equation}
\textrm{softmax}_i(s) = \frac{e^{s_i}}{\sum_k e^{s_k}}
\end{equation}
(One answer!!!!!!) | ['$(0, 0, 0, 0, 0)$', '$(\\frac{1}{5}, \\frac{1}{5}, \\frac{1}{5}, \\frac{1}{5}, \\frac{1}{5})$', '$(\\frac{1}{3}, \\frac{1}{3}, \\frac{1}{3})$', '$(0, 0, 0)$'] | C | null | Document 1:::
Stochastic gradient descent
Stochastic gradient descent (often abbreviated SGD) is an iterative method for optimizing an objective function with suitable smoothness properties (e.g. differentiable or subdifferentiable). It can be regarded as a stochastic approximation of gradient descent optimization, since it replaces the actual gradient (calculated from the entire data set) by an estimate thereof (calculated from a randomly selected subset of the data). Especially in high-dimensional optimization problems this reduces the very high computational burden, achieving faster iterations in exchange for a lower convergence rate.While the basic idea behind stochastic approximation can be traced back to the Robbins–Monro algorithm of the 1950s, stochastic gradient descent has become an important optimization method in machine learning.
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Stability (learning theory)
2010 - Shalev Shwartz et al. noticed problems with the original results of Vapnik due to the complex relations between hypothesis space and loss class. They discuss stability notions that capture different loss classes and different types of learning, supervised and unsupervised. 2016 - Moritz Hardt et al. proved stability of gradient descent given certain assumption on the hypothesis and number of times each instance is used to update the model.
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Stability (learning theory)
2010 - Shalev Shwartz et al. noticed problems with the original results of Vapnik due to the complex relations between hypothesis space and loss class. They discuss stability notions that capture different loss classes and different types of learning, supervised and unsupervised. 2016 - Moritz Hardt et al. proved stability of gradient descent given certain assumption on the hypothesis and number of times each instance is used to update the model.
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Extreme learning machine
Extreme learning machines are feedforward neural networks for classification, regression, clustering, sparse approximation, compression and feature learning with a single layer or multiple layers of hidden nodes, where the parameters of hidden nodes (not just the weights connecting inputs to hidden nodes) need to be tuned. These hidden nodes can be randomly assigned and never updated (i.e. they are random projection but with nonlinear transforms), or can be inherited from their ancestors without being changed. In most cases, the output weights of hidden nodes are usually learned in a single step, which essentially amounts to learning a linear model.
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Loss functions for classification
Given the binary nature of classification, a natural selection for a loss function (assuming equal cost for false positives and false negatives) would be the 0-1 loss function (0–1 indicator function), which takes the value of 0 if the predicted classification equals that of the true class or a 1 if the predicted classification does not match the true class. This selection is modeled by V ( f ( x → ) , y ) = H ( − y f ( x → ) ) {\displaystyle V(f({\vec {x}}),y)=H(-yf({\vec {x}}))} where H {\displaystyle H} indicates the Heaviside step function. However, this loss function is non-convex and non-smooth, and solving for the optimal solution is an NP-hard combinatorial optimization problem. |
epfl-collab | Consider the following PyTorch code:
class ThreeLayerNet (nn.Module):
def __init__():
super().__init__()
def forward(x):
x = nn.Linear(100, 10)(x)
x = nn.ReLU()(x)
x = nn.Linear(10, 200)(x)
x = nn.ReLU()(x)
x = nn.Linear(200, 1)(x)
return x
Suppose that inputs are 100-dimensional, and outputs are 1-dimensional. What will happen if we try to train this network? | ['Everything is fine.', 'There will be an error because the second layer has more neurons than the first. The number of neurons must never increase from one layer to the next.', 'The model will not train properly. The performance will be the same at the beginning of the first epoch and at the end of the last epoch.', 'There will be an error because we are re-using the variable x throughout the forward() method.'] | C | null | Document 1:::
Tensor (machine learning)
Operations on data tensors can be expressed in terms of matrix multiplication and the Kronecker product. The computation of gradients, an important aspect of the backpropagation algorithm, can be performed using PyTorch and TensorFlow.Computations are often performed on graphics processing units (GPUs) using CUDA and on dedicated hardware such as Google's Tensor Processing Unit or Nvidia's Tensor core. These developments have greatly accelerated neural network architectures and increased the size and complexity of models that can be trained.
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Neural Network
Neurons may have a threshold such that a signal is sent only if the aggregate signal crosses that threshold. Typically, neurons are aggregated into layers. Different layers may perform different transformations on their inputs. Signals travel from the first layer (the input layer), to the last layer (the output layer), possibly after traversing the layers multiple times.
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Tensor (machine learning)
Tensors may be used as the unit values of neural networks which extend the concept of scalar, vector and matrix values to multiple dimensions. The output value of single layer unit y m {\displaystyle y_{m}} is the sum-product of its input units and the connection weights filtered through the activation function f {\displaystyle f}: y m = f ( ∑ n x n u m , n ) , {\displaystyle y_{m}=f\left(\sum _{n}x_{n}u_{m,n}\right),} where y m ∈ R . {\displaystyle y_{m}\in \mathbb {R} .} If each output element of y m {\displaystyle y_{m}} is a scalar, then we have the classical definition of an artificial neural network.
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Tensor (machine learning)
In this example the unit values are scalars while the tensor takes on the dimensions of the network layers: x n ↦ X ∈ R 1 × N , {\displaystyle x_{n}\mapsto {\mathcal {X}}\in \mathbb {R} ^{1\times N},} y n ↦ Y ∈ R M × 1 , {\displaystyle y_{n}\mapsto {\mathcal {Y}}\in \mathbb {R} ^{M\times 1},} u n ↦ U ∈ R N × M . {\displaystyle u_{n}\mapsto {\mathcal {U}}\in \mathbb {R} ^{N\times M}.} In this notation, the output values can be computed as a tensor product of the input and weight tensors: Y = f ( X U ) .
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Tensor (machine learning)
Tensors may also be used to compute the layers of a fully connected neural network, where the tensor is applied to the entire layer instead of individual unit values. The output value of single layer unit y m {\displaystyle y_{m}} is the sum-product of its input units and the connection weights filtered through the activation function f {\displaystyle f}: y m = f ( ∑ n x n u m , n ) . {\displaystyle y_{m}=f\left(\sum _{n}x_{n}u_{m,n}\right).} The vectors x {\displaystyle x} and y {\displaystyle y} of output values can be expressed as a mode-1 tensors, while the hidden weights can be expressed as a mode-2 tensor. |
epfl-collab | You are using a 3-layer fully-connected neural net with \textbf{ReLU activations}. Your input data has components in [0, 1]. \textbf{You initialize your weights by sampling from $\mathcal{N}(-10, 0.1)$ (Gaussians of mean -10 and variance 0.1)}, and set all the bias terms to 0. You start optimizing using SGD. What will likely happen? | ['Training is fine, but our neural net does only as well as a linear model', 'Everything is fine', 'The gradient is 0 so nothing happens', "The gradient is very large so the model can't converge"] | C | null | Document 1:::
Stochastic gradient descent
Stochastic gradient descent (often abbreviated SGD) is an iterative method for optimizing an objective function with suitable smoothness properties (e.g. differentiable or subdifferentiable). It can be regarded as a stochastic approximation of gradient descent optimization, since it replaces the actual gradient (calculated from the entire data set) by an estimate thereof (calculated from a randomly selected subset of the data). Especially in high-dimensional optimization problems this reduces the very high computational burden, achieving faster iterations in exchange for a lower convergence rate.While the basic idea behind stochastic approximation can be traced back to the Robbins–Monro algorithm of the 1950s, stochastic gradient descent has become an important optimization method in machine learning.
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Pooling (neural networks)
They have applications in: image and video recognition, recommender systems, image classification, image segmentation, medical image analysis, natural language processing, brain–computer interfaces, and financial time series.CNNs are also known as Shift Invariant or Space Invariant Artificial Neural Networks (SIANN), based on the shared-weight architecture of the convolution kernels or filters that slide along input features and provide translation-equivariant responses known as feature maps. Counter-intuitively, most convolutional neural networks are not invariant to translation, due to the downsampling operation they apply to the input.Feed-forward neural networks are usually fully connected networks, that is, each neuron in one layer is connected to all neurons in the next layer. The "full connectivity" of these networks make them prone to overfitting data.
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Fine-tuning (deep learning)
In deep learning, fine-tuning is an approach to transfer learning in which the weights of a pre-trained model are trained on new data. Fine-tuning can be done on the entire neural network, or on only a subset of its layers, in which case the layers that are not being fine-tuned are "frozen" (not updated during the backpropagation step). A model may also be augmented with "adapters" that consist of far fewer parameters than the original model, and fine-tuned in a parameter-efficient way by tuning the weights of the adapters and leaving the rest of the model's weights frozen.For some architectures, such as convolutional neural networks, it is common to keep the earlier layers (those closest to the input layer) frozen because they capture lower-level features, while later layers often discern high-level features that can be more related to the task that the model is trained on.Models that are pre-trained on large and general corpora are usually fine-tuned by reusing the model's parameters as a starting point and adding a task-specific layer trained from scratch. Fine-tuning the full model is common as well and often yields better results, but it is more computationally expensive.Fine-tuning is typically accomplished with supervised learning, but there are also techniques to fine-tune a model using weak supervision. Fine-tuning can be combined with a reinforcement learning from human feedback-based objective to produce language models like ChatGPT (a fine-tuned version of GPT-3) and Sparrow.
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Fine-tuning (deep learning)
In deep learning, fine-tuning is an approach to transfer learning in which the weights of a pre-trained model are trained on new data. Fine-tuning can be done on the entire neural network, or on only a subset of its layers, in which case the layers that are not being fine-tuned are "frozen" (not updated during the backpropagation step). A model may also be augmented with "adapters" that consist of far fewer parameters than the original model, and fine-tuned in a parameter-efficient way by tuning the weights of the adapters and leaving the rest of the model's weights frozen.For some architectures, such as convolutional neural networks, it is common to keep the earlier layers (those closest to the input layer) frozen because they capture lower-level features, while later layers often discern high-level features that can be more related to the task that the model is trained on.Models that are pre-trained on large and general corpora are usually fine-tuned by reusing the model's parameters as a starting point and adding a task-specific layer trained from scratch. Fine-tuning the full model is common as well and often yields better results, but it is more computationally expensive.Fine-tuning is typically accomplished with supervised learning, but there are also techniques to fine-tune a model using weak supervision. Fine-tuning can be combined with a reinforcement learning from human feedback-based objective to produce language models like ChatGPT (a fine-tuned version of GPT-3) and Sparrow.
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Tensor (machine learning)
Tensors may also be used to compute the layers of a fully connected neural network, where the tensor is applied to the entire layer instead of individual unit values. The output value of single layer unit y m {\displaystyle y_{m}} is the sum-product of its input units and the connection weights filtered through the activation function f {\displaystyle f}: y m = f ( ∑ n x n u m , n ) . {\displaystyle y_{m}=f\left(\sum _{n}x_{n}u_{m,n}\right).} The vectors x {\displaystyle x} and y {\displaystyle y} of output values can be expressed as a mode-1 tensors, while the hidden weights can be expressed as a mode-2 tensor. |
epfl-collab | We saw in class that we can quickly decrease the spatial size of the representation using pooling layers. Is there another way to do this without pooling? | ['Yes, by increasing the amount of padding.', 'Yes, by increasing the stride.', 'No, pooling is necessary.', 'Yes, by increasing the number of filters.'] | B | null | Document 1:::
Spatial embedding
Spatial embedding is one of feature learning techniques used in spatial analysis where points, lines, polygons or other spatial data types. representing geographic locations are mapped to vectors of real numbers. Conceptually it involves a mathematical embedding from a space with many dimensions per geographic object to a continuous vector space with a much lower dimension. Such embedding methods allow complex spatial data to be used in neural networks and have been shown to improve performance in spatial analysis tasks
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Pooling (neural networks)
Typical ways of regularization, or preventing overfitting, include: penalizing parameters during training (such as weight decay) or trimming connectivity (skipped connections, dropout, etc.) Robust datasets also increases the probability that CNNs will learn the generalized principles that characterize a given dataset rather than the biases of a poorly-populated set.Convolutional networks were inspired by biological processes in that the connectivity pattern between neurons resembles the organization of the animal visual cortex. Individual cortical neurons respond to stimuli only in a restricted region of the visual field known as the receptive field. The receptive fields of different neurons partially overlap such that they cover the entire visual field.
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Pooling (neural networks)
Convolutional neural network (CNN) is a regularized type of feed-forward neural network that learns feature engineering by itself via filters (or kernel) optimization. Vanishing gradients and exploding gradients, seen during backpropagation in earlier neural networks, are prevented by using regularized weights over fewer connections. For example, for each neuron in the fully-connected layer 10,000 weights would be required for processing an image sized 100 × 100 pixels. However, applying cascaded convolution (or cross-correlation) kernels, only 25 neurons are required to process 5x5-sized tiles Higher-layer features are extracted from wider context windows, compared to lower-layer features.
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Dimensionality reduction
Dimensionality reduction, or dimension reduction, is the transformation of data from a high-dimensional space into a low-dimensional space so that the low-dimensional representation retains some meaningful properties of the original data, ideally close to its intrinsic dimension. Working in high-dimensional spaces can be undesirable for many reasons; raw data are often sparse as a consequence of the curse of dimensionality, and analyzing the data is usually computationally intractable (hard to control or deal with). Dimensionality reduction is common in fields that deal with large numbers of observations and/or large numbers of variables, such as signal processing, speech recognition, neuroinformatics, and bioinformatics.Methods are commonly divided into linear and nonlinear approaches. Approaches can also be divided into feature selection and feature extraction. Dimensionality reduction can be used for noise reduction, data visualization, cluster analysis, or as an intermediate step to facilitate other analyses.
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Pooling (neural networks)
CNNs use relatively little pre-processing compared to other image classification algorithms. This means that the network learns to optimize the filters (or kernels) through automated learning, whereas in traditional algorithms these filters are hand-engineered. This independence from prior knowledge and human intervention in feature extraction is a major advantage. |
epfl-collab | The \textbf{parameters} (weights \textbf{W}) are learned with ...
(One answer) | [' test ', ' validation ', ' training ', ' all the data together '] | C | null | Document 1:::
Trainable parameter
In particular, three data sets are commonly used in different stages of the creation of the model: training, validation, and test sets. The model is initially fit on a training data set, which is a set of examples used to fit the parameters (e.g. weights of connections between neurons in artificial neural networks) of the model. The model (e.g. a naive Bayes classifier) is trained on the training data set using a supervised learning method, for example using optimization methods such as gradient descent or stochastic gradient descent.
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Trainable parameter
In practice, the training data set often consists of pairs of an input vector (or scalar) and the corresponding output vector (or scalar), where the answer key is commonly denoted as the target (or label). The current model is run with the training data set and produces a result, which is then compared with the target, for each input vector in the training data set. Based on the result of the comparison and the specific learning algorithm being used, the parameters of the model are adjusted.
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Linear classification
If the input feature vector to the classifier is a real vector x → {\displaystyle {\vec {x}}} , then the output score is y = f ( w → ⋅ x → ) = f ( ∑ j w j x j ) , {\displaystyle y=f({\vec {w}}\cdot {\vec {x}})=f\left(\sum _{j}w_{j}x_{j}\right),} where w → {\displaystyle {\vec {w}}} is a real vector of weights and f is a function that converts the dot product of the two vectors into the desired output. (In other words, w → {\displaystyle {\vec {w}}} is a one-form or linear functional mapping x → {\displaystyle {\vec {x}}} onto R.) The weight vector w → {\displaystyle {\vec {w}}} is learned from a set of labeled training samples.
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Weighted majority algorithm (machine learning)
In machine learning, weighted majority algorithm (WMA) is a meta learning algorithm used to construct a compound algorithm from a pool of prediction algorithms, which could be any type of learning algorithms, classifiers, or even real human experts. The algorithm assumes that we have no prior knowledge about the accuracy of the algorithms in the pool, but there are sufficient reasons to believe that one or more will perform well. Assume that the problem is a binary decision problem. To construct the compound algorithm, a positive weight is given to each of the algorithms in the pool.
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Learning rule
Depending on the complexity of actual model being simulated, the learning rule of the network can be as simple as an XOR gate or mean squared error, or as complex as the result of a system of differential equations. The learning rule is one of the factors which decides how fast or how accurately the artificial network can be developed. Depending upon the process to develop the network there are three main models of machine learning: Unsupervised learning Supervised learning Reinforcement learning |
epfl-collab | The \textbf{hyperparameters} are learned with ...
(One answer) | [' test ', ' all the data together ', ' training ', ' validation '] | D | null | Document 1:::
Hyperparameter (machine learning)
In machine learning, a hyperparameter is a parameter whose value is used to control the learning process. By contrast, the values of other parameters (typically node weights) are derived via training. Hyperparameters can be classified as model hyperparameters, that cannot be inferred while fitting the machine to the training set because they refer to the model selection task, or algorithm hyperparameters, that in principle have no influence on the performance of the model but affect the speed and quality of the learning process. An example of a model hyperparameter is the topology and size of a neural network.
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Hyperparameter (machine learning)
In machine learning, a hyperparameter is a parameter whose value is used to control the learning process. By contrast, the values of other parameters (typically node weights) are derived via training. Hyperparameters can be classified as model hyperparameters, that cannot be inferred while fitting the machine to the training set because they refer to the model selection task, or algorithm hyperparameters, that in principle have no influence on the performance of the model but affect the speed and quality of the learning process. An example of a model hyperparameter is the topology and size of a neural network.
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Grid search
In machine learning, hyperparameter optimization or tuning is the problem of choosing a set of optimal hyperparameters for a learning algorithm. A hyperparameter is a parameter whose value is used to control the learning process. By contrast, the values of other parameters (typically node weights) are learned.
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Grid search
The same kind of machine learning model can require different constraints, weights or learning rates to generalize different data patterns. These measures are called hyperparameters, and have to be tuned so that the model can optimally solve the machine learning problem. Hyperparameter optimization finds a tuple of hyperparameters that yields an optimal model which minimizes a predefined loss function on given independent data. The objective function takes a tuple of hyperparameters and returns the associated loss. Cross-validation is often used to estimate this generalization performance, and therefore choose the set of values for hyperparameters that maximize it.
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Hyperparameter (machine learning)
Examples of algorithm hyperparameters are learning rate and batch size as well as mini-batch size. Batch size can refer to the full data sample where mini-batch size would be a smaller sample set. Different model training algorithms require different hyperparameters, some simple algorithms (such as ordinary least squares regression) require none. Given these hyperparameters, the training algorithm learns the parameters from the data. For instance, LASSO is an algorithm that adds a regularization hyperparameter to ordinary least squares regression, which has to be set before estimating the parameters through the training algorithm. |
epfl-collab | We report the final performance (e.g., accuracy) on the ...
(One answer) | [' test ', ' validation ', ' training ', ' all the data together '] | A | null | Document 1:::
Stats Perform
Stats Perform (formerly STATS, LLC and STATS, Inc.) is a sports data and analytics company formed through the combination of Stats and Perform.The company is involved in sports data collection and predictive analysis for use across various sports sectors including professional team performance, digital, media, broadcast and betting. The company has also been increasingly involved in artificial intelligence and machine learning. Its clients include media outlets, sports leagues and teams, fantasy sports and sports betting services. As of 2023, the company covers 500,000+ matches and 3900 competition annually from over 300 leagues and competitions.Stats Perform is headquartered in London with other office locations in Amsterdam, Chicago, Limerick, Aveiro, Buenos Aires, Castelfranco Veneto, Beijing, Bangalore, Chennai, Düsseldorf, Graz, Istanbul, Katowice, Kuala Lampur, Madrid, Nice, Paris, Prague, Sao Paulo, and Sydney. In addition to their offices, Stats Perform have full-time staff all over the globe operating in local markets including Cape Town, Dubai, Seoul, Tokyo, and more.
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Performance measurement
Performance measurement is the process of collecting, analyzing and/or reporting information regarding the performance of an individual, group, organization, system or component.Definitions of performance measurement tend to be predicated upon an assumption about why the performance is being measured. Moullin defines the term with a forward looking organisational focus—"the process of evaluating how well organisations are managed and the value they deliver for customers and other stakeholders". Neely et al. use a more operational retrospective focus—"the process of quantifying the efficiency and effectiveness of past actions". In 2007 the Office of the Chief Information Officer in the USA defined it using a more evaluative focus—"Performance measurement estimates the parameters under which programs, investments, and acquisitions are reaching the targeted results". Defining performance measures or methods by which they can be chosen is also a popular activity for academics—for example a list of railway infrastructure indicators is offered by Stenström et al., a novel method for measure selection is proposed by Mendibil et al.
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Data reporting
Data reporting is the process of collecting and submitting data which gives rise to accurate analyses of the facts on the ground; inaccurate data reporting can lead to vastly uninformed decision-making based on erroneous evidence. Different from data analysis that transforms data and information into insights, data reporting is the previous step that translates raw data into information. When data is not reported, the problem is known as underreporting; the opposite problem leads to false positives. Data reporting can be an incredibly difficult endeavor.
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System test
The actual result is the behavior produced or observed when a component or system is tested.System testing is performed on the entire system in the context of either functional requirement specifications (FRS) or system requirement specification (SRS), or both. System testing tests not only the design, but also the behaviour and even the believed expectations of the customer. It is also intended to test up to and beyond the bounds defined in the software or hardware requirements specification(s).
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Performance tuning
Measure the performance of the system after modification. If the modification makes the performance better, adopt it. If the modification makes the performance worse, put it back the way it was.This is an instance of the measure-evaluate-improve-learn cycle from quality assurance. |
epfl-collab | We consider a classification problem on linearly separable data. Our dataset had an outlier---a point that is very far from the other datapoints in distance (and also far from margins in SVM but still correctly classified by the SVM classifier).
We trained the SVM, logistic regression and 1-nearest-neighbour models on this dataset.
We tested trained models on a test set that comes from the same distribution as training set, but doesn't have any outlier points.
For any vector $ v \in \R^D$ let $\| v\|_2 := \sqrt{v_1^2 + \dots + v_D^2}$ denote the Euclidean norm. The hard-margin SVM problem for linearly separable points in $\R^D$ is to minimize the Euclidean norm $\| \wv \|_2$ under some constraints.
What are the additional constraints for this optimization problem? | ['$y_n \\ww^\top x_n \\geq 1 ~ \x0corall n \\in \\{1,\\cdots,N\\}$', '$\\ww^\top x_n \\geq 1 ~ \x0corall n \\in\\{1,\\cdots,N\\}$', '$y_n + \\ww^\top x_n \\geq 1 ~ \x0corall n \\in \\{1,\\cdots,N\\}$', '$\x0crac{y_n}{\\ww^\top x_n }\\geq 1 ~\x0corall n \\in \\{1,\\cdots,N\\}$'] | A | null | Document 1:::
Margin classifier
In machine learning, a margin classifier is a classifier which is able to give an associated distance from the decision boundary for each example. For instance, if a linear classifier (e.g. perceptron or linear discriminant analysis) is used, the distance (typically euclidean distance, though others may be used) of an example from the separating hyperplane is the margin of that example. The notion of margin is important in several machine learning classification algorithms, as it can be used to bound the generalization error of the classifier. These bounds are frequently shown using the VC dimension. Of particular prominence is the generalization error bound on boosting algorithms and support vector machines.
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L1-norm principal component analysis
However, in modern big data sets, data often include corrupted, faulty points, commonly referred to as outliers. Standard PCA is known to be sensitive to outliers, even when they appear as a small fraction of the processed data. The reason is that the L2-norm formulation of L2-PCA places squared emphasis on the magnitude of each coordinate of each data point, ultimately overemphasizing peripheral points, such as outliers. On the other hand, following an L1-norm formulation, L1-PCA places linear emphasis on the coordinates of each data point, effectively restraining outliers.
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Regularization perspectives on support vector machines
Specifically, Tikhonov regularization algorithms produce a decision boundary that minimizes the average training-set error and constrain the Decision boundary not to be excessively complicated or overfit the training data via a L2 norm of the weights term. The training and test-set errors can be measured without bias and in a fair way using accuracy, precision, Auc-Roc, precision-recall, and other metrics. Regularization perspectives on support-vector machines interpret SVM as a special case of Tikhonov regularization, specifically Tikhonov regularization with the hinge loss for a loss function.
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Regularization perspectives on support vector machines
Within mathematical analysis, Regularization perspectives on support-vector machines provide a way of interpreting support-vector machines (SVMs) in the context of other regularization-based machine-learning algorithms. SVM algorithms categorize binary data, with the goal of fitting the training set data in a way that minimizes the average of the hinge-loss function and L2 norm of the learned weights. This strategy avoids overfitting via Tikhonov regularization and in the L2 norm sense and also corresponds to minimizing the bias and variance of our estimator of the weights. Estimators with lower Mean squared error predict better or generalize better when given unseen data.
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Support-vector machine
In machine learning, support vector machines (SVMs, also support vector networks) are supervised learning models with associated learning algorithms that analyze data for classification and regression analysis. Developed at AT&T Bell Laboratories by Vladimir Vapnik with colleagues (Boser et al., 1992, Guyon et al., 1993, Cortes and Vapnik, 1995, Vapnik et al., 1997) SVMs are one of the most robust prediction methods, being based on statistical learning frameworks or VC theory proposed by Vapnik (1982, 1995) and Chervonenkis (1974). Given a set of training examples, each marked as belonging to one of two categories, an SVM training algorithm builds a model that assigns new examples to one category or the other, making it a non-probabilistic binary linear classifier (although methods such as Platt scaling exist to use SVM in a probabilistic classification setting). SVM maps training examples to points in space so as to maximise the width of the gap between the two categories. |
epfl-collab | Which of the following statements is correct? | ['When applying stochastic gradient descent on the objective function $f(\\boldsymbol{w}):=\\sum_{n=1}^{30}\\left\\|\\boldsymbol{w}-\\boldsymbol{x}_{n}\\right\\|^{2}$ where $\\boldsymbol{x}_{n}$ are the datapoints, a stochastic gradient step is roughly $30 \\times$ faster than a full gradient step.', 'When applying stochastic gradient descent on the objective function $f(\\boldsymbol{w}):=\\sum_{n=1}^{30} n \\cdot\\|\\boldsymbol{w}\\|^{2}$, a stochastic gradient (for the $n$-th summand) is given by $2 n \\cdot \\boldsymbol{w}$.', 'In practice, it could be good to let your model first overfit your task, and then apply drop-out or other regularization techniques.', 'The function $f(\\boldsymbol{u} ; \\boldsymbol{v}):=g\\left(\\boldsymbol{u} \\boldsymbol{v}^{\\top}\\right)$ is convex over the set of pairs of vectors $(\\boldsymbol{u} ; \\boldsymbol{v}) \\in \\mathbb{R}^{2} \\times \\mathbb{R}^{2}$, when $g: \\mathbb{R}^{2 \\times 2} \\rightarrow \\mathbb{R}$ is defined as $g(\\boldsymbol{X}):=X_{12}+X_{21}$.'] | A | null | Document 1:::
Statement (logic)
In logic and semantics, the term statement is variously understood to mean either: a meaningful declarative sentence that is true or false, or a proposition. Which is the assertion that is made by (i.e., the meaning of) a true or false declarative sentence.In the latter case, a statement is distinct from a sentence in that a sentence is only one formulation of a statement, whereas there may be many other formulations expressing the same statement. By a statement, I mean "that which one states", not one's stating of it. There are many interpretations of what the term statement means, but generally, it indicates either a meaningful declarative sentence that is either true or false (bivalence).
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Statement (logic)
In logic and semantics, the term statement is variously understood to mean either: a meaningful declarative sentence that is true or false, or a proposition. Which is the assertion that is made by (i.e., the meaning of) a true or false declarative sentence.In the latter case, a statement is distinct from a sentence in that a sentence is only one formulation of a statement, whereas there may be many other formulations expressing the same statement. By a statement, I mean "that which one states", not one's stating of it. There are many interpretations of what the term statement means, but generally, it indicates either a meaningful declarative sentence that is either true or false (bivalence).
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Atomic fact
In logic and analytic philosophy, an atomic sentence is a type of declarative sentence which is either true or false (may also be referred to as a proposition, statement or truthbearer) and which cannot be broken down into other simpler sentences. For example, "The dog ran" is an atomic sentence in natural language, whereas "The dog ran and the cat hid" is a molecular sentence in natural language. From a logical analysis point of view, the truth or falsity of sentences in general is determined by only two things: the logical form of the sentence and the truth or falsity of its simple sentences. This is to say, for example, that the truth of the sentence "John is Greek and John is happy" is a function of the meaning of "and", and the truth values of the atomic sentences "John is Greek" and "John is happy".
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Characterization theorem
Common mathematical expressions for a characterization of X in terms of P include "P is necessary and sufficient for X", and "X holds if and only if P". It is also common to find statements such as "Property Q characterizes Y up to isomorphism". The first type of statement says in different words that the extension of P is a singleton set, while the second says that the extension of Q is a single equivalence class (for isomorphism, in the given example — depending on how up to is being used, some other equivalence relation might be involved).
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Judgment (mathematical logic)
In mathematical logic, a judgment (or judgement) or assertion is a statement or enunciation in a metalanguage. For example, typical judgments in first-order logic would be that a string is a well-formed formula, or that a proposition is true. Similarly, a judgment may assert the occurrence of a free variable in an expression of the object language, or the provability of a proposition. In general, a judgment may be any inductively definable assertion in the metatheory. |
epfl-collab | Consider the function $f(x)=-x^{2}$. Which of the following statements are true regarding subgradients of $f(x)$ at $x=0$ ? | ['A subgradient exists but is not unique.', 'A subgradient exists and is unique.', 'A subgradient does not exist even though $f(x)$ is differentiable at $x=0$.', 'A subgradient does not exist as $f(x)$ is differentiable at $x=0$.'] | C | null | Document 1:::
Subderivative
Rigorously, a subderivative of a convex function f: I → R {\displaystyle f:I\to \mathbb {R} } at a point x 0 {\displaystyle x_{0}} in the open interval I {\displaystyle I} is a real number c {\displaystyle c} such that for all x ∈ I {\displaystyle x\in I} . By the converse of the mean value theorem, the set of subderivatives at x 0 {\displaystyle x_{0}} for a convex function is a nonempty closed interval {\displaystyle } , where a {\displaystyle a} and b {\displaystyle b} are the one-sided limits The set {\displaystyle } of all subderivatives is called the subdifferential of the function f {\displaystyle f} at x 0 {\displaystyle x_{0}} , denoted by ∂ f ( x 0 ) {\displaystyle \partial f(x_{0})} . If f {\displaystyle f} is convex, then its subdifferential at any point is non-empty. Moreover, if its subdifferential at x 0 {\displaystyle x_{0}} contains exactly one subderivative, then ∂ f ( x 0 ) = { f ′ ( x 0 ) } {\displaystyle \partial f(x_{0})=\{f'(x_{0})\}} and f {\displaystyle f} is differentiable at x 0 {\displaystyle x_{0}} .
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Subderivative
In mathematics, the subderivative, subgradient, and subdifferential generalize the derivative to convex functions which are not necessarily differentiable. Subderivatives arise in convex analysis, the study of convex functions, often in connection to convex optimization. Let f: I → R {\displaystyle f:I\to \mathbb {R} } be a real-valued convex function defined on an open interval of the real line. Such a function need not be differentiable at all points: For example, the absolute value function f ( x ) = | x | {\displaystyle f(x)=|x|} is non-differentiable when x = 0 {\displaystyle x=0} . However, as seen in the graph on the right (where f ( x ) {\displaystyle f(x)} in blue has non-differentiable kinks similar to the absolute value function), for any x 0 {\displaystyle x_{0}} in the domain of the function one can draw a line which goes through the point ( x 0 , f ( x 0 ) ) {\displaystyle (x_{0},f(x_{0}))} and which is everywhere either touching or below the graph of f. The slope of such a line is called a subderivative.
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Sublinear function
{\displaystyle x,y\in X.} This subadditivity condition requires p {\displaystyle p} to be real-valued.A function p: X → R {\displaystyle p:X\to \mathbb {R} } is called positive or nonnegative if p ( x ) ≥ 0 {\displaystyle p(x)\geq 0} for all x ∈ X , {\displaystyle x\in X,} although some authors define positive to instead mean that p ( x ) ≠ 0 {\displaystyle p(x)\neq 0} whenever x ≠ 0 ; {\displaystyle x\neq 0;} these definitions are not equivalent. It is a symmetric function if p ( − x ) = p ( x ) {\displaystyle p(-x)=p(x)} for all x ∈ X .
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Convex functions
In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain. Well-known examples of convex functions of a single variable include a linear function f ( x ) = c x {\displaystyle f(x)=cx} (where c {\displaystyle c} is a real number), a quadratic function c x 2 {\displaystyle cx^{2}} ( c {\displaystyle c} as a nonnegative real number) and a exponential function c e x {\displaystyle ce^{x}} ( c {\displaystyle c} as a nonnegative real number).
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Legendre transformation
For a differentiable convex function f {\displaystyle f} on the real line with the first derivative f ′ {\displaystyle f'} and its inverse ( f ′ ) − 1 {\displaystyle (f')^{-1}} , the Legendre transform of f {\displaystyle f} , f ∗ {\displaystyle f^{*}} , can be specified, up to an additive constant, by the condition that the functions' first derivatives are inverse functions of each other, i.e., f ′ = ( ( f ∗ ) ′ ) − 1 {\displaystyle f'=((f^{*})')^{-1}} and ( f ∗ ) ′ = ( f ′ ) − 1 {\displaystyle (f^{*})'=(f')^{-1}} . To see this, first note that if f {\displaystyle f} as a convex function on the real line is differentiable and x ¯ {\displaystyle {\overline {x}}} is a critical point of the function of x ↦ p ⋅ x − f ( x ) {\displaystyle x\mapsto p\cdot x-f(x)} , then the supremum is achieved at x ¯ {\displaystyle {\overline {x}}} (by convexity, see the first figure in this Wikipedia page). Therefore, the Legendre transform of f {\displaystyle f} is f ∗ ( p ) = p ⋅ x ¯ − f ( x ¯ ) {\displaystyle f^{*}(p)=p\cdot {\overline {x}}-f({\overline {x}})} . Then, suppose that the first derivative f ′ {\displaystyle f'} is invertible and let the inverse be g = ( f ′ ) − 1 {\displaystyle g=(f')^{-1}} . |
epfl-collab | In Text Representation learning, which of the following statements are correct? | ['The skip-gram model for learning original word2vec embeddings does learn a binary classifier for each word.', 'FastText as discussed in the course learns word vectors and sentence representations which are specific to a supervised classification task.', 'Logistic regression used for text classification is faster at test time when using word vectors as opposed to bag-of-word representation of the input.', 'Learning GloVe word vectors can be done using the singular value decomposition, if the $f_{d n}$ weights are set to 1 for all observed entries.'] | A | null | Document 1:::
Sequence labeling
In machine learning, sequence labeling is a type of pattern recognition task that involves the algorithmic assignment of a categorical label to each member of a sequence of observed values. A common example of a sequence labeling task is part of speech tagging, which seeks to assign a part of speech to each word in an input sentence or document. Sequence labeling can be treated as a set of independent classification tasks, one per member of the sequence. However, accuracy is generally improved by making the optimal label for a given element dependent on the choices of nearby elements, using special algorithms to choose the globally best set of labels for the entire sequence at once.
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Sequence labeling
In machine learning, sequence labeling is a type of pattern recognition task that involves the algorithmic assignment of a categorical label to each member of a sequence of observed values. A common example of a sequence labeling task is part of speech tagging, which seeks to assign a part of speech to each word in an input sentence or document. Sequence labeling can be treated as a set of independent classification tasks, one per member of the sequence. However, accuracy is generally improved by making the optimal label for a given element dependent on the choices of nearby elements, using special algorithms to choose the globally best set of labels for the entire sequence at once.
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Feature learning
In machine learning, feature learning or representation learning is a set of techniques that allows a system to automatically discover the representations needed for feature detection or classification from raw data. This replaces manual feature engineering and allows a machine to both learn the features and use them to perform a specific task. Feature learning is motivated by the fact that machine learning tasks such as classification often require input that is mathematically and computationally convenient to process.
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Feature learning
In machine learning, feature learning or representation learning is a set of techniques that allows a system to automatically discover the representations needed for feature detection or classification from raw data. This replaces manual feature engineering and allows a machine to both learn the features and use them to perform a specific task. Feature learning is motivated by the fact that machine learning tasks such as classification often require input that is mathematically and computationally convenient to process.
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Feature learning
In supervised feature learning, features are learned using labeled input data. Labeled data includes input-label pairs where the input is given to the model and it must produce the ground truth label as the correct answer. This can be leveraged to generate feature representations with the model which result in high label prediction accuracy. |
epfl-collab | When constructing a word embedding, what is true regarding negative samples? | ['Their frequency is decreased down to its logarithm', 'They are oversampled if less frequent', 'They are words that do not appear as context words', 'They are selected among words which are not stop words'] | B | null | Document 1:::
Precision and recall
For classification tasks, the terms true positives, true negatives, false positives, and false negatives (see Type I and type II errors for definitions) compare the results of the classifier under test with trusted external judgments. The terms positive and negative refer to the classifier's prediction (sometimes known as the expectation), and the terms true and false refer to whether that prediction corresponds to the external judgment (sometimes known as the observation). Let us define an experiment from P positive instances and N negative instances for some condition. The four outcomes can be formulated in a 2×2 contingency table or confusion matrix, as follows: Precision and recall are then defined as: Recall in this context is also referred to as the true positive rate or sensitivity, and precision is also referred to as positive predictive value (PPV); other related measures used in classification include true negative rate and accuracy. True negative rate is also called specificity.
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Winnow (algorithm)
During training, Winnow is shown a sequence of positive and negative examples. From these it learns a decision hyperplane that can then be used to label novel examples as positive or negative. The algorithm can also be used in the online learning setting, where the learning and the classification phase are not clearly separated.
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Strong and weak sampling
Strong and weak sampling are two sampling approach in Statistics, and are popular in computational cognitive science and language learning. In strong sampling, it is assumed that the data are intentionally generated as positive examples of a concept, while in weak sampling, it is assumed that the data are generated without any restrictions.
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Graph Laplacian
Negative weights present several challenges for normalisation: The presence of negative weights may naturally result in zero row- and/or column-sums for non-isolated vertices. A vertex with a large row-sum of positive weights and equally negatively large row-sum of negative weights, together summing up to zero, could be considered a heavy node and both large values scaled, while the diagonal entry remains zero, like for a isolated vertex. Negative weights may also give negative row- and/or column-sums, so that the corresponding diagonal entry in the non-normalized Laplacian matrix would be negative and a positive square root needed for the symmetric normalization would not exist. Arguments can be made to take the absolute value of the row- and/or column-sums for the purpose of normalization, thus treating a possible value -1 as a legitimate unit entry of the main diagonal of the normalized Laplacian matrix.
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Sample-complexity bounds
The sample complexity of a machine learning algorithm represents the number of training-samples that it needs in order to successfully learn a target function. More precisely, the sample complexity is the number of training-samples that we need to supply to the algorithm, so that the function returned by the algorithm is within an arbitrarily small error of the best possible function, with probability arbitrarily close to 1. There are two variants of sample complexity: The weak variant fixes a particular input-output distribution; The strong variant takes the worst-case sample complexity over all input-output distributions.The No free lunch theorem, discussed below, proves that, in general, the strong sample complexity is infinite, i.e. that there is no algorithm that can learn the globally-optimal target function using a finite number of training samples. However, if we are only interested in a particular class of target functions (e.g, only linear functions) then the sample complexity is finite, and it depends linearly on the VC dimension on the class of target functions. |
epfl-collab | If the first column of matrix L is (0,1,1,1) and all other entries are 0 then the authority values | ['(0, 1/sqrt(3), 1/sqrt(3), 1/sqrt(3))', '(1, 0, 0, 0)', '(1, 1/sqrt(3), 1/sqrt(3), 1/sqrt(3))', '(0, 1, 1, 1)'] | A | null | Document 1:::
Zero matrix
In mathematics, particularly linear algebra, a zero matrix or null matrix is a matrix all of whose entries are zero. It also serves as the additive identity of the additive group of m × n {\displaystyle m\times n} matrices, and is denoted by the symbol O {\displaystyle O} or 0 {\displaystyle 0} followed by subscripts corresponding to the dimension of the matrix as the context sees fit. Some examples of zero matrices are 0 1 , 1 = , 0 2 , 2 = , 0 2 , 3 = . {\displaystyle 0_{1,1}={\begin{bmatrix}0\end{bmatrix}},\ 0_{2,2}={\begin{bmatrix}0&0\\0&0\end{bmatrix}},\ 0_{2,3}={\begin{bmatrix}0&0&0\\0&0&0\end{bmatrix}}.\ }
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Gauss elimination
For each row in a matrix, if the row does not consist of only zeros, then the leftmost nonzero entry is called the leading coefficient (or pivot) of that row. So if two leading coefficients are in the same column, then a row operation of type 3 could be used to make one of those coefficients zero. Then by using the row swapping operation, one can always order the rows so that for every non-zero row, the leading coefficient is to the right of the leading coefficient of the row above. If this is the case, then matrix is said to be in row echelon form.
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Matrix unit
In linear algebra, a matrix unit is a matrix with only one nonzero entry with value 1. The matrix unit with a 1 in the ith row and jth column is denoted as E i j {\displaystyle E_{ij}} . For example, the 3 by 3 matrix unit with i = 1 and j = 2 is A vector unit is a standard unit vector. A single-entry matrix generalizes the matrix unit for matrices with only one nonzero entry of any value, not necessarily of value 1.
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Manin matrices
Cramer's rule. The inverse matrix is given by the standard formula M − 1 = 1 a d − c b ( d − b − c a ) {\displaystyle M^{-1}={\frac {1}{ad-cb}}{\begin{pmatrix}d&-b\\-c&a\end{pmatrix}}} if and only if M is a Manin matrix. Proof: ( d − b − c a ) ( a b c d ) = ( d a − b c d b − b d − c a + a c − c b + a d ) = if and only if M is a Manin matrix = ( a d − c b 0 0 a d − c b ) .
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Logical matrix
A logical matrix, binary matrix, relation matrix, Boolean matrix, or (0, 1)-matrix is a matrix with entries from the Boolean domain B = {0, 1}. Such a matrix can be used to represent a binary relation between a pair of finite sets. It is an important tool in combinatorial mathematics and theoretical computer science. |
epfl-collab | If the top 100 documents contain 50 relevant documents | ['the precision of the system at 100 is 0.5', 'the precision of the system at 50 is 0.25', 'the recall of the system is 0.5', 'All of the above'] | A | null | Document 1:::
Relevance (information retrieval)
In information science and information retrieval, relevance denotes how well a retrieved document or set of documents meets the information need of the user. Relevance may include concerns such as timeliness, authority or novelty of the result.
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Uncertain inference
Rather than retrieving a document that exactly matches the query we should rank the documents based on their plausibility in regards to that query. Since d and q are both generated by users, they are error prone; thus d → q {\displaystyle d\to q} is uncertain. This will affect the plausibility of a given query.
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Uncertain inference
Rather than retrieving a document that exactly matches the query we should rank the documents based on their plausibility in regards to that query. Since d and q are both generated by users, they are error prone; thus d → q {\displaystyle d\to q} is uncertain. This will affect the plausibility of a given query.
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Text categorization
Document classification or document categorization is a problem in library science, information science and computer science. The task is to assign a document to one or more classes or categories. This may be done "manually" (or "intellectually") or algorithmically. The intellectual classification of documents has mostly been the province of library science, while the algorithmic classification of documents is mainly in information science and computer science.
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Document retrieval
Document retrieval is defined as the matching of some stated user query against a set of free-text records. These records could be any type of mainly unstructured text, such as newspaper articles, real estate records or paragraphs in a manual. User queries can range from multi-sentence full descriptions of an information need to a few words. |
epfl-collab | What is WRONG regarding the Transformer model? | ['It uses a self-attention mechanism to compute representations of the input and output.', 'Its complexity is quadratic to the input size.', 'It captures the semantic context of the input.', 'Its computation cannot be parallelized compared to LSTMs and other sequential models.'] | D | null | Document 1:::
Parametric transformer
The Parametric transformer (or paraformer) is a particular type of transformer. It transfers the power from primary to secondary windings not by mutual inductance coupling but by a variation of a parameter in its magnetic circuit. First described by Wanlass, et al., 1968.
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Transformer utilization factor
m . s I ˙ r . m .
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Step-up transformer
A transformer is a passive component that transfers electrical energy from one electrical circuit to another circuit, or multiple circuits. A varying current in any coil of the transformer produces a varying magnetic flux in the transformer's core, which induces a varying electromotive force (EMF) across any other coils wound around the same core. Electrical energy can be transferred between separate coils without a metallic (conductive) connection between the two circuits. Faraday's law of induction, discovered in 1831, describes the induced voltage effect in any coil due to a changing magnetic flux encircled by the coil.
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Transformer (neural network)
A transformer is a deep learning architecture that relies on the parallel multi-head attention mechanism. The modern transformer was proposed in the 2017 paper titled 'Attention Is All You Need' by Ashish Vaswani et al., Google Brain team. It is notable for requiring less training time than previous recurrent neural architectures, such as long short-term memory (LSTM), and its later variation has been prevalently adopted for training large language models on large (language) datasets, such as the Wikipedia corpus and Common Crawl, by virtue of the parallelized processing of input sequence. Input text is split into n-grams encoded as tokens and each token is converted into a vector via looking up from a word embedding table.
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Modulation transformer
A modulation transformer is an audio-frequency transformer that forms a major part of most AM transmitters. The primary winding of a modulation transformer is fed by an audio amplifier that has about 1/2 of the rated input power of the transmitter's final amplifier stage. The secondary winding is in series with the power supply of that final radio-frequency amplifier stage, thereby allowing the audio signal to lower and raise the instantaneous DC supply voltage of the power amplifier (PA) tube or transistor. |
epfl-collab | Which of the following statements about index merging (when constructing inverted files) is correct? | ['While merging two partial indices on disk, the vocabularies are concatenated without sorting', 'While merging two partial indices on disk, the inverted lists of a term are concatenated without sorting', 'The size of the final merged index file is O (n log2 (n) M )), where M is the size of the available memory', 'Index merging is used when the vocabulary does no longer fit into the main memory'] | B | null | Document 1:::
Inverted index
In computer science, an inverted index (also referred to as a postings list, postings file, or inverted file) is a database index storing a mapping from content, such as words or numbers, to its locations in a table, or in a document or a set of documents (named in contrast to a forward index, which maps from documents to content). The purpose of an inverted index is to allow fast full-text searches, at a cost of increased processing when a document is added to the database. The inverted file may be the database file itself, rather than its index. It is the most popular data structure used in document retrieval systems, used on a large scale for example in search engines.
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Inverted index
Additionally, several significant general-purpose mainframe-based database management systems have used inverted list architectures, including ADABAS, DATACOM/DB, and Model 204. There are two main variants of inverted indexes: A record-level inverted index (or inverted file index or just inverted file) contains a list of references to documents for each word. A word-level inverted index (or full inverted index or inverted list) additionally contains the positions of each word within a document. The latter form offers more functionality (like phrase searches), but needs more processing power and space to be created.
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External Sorting
External merge sort typically uses a hybrid sort-merge strategy. In the sorting phase, chunks of data small enough to fit in main memory are read, sorted, and written out to a temporary file. In the merge phase, the sorted subfiles are combined into a single larger file.
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Loop inversion
In computer science, loop inversion is a compiler optimization and loop transformation in which a while loop is replaced by an if block containing a do..while loop. When used correctly, it may improve performance due to instruction pipelining.
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Block merge sort
Block sort, or block merge sort, is a sorting algorithm combining at least two merge operations with an insertion sort to arrive at O(n log n) in-place stable sorting. It gets its name from the observation that merging two sorted lists, A and B, is equivalent to breaking A into evenly sized blocks, inserting each A block into B under special rules, and merging AB pairs. One practical algorithm for O(log n) in place merging was proposed by Pok-Son Kim and Arne Kutzner in 2008. |
epfl-collab | Which of the following statements on Latent Semantic Indexing (LSI) and Word Embeddings (WE) is false? | ['The dimensions of LSI can be interpreted as concepts, whereas those of WE cannot', 'LSI is deterministic (given the dimension), whereas WE is not', 'LSI does take into account the frequency of words in the documents, whereas WE with negative sampling does not', 'LSI does not depend on the order of words in the document, whereas WE does'] | C | null | Document 1:::
Semantic analysis (machine learning)
A prominent example is PLSI. Latent Dirichlet allocation involves attributing document terms to topics. n-grams and hidden Markov models work by representing the term stream as a Markov chain where each term is derived from the few terms before it.
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Semantic analysis (machine learning)
If language is grounded, it is equal to recognizing a machine readable meaning. For the restricted domain of spatial analysis, a computer based language understanding system was demonstrated. : 123 Latent semantic analysis (sometimes latent semantic indexing), is a class of techniques where documents are represented as vectors in term space.
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Random indexing
Random indexing is a dimensionality reduction method and computational framework for distributional semantics, based on the insight that very-high-dimensional vector space model implementations are impractical, that models need not grow in dimensionality when new items (e.g. new terminology) are encountered, and that a high-dimensional model can be projected into a space of lower dimensionality without compromising L2 distance metrics if the resulting dimensions are chosen appropriately. This is the original point of the random projection approach to dimension reduction first formulated as the Johnson–Lindenstrauss lemma, and locality-sensitive hashing has some of the same starting points. Random indexing, as used in representation of language, originates from the work of Pentti Kanerva on sparse distributed memory, and can be described as an incremental formulation of a random projection.It can be also verified that random indexing is a random projection technique for the construction of Euclidean spaces—i.e. L2 normed vector spaces.
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Natural Language Semantics
Natural Language Semantics is a quarterly peer-reviewed academic journal of semantics published by Springer Science+Business Media. It covers semantics and its interfaces in grammar, especially in syntax. The founding editors-in-chief were Irene Heim (MIT) and Angelika Kratzer (University of Massachusetts Amherst). The current editor-in-chief is Amy Rose Deal (University of California, Berkeley).
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Lesk algorithm
Variations, such as the Simplified Lesk algorithm, have demonstrated improved precision and efficiency. However, the Lesk algorithm has faced criticism for its sensitivity to definition wording and its reliance on brief glosses. Researchers have sought to enhance its accuracy by incorporating additional resources like thesauruses and syntactic models. |
epfl-collab | The number of non-zero entries in a column of a term-document matrix indicates: | ['how often a term of the vocabulary occurs in a document', 'how many terms of the vocabulary a document contains', 'none of the other responses is correct', 'how relevant a term is for a document'] | C | null | Document 1:::
Zero matrix
In mathematics, particularly linear algebra, a zero matrix or null matrix is a matrix all of whose entries are zero. It also serves as the additive identity of the additive group of m × n {\displaystyle m\times n} matrices, and is denoted by the symbol O {\displaystyle O} or 0 {\displaystyle 0} followed by subscripts corresponding to the dimension of the matrix as the context sees fit. Some examples of zero matrices are 0 1 , 1 = , 0 2 , 2 = , 0 2 , 3 = . {\displaystyle 0_{1,1}={\begin{bmatrix}0\end{bmatrix}},\ 0_{2,2}={\begin{bmatrix}0&0\\0&0\end{bmatrix}},\ 0_{2,3}={\begin{bmatrix}0&0&0\\0&0&0\end{bmatrix}}.\ }
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Pascal matrix
The non-zero elements of a Pascal matrix are given by the binomial coefficients: such that the indices i, j start at 0, and ! denotes the factorial.
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Zernike polynomials
Applications often involve linear algebra, where an integral over a product of Zernike polynomials and some other factor builds a matrix elements. To enumerate the rows and columns of these matrices by a single index, a conventional mapping of the two indices n and l to a single index j has been introduced by Noll. The table of this association Z n l → Z j {\displaystyle Z_{n}^{l}\rightarrow Z_{j}} starts as follows (sequence A176988 in the OEIS). j = n ( n + 1 ) 2 + | l | + { 0 , l > 0 ∧ n ≡ { 0 , 1 } ( mod 4 ) ; 0 , l < 0 ∧ n ≡ { 2 , 3 } ( mod 4 ) ; 1 , l ≥ 0 ∧ n ≡ { 2 , 3 } ( mod 4 ) ; 1 , l ≤ 0 ∧ n ≡ { 0 , 1 } ( mod 4 ) .
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Boolean model of information retrieval
An index term is a word or expression, which may be stemmed, describing or characterizing a document, such as a keyword given for a journal article. Letbe the set of all such index terms. A document is any subset of T {\displaystyle T} . Letbe the set of all documents.
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Logical matrix
A logical matrix, binary matrix, relation matrix, Boolean matrix, or (0, 1)-matrix is a matrix with entries from the Boolean domain B = {0, 1}. Such a matrix can be used to represent a binary relation between a pair of finite sets. It is an important tool in combinatorial mathematics and theoretical computer science. |
epfl-collab | Which of the following statements on Latent Semantic Indexing (LSI) and Word Embeddings (WE) is incorrect | ['LSI is deterministic (given the dimension), whereas WE is not', 'The dimensions of LSI can be interpreted as concepts, whereas those of WE cannot', 'LSI does take into account the frequency of words in the documents, whereas WE does not.', 'LSI does not take into account the order of words in the document, whereas WE does'] | C | null | Document 1:::
Semantic analysis (machine learning)
A prominent example is PLSI. Latent Dirichlet allocation involves attributing document terms to topics. n-grams and hidden Markov models work by representing the term stream as a Markov chain where each term is derived from the few terms before it.
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Semantic analysis (machine learning)
If language is grounded, it is equal to recognizing a machine readable meaning. For the restricted domain of spatial analysis, a computer based language understanding system was demonstrated. : 123 Latent semantic analysis (sometimes latent semantic indexing), is a class of techniques where documents are represented as vectors in term space.
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Natural Language Semantics
Natural Language Semantics is a quarterly peer-reviewed academic journal of semantics published by Springer Science+Business Media. It covers semantics and its interfaces in grammar, especially in syntax. The founding editors-in-chief were Irene Heim (MIT) and Angelika Kratzer (University of Massachusetts Amherst). The current editor-in-chief is Amy Rose Deal (University of California, Berkeley).
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Lesk algorithm
Variations, such as the Simplified Lesk algorithm, have demonstrated improved precision and efficiency. However, the Lesk algorithm has faced criticism for its sensitivity to definition wording and its reliance on brief glosses. Researchers have sought to enhance its accuracy by incorporating additional resources like thesauruses and syntactic models.
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Random indexing
Random indexing is a dimensionality reduction method and computational framework for distributional semantics, based on the insight that very-high-dimensional vector space model implementations are impractical, that models need not grow in dimensionality when new items (e.g. new terminology) are encountered, and that a high-dimensional model can be projected into a space of lower dimensionality without compromising L2 distance metrics if the resulting dimensions are chosen appropriately. This is the original point of the random projection approach to dimension reduction first formulated as the Johnson–Lindenstrauss lemma, and locality-sensitive hashing has some of the same starting points. Random indexing, as used in representation of language, originates from the work of Pentti Kanerva on sparse distributed memory, and can be described as an incremental formulation of a random projection.It can be also verified that random indexing is a random projection technique for the construction of Euclidean spaces—i.e. L2 normed vector spaces. |
epfl-collab | Suppose that in a given FP Tree, an item in a leaf node N exists in every path. Which of the following is true? | ['For every node P that is a parent of N in the FP tree, confidence(P->N) = 1', 'N co-occurs with its prefixes in every transaction', '{N}’s minimum possible support is equal to the number of paths', 'The item N exists in every candidate set'] | C | null | Document 1:::
Tree (automata theory)
If every node of a tree has finitely many successors, then it is called a finitely, otherwise an infinitely branching tree. A path π is a subset of T such that ε ∈ π and for every t ∈ T, either t is a leaf or there exists a unique c ∈ N {\displaystyle \mathbb {N} } such that t.c ∈ π. A path may be a finite or infinite set. If all paths of a tree are finite then the tree is called finite, otherwise infinite.
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Locally finite rooted tree
In graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by at most one path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees.A polytree (or directed tree or oriented tree or singly connected network) is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. The various kinds of data structures referred to as trees in computer science have underlying graphs that are trees in graph theory, although such data structures are generally rooted trees.
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Directed tree
A tree is an undirected graph G that satisfies any of the following equivalent conditions: G is connected and acyclic (contains no cycles). G is acyclic, and a simple cycle is formed if any edge is added to G. G is connected, but would become disconnected if any single edge is removed from G. G is connected and the 3-vertex complete graph K3 is not a minor of G. Any two vertices in G can be connected by a unique simple path.If G has finitely many vertices, say n of them, then the above statements are also equivalent to any of the following conditions: G is connected and has n − 1 edges. G is connected, and every subgraph of G includes at least one vertex with zero or one incident edges. (That is, G is connected and 1-degenerate.)
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Rooted tree
A labeled tree is a tree in which each vertex is given a unique label. The vertices of a labeled tree on n vertices (for nonnegative integers n) are typically given the labels 1, 2, …, n. A recursive tree is a labeled rooted tree where the vertex labels respect the tree order (i.e., if u < v for two vertices u and v, then the label of u is smaller than the label of v). In a rooted tree, the parent of a vertex v is the vertex connected to v on the path to the root; every vertex has a unique parent, except the root has no parent.
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Arborescence (graph theory)
In graph theory, an arborescence is a directed graph in which, for a vertex u (called the root) and any other vertex v, there is exactly one directed path from u to v. An arborescence is thus the directed-graph form of a rooted tree, understood here as an undirected graph.Equivalently, an arborescence is a directed, rooted tree in which all edges point away from the root; a number of other equivalent characterizations exist. Every arborescence is a directed acyclic graph (DAG), but not every DAG is an arborescence. An arborescence can equivalently be defined as a rooted digraph in which the path from the root to any other vertex is unique. |
epfl-collab | Which of the following statements regarding topic models is false? | ['Topic models map documents to dense vectors', 'LDA assumes that each document is generated from a mixture of topics with a probability distribution', 'In LDA, topics are modeled as distributions over documents', 'Topics can serve as features for document classification'] | C | null | Document 1:::
Boolean model of information retrieval
The (standard) Boolean model of information retrieval (BIR) is a classical information retrieval (IR) model and, at the same time, the first and most-adopted one. It is used by many IR systems to this day. The BIR is based on Boolean logic and classical set theory in that both the documents to be searched and the user's query are conceived as sets of terms (a bag-of-words model). Retrieval is based on whether or not the documents contain the query terms.
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Neural net language model
A language model is a probabilistic model of a natural language that can generate probabilities of a series of words, based on text corpora in one or multiple languages it was trained on. Large language models, as their most advanced form, are a combination of feedforward neural networks and transformers. They have superseded recurrent neural network-based models, which had previously superseded the pure statistical models, such as word n-gram language model. Language models are useful for a variety of tasks, including speech recognition (helping prevent predictions of low-probability (e.g. nonsense) sequences), machine translation, natural language generation (generating more human-like text), optical character recognition, handwriting recognition, grammar induction, information retrieval, and other.
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Information structure
Focus "indicates the presence of alternatives that are relevant for the interpretation of linguistic expressions", givenness indicates that "the denotation of an expression is present" in the immediate context of the utterance, and topic is "the entity that a speaker identifies, about which then information, the comment, is given". Additional notions in information structure may include contrast and exhaustivity, but there is no general agreement in the linguistic literature about extensions of the basic three notions. There are many different approaches, such as cognitive, generative or functional architectures, to information structure. The concept has also been used in studies measuring information density in cognitive linguistics.
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Word n-gram language model
A word n-gram language model is a purely statistical model of language. It has been superseded by recurrent neural network-based models, which has been superseded by large language models. It is based on an assumption that the probability of the next word in a sequence depends only on a fixed size window of previous words.
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Word n-gram language model
A word n-gram language model is a purely statistical model of language. It has been superseded by recurrent neural network-based models, which has been superseded by large language models. It is based on an assumption that the probability of the next word in a sequence depends only on a fixed size window of previous words. |
epfl-collab | Modularity of a social network always: | ['Decreases when new nodes are added to the social network that form their own communities', 'Increases when an edge is added between two members of the same community', 'Increases with the number of communities', 'Decreases if an edge is removed'] | B | null | Document 1:::
Modularity (networks)
Modularity is a measure of the structure of networks or graphs which measures the strength of division of a network into modules (also called groups, clusters or communities). Networks with high modularity have dense connections between the nodes within modules but sparse connections between nodes in different modules. Modularity is often used in optimization methods for detecting community structure in networks.
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Modularity (networks)
Modularity is the fraction of the edges that fall within the given groups minus the expected fraction if edges were distributed at random. The value of the modularity for unweighted and undirected graphs lies in the range {\displaystyle } . It is positive if the number of edges within groups exceeds the number expected on the basis of chance. For a given division of the network's vertices into some modules, modularity reflects the concentration of edges within modules compared with random distribution of links between all nodes regardless of modules.
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Modularity (networks)
Thus A v w = A w v {\displaystyle A_{vw}=A_{wv}} . (It is important to note that multiple edges may exist between two nodes, but here we assess the simplest case). Modularity Q {\displaystyle Q} is then defined as the fraction of edges that fall within group 1 or 2, minus the expected number of edges within groups 1 and 2 for a random graph with the same node degree distribution as the given network.
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Modularity (networks)
If a node v {\displaystyle v} belongs to community 1, s v = 1 {\displaystyle s_{v}=1} , or if v {\displaystyle v} belongs to community 2, s v = − 1 {\displaystyle s_{v}=-1} . Let the adjacency matrix for the network be represented by A {\displaystyle A} , where A v w = 0 {\displaystyle A_{vw}=0} means there's no edge (no interaction) between nodes v {\displaystyle v} and w {\displaystyle w} and A v w = 1 {\displaystyle A_{vw}=1} means there is an edge between the two. Also for simplicity we consider an undirected network.
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Modularity (networks)
Biological networks, including animal brains, exhibit a high degree of modularity. However, modularity maximization is not statistically consistent, and finds communities in its own null model, i.e. fully random graphs, and therefore it cannot be used to find statistically significant community structures in empirical networks. Furthermore, it has been shown that modularity suffers a resolution limit and, therefore, it is unable to detect small communities. |
epfl-collab | Which of the following is wrong regarding Ontologies? | ['We can create more than one ontology that conceptualizes the same real-world entities', 'Ontologies support domain-specific vocabularies', 'Ontologies help in the integration of data expressed in different models', 'Ontologies dictate how semi-structured data are serialized'] | D | null | Document 1:::
Class (knowledge representation)
The first definition of class results in ontologies in which a class is a subclass of collection. The second definition of class results in ontologies in which collections and classes are more fundamentally different. Classes may classify individuals, other classes, or a combination of both.
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Class (knowledge representation)
While extensional classes are more well-behaved and well understood mathematically, as well as less problematic philosophically, they do not permit the fine grained distinctions that ontologies often need to make. For example, an ontology may want to distinguish between the class of all creatures with a kidney and the class of all creatures with a heart, even if these classes happen to have exactly the same members. In most upper ontologies, the classes are defined intensionally. Intensionally defined classes usually have necessary conditions associated with membership in each class. Some classes may also have sufficient conditions, and in those cases the combination of necessary and sufficient conditions make that class a fully defined class.
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Plant ontology
Plant ontology (PO) is a collection of ontologies developed by the Plant Ontology Consortium. These ontologies describe anatomical structures and growth and developmental stages across Viridiplantae. The PO is intended for multiple applications, including genetics, genomics, phenomics, and development, taxonomy and systematics, semantic applications and education.
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Class (knowledge representation)
The classes of an ontology may be extensional or intensional in nature. A class is extensional if and only if it is characterized solely by its membership. More precisely, a class C is extensional if and only if for any class C', if C' has exactly the same members as C, then C and C' are identical. If a class does not satisfy this condition, then it is intensional.
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Disease Ontology
The Disease Ontology (DO) is a formal ontology of human disease. The Disease Ontology project is hosted at the Institute for Genome Sciences at the University of Maryland School of Medicine. The Disease Ontology project was initially developed in 2003 at Northwestern University to address the need for a purpose-built ontology that covers the full spectrum of disease concepts annotated within biomedical repositories within an ontological framework that is extensible to meet community needs. |
epfl-collab | Which of the following statements is correct concerning the use of Pearson’s Correlation for user- based collaborative filtering? | ['It measures how much a user’s ratings deviate from the average ratings I', 'It measures whether different users have similar preferences for the same items', 't measures how well the recommendations match the user’s preferences', 'It measures whether a user has similar preferences for different items'] | B | null | Document 1:::
Statistical correlation
The most common of these is the Pearson correlation coefficient, which is sensitive only to a linear relationship between two variables (which may be present even when one variable is a nonlinear function of the other). Other correlation coefficients – such as Spearman's rank correlation – have been developed to be more robust than Pearson's, that is, more sensitive to nonlinear relationships. Mutual information can also be applied to measure dependence between two variables.
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Pearson's correlation
Pearson's correlation coefficient, when applied to a sample, is commonly represented by r x y {\displaystyle r_{xy}} and may be referred to as the sample correlation coefficient or the sample Pearson correlation coefficient. We can obtain a formula for r x y {\displaystyle r_{xy}} by substituting estimates of the covariances and variances based on a sample into the formula above. Given paired data { ( x 1 , y 1 ) , … , ( x n , y n ) } {\displaystyle \left\{(x_{1},y_{1}),\ldots ,(x_{n},y_{n})\right\}} consisting of n {\displaystyle n} pairs, r x y {\displaystyle r_{xy}} is defined as where n {\displaystyle n} is sample size x i , y i {\displaystyle x_{i},y_{i}} are the individual sample points indexed with i x ¯ = 1 n ∑ i = 1 n x i {\textstyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}} (the sample mean); and analogously for y ¯ {\displaystyle {\bar {y}}} .Rearranging gives us this formula for r x y {\displaystyle r_{xy}}: r x y = n ∑ x i y i − ∑ x i ∑ y i n ∑ x i 2 − ( ∑ x i ) 2 n ∑ y i 2 − ( ∑ y i ) 2 . {\displaystyle r_{xy}={\frac {n\sum x_{i}y_{i}-\sum x_{i}\sum y_{i}}{{\sqrt {n\sum x_{i}^{2}-\left(\sum x_{i}\right)^{2}}}~{\sqrt {n\sum y_{i}^{2}-\left(\sum y_{i}\right)^{2}}}}}.}
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Pearson's correlation
where n , x i , y i {\displaystyle n,x_{i},y_{i}} are defined as above. This formula suggests a convenient single-pass algorithm for calculating sample correlations, though depending on the numbers involved, it can sometimes be numerically unstable. Rearranging again gives us this formula for r x y {\displaystyle r_{xy}}: r x y = ∑ i x i y i − n x ¯ y ¯ ∑ i x i 2 − n x ¯ 2 ∑ i y i 2 − n y ¯ 2 .
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Statistical correlation
In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics it usually refers to the degree to which a pair of variables are linearly related. Familiar examples of dependent phenomena include the correlation between the height of parents and their offspring, and the correlation between the price of a good and the quantity the consumers are willing to purchase, as it is depicted in the so-called demand curve. Correlations are useful because they can indicate a predictive relationship that can be exploited in practice.
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Distance correlation
In statistics and in probability theory, distance correlation or distance covariance is a measure of dependence between two paired random vectors of arbitrary, not necessarily equal, dimension. The population distance correlation coefficient is zero if and only if the random vectors are independent. Thus, distance correlation measures both linear and nonlinear association between two random variables or random vectors. This is in contrast to Pearson's correlation, which can only detect linear association between two random variables. Distance correlation can be used to perform a statistical test of dependence with a permutation test. One first computes the distance correlation (involving the re-centering of Euclidean distance matrices) between two random vectors, and then compares this value to the distance correlations of many shuffles of the data. |
epfl-collab | After the join step, the number of k+1-itemsets | ['is equal to the number of frequent k-itemsets', 'is always higher than the number of frequent k-itemsets', 'is always lower than the number of frequent k-itemsets', 'can be equal, lower or higher than the number of frequent k-itemsets'] | D | null | Document 1:::
Karmarkar–Karp bin packing algorithms
They also devised several other algorithms with slightly different approximation guarantees and run-time bounds. The KK algorithms were considered a breakthrough in the study of bin packing: the previously-known algorithms found multiplicative approximation, where the number of bins was at most r ⋅ O P T + s {\displaystyle r\cdot \mathrm {OPT} +s} for some constants r > 1 , s > 0 {\displaystyle r>1,s>0} , or at most ( 1 + ε ) O P T + 1 {\displaystyle (1+\varepsilon )\mathrm {OPT} +1} . The KK algorithms were the first ones to attain an additive approximation.
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Combination
If the set has n elements, the number of k-combinations, denoted by C ( n , k ) {\displaystyle C(n,k)} or C k n {\displaystyle C_{k}^{n}} , is equal to the binomial coefficient which can be written using factorials as n ! k ! ( n − k ) !
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Binomial coefficient
For natural numbers (taken to include 0) n and k, the binomial coefficient ( n k ) {\displaystyle {\tbinom {n}{k}}} can be defined as the coefficient of the monomial Xk in the expansion of (1 + X)n. The same coefficient also occurs (if k ≤ n) in the binomial formula (valid for any elements x, y of a commutative ring), which explains the name "binomial coefficient". Another occurrence of this number is in combinatorics, where it gives the number of ways, disregarding order, that k objects can be chosen from among n objects; more formally, the number of k-element subsets (or k-combinations) of an n-element set. This number can be seen as equal to the one of the first definition, independently of any of the formulas below to compute it: if in each of the n factors of the power (1 + X)n one temporarily labels the term X with an index i (running from 1 to n), then each subset of k indices gives after expansion a contribution Xk, and the coefficient of that monomial in the result will be the number of such subsets. This shows in particular that ( n k ) {\displaystyle {\tbinom {n}{k}}} is a natural number for any natural numbers n and k. There are many other combinatorial interpretations of binomial coefficients (counting problems for which the answer is given by a binomial coefficient expression), for instance the number of words formed of n bits (digits 0 or 1) whose sum is k is given by ( n k ) {\displaystyle {\tbinom {n}{k}}} , while the number of ways to write k = a 1 + a 2 + ⋯ + a n {\displaystyle k=a_{1}+a_{2}+\cdots +a_{n}} where every ai is a nonnegative integer is given by ( n + k − 1 n − 1 ) {\displaystyle {\tbinom {n+k-1}{n-1}}} . Most of these interpretations can be shown to be equivalent to counting k-combinations.
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Sort-Merge Join
The sort-merge join (also known as merge join) is a join algorithm and is used in the implementation of a relational database management system. The basic problem of a join algorithm is to find, for each distinct value of the join attribute, the set of tuples in each relation which display that value. The key idea of the sort-merge algorithm is to first sort the relations by the join attribute, so that interleaved linear scans will encounter these sets at the same time. In practice, the most expensive part of performing a sort-merge join is arranging for both inputs to the algorithm to be presented in sorted order.
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Affinity analysis
The first step in the process is to count the co-occurrence of attributes in the data set. Next, a subset is created called the frequent itemset. The association rules mining takes the form of if a condition or feature (A) is present then another condition or feature (B) exists. |
epfl-collab | Which is true about the use of entropy in decision tree induction? | ['The entropy of the set of class labels of the samples from the training set at the leaf level can be 1', 'The entropy of the set of class labels of the samples from the training set at the leaf level is always 0', 'We split on the attribute that has the highest entropy', 'We split on the attribute that has the lowest entropy'] | A | null | Document 1:::
Information gain in decision trees
In information theory and machine learning, information gain is a synonym for Kullback–Leibler divergence; the amount of information gained about a random variable or signal from observing another random variable. However, in the context of decision trees, the term is sometimes used synonymously with mutual information, which is the conditional expected value of the Kullback–Leibler divergence of the univariate probability distribution of one variable from the conditional distribution of this variable given the other one. The information gain of a random variable X obtained from an observation of a random variable A taking value A = a {\displaystyle A=a} is defined the Kullback–Leibler divergence of the prior distribution P X ( x | I ) {\displaystyle P_{X}{(x|I)}} for x from the posterior distribution P X | A ( x | a ) {\displaystyle P_{X|A}{(x|a)}} for x given a. The expected value of the information gain is the mutual information I ( X ; A ) {\displaystyle I(X;A)} of X and A – i.e. the reduction in the entropy of X achieved by learning the state of the random variable A. In machine learning, this concept can be used to define a preferred sequence of attributes to investigate to most rapidly narrow down the state of X. Such a sequence (which depends on the outcome of the investigation of previous attributes at each stage) is called a decision tree and applied in the area of machine learning known as decision tree learning. Usually an attribute with high mutual information should be preferred to other attributes.
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Classification and regression tree
Decision tree learning is a supervised learning approach used in statistics, data mining and machine learning. In this formalism, a classification or regression decision tree is used as a predictive model to draw conclusions about a set of observations. Tree models where the target variable can take a discrete set of values are called classification trees; in these tree structures, leaves represent class labels and branches represent conjunctions of features that lead to those class labels. Decision trees where the target variable can take continuous values (typically real numbers) are called regression trees. More generally, the concept of regression tree can be extended to any kind of object equipped with pairwise dissimilarities such as categorical sequences.Decision trees are among the most popular machine learning algorithms given their intelligibility and simplicity.In decision analysis, a decision tree can be used to visually and explicitly represent decisions and decision making. In data mining, a decision tree describes data (but the resulting classification tree can be an input for decision making).
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Iterative Dichotomiser 3
Calculate the entropy of every attribute a {\displaystyle a} of the data set S {\displaystyle S} . Partition ("split") the set S {\displaystyle S} into subsets using the attribute for which the resulting entropy after splitting is minimized; or, equivalently, information gain is maximum. Make a decision tree node containing that attribute. Recurse on subsets using the remaining attributes.
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Rule induction
Rule induction is an area of machine learning in which formal rules are extracted from a set of observations. The rules extracted may represent a full scientific model of the data, or merely represent local patterns in the data. Data mining in general and rule induction in detail are trying to create algorithms without human programming but with analyzing existing data structures. : 415- In the easiest case, a rule is expressed with “if-then statements” and was created with the ID3 algorithm for decision tree learning.
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Decision tree complexity
In computational complexity the decision tree model is the model of computation in which an algorithm is considered to be basically a decision tree, i.e., a sequence of queries or tests that are done adaptively, so the outcome of previous tests can influence the tests performed next. Typically, these tests have a small number of outcomes (such as a yes–no question) and can be performed quickly (say, with unit computational cost), so the worst-case time complexity of an algorithm in the decision tree model corresponds to the depth of the corresponding decision tree. This notion of computational complexity of a problem or an algorithm in the decision tree model is called its decision tree complexity or query complexity. Decision trees models are instrumental in establishing lower bounds for complexity theory for certain classes of computational problems and algorithms. |
epfl-collab | Modularity clustering will end up always with the same community structure? | ['True', 'Only for cliques', 'False', 'Only for connected graphs'] | C | null | Document 1:::
Modularity (networks)
Biological networks, including animal brains, exhibit a high degree of modularity. However, modularity maximization is not statistically consistent, and finds communities in its own null model, i.e. fully random graphs, and therefore it cannot be used to find statistically significant community structures in empirical networks. Furthermore, it has been shown that modularity suffers a resolution limit and, therefore, it is unable to detect small communities.
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Modularity (networks)
Modularity is a measure of the structure of networks or graphs which measures the strength of division of a network into modules (also called groups, clusters or communities). Networks with high modularity have dense connections between the nodes within modules but sparse connections between nodes in different modules. Modularity is often used in optimization methods for detecting community structure in networks.
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Modularity (networks)
There are different methods for calculating modularity. In the most common version of the concept, the randomization of the edges is done so as to preserve the degree of each vertex. Consider a graph with n {\displaystyle n} nodes and m {\displaystyle m} links (edges) such that the graph can be partitioned into two communities using a membership variable s {\displaystyle s} .
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Modularity (biology)
Modularity refers to the ability of a system to organize discrete, individual units that can overall increase the efficiency of network activity and, in a biological sense, facilitates selective forces upon the network. Modularity is observed in all model systems, and can be studied at nearly every scale of biological organization, from molecular interactions all the way up to the whole organism.
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Modularity (networks)
If a node v {\displaystyle v} belongs to community 1, s v = 1 {\displaystyle s_{v}=1} , or if v {\displaystyle v} belongs to community 2, s v = − 1 {\displaystyle s_{v}=-1} . Let the adjacency matrix for the network be represented by A {\displaystyle A} , where A v w = 0 {\displaystyle A_{vw}=0} means there's no edge (no interaction) between nodes v {\displaystyle v} and w {\displaystyle w} and A v w = 1 {\displaystyle A_{vw}=1} means there is an edge between the two. Also for simplicity we consider an undirected network. |
epfl-collab | When searching for an entity 𝑒𝑛𝑒𝑤 that has a given relationship 𝑟 with a given entity 𝑒 | ['We search for pairs (𝑒𝑛𝑒𝑤, 𝑒) that have similar embedding to (𝑒𝑜𝑙𝑑, 𝑒) for 𝑒𝑜𝑙𝑑 which has relationship 𝑟 with 𝑒', 'We search for 𝑒𝑛𝑒𝑤 that have a similar embedding vector to 𝑒', 'We search for 𝑒𝑛𝑒𝑤 that have a similar embedding vector to 𝑒𝑜𝑙𝑑 which has relationship 𝑟 with 𝑒', 'We search for pairs (𝑒𝑛𝑒𝑤, 𝑒) that have similar embedding to (𝑒𝑜𝑙𝑑, 𝑒)'] | D | null | Document 1:::
Trigram search
Trigram search is a method of searching for text when the exact syntax or spelling of the target object is not precisely known or when queries may be regular expressions. It finds objects which match the maximum number of three consecutive character strings (i.e. trigrams) in the entered search terms, which are generally near matches. Two strings with many shared trigrams can be expected to be very similar. Trigrams also allow for efficiently creating search engine indexes for searches that are regular expressions or match the text inexactly. Indexes can significantly accelerate searches. A threshold for number of trigram matches can be specified as a cutoff point, after which a result is no longer considered a match.Using trigrams for accelerating searches is a technique used in some systems for code searching, in situations in which queries that are regular expressions may be useful, in search engines such as Elasticsearch, as well as in databases such as PostgreSQL.
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Boolean conjunctive query
In the theory of relational databases, a Boolean conjunctive query is a conjunctive query without distinguished predicates, i.e., a query in the form R 1 ( t 1 ) ∧ ⋯ ∧ R n ( t n ) {\displaystyle R_{1}(t_{1})\wedge \cdots \wedge R_{n}(t_{n})} , where each R i {\displaystyle R_{i}} is a relation symbol and each t i {\displaystyle t_{i}} is a tuple of variables and constants; the number of elements in t i {\displaystyle t_{i}} is equal to the arity of R i {\displaystyle R_{i}} . Such a query evaluates to either true or false depending on whether the relations in the database contain the appropriate tuples of values, i.e. the conjunction is valid according to the facts in the database. As an example, if a database schema contains the relation symbols Father (binary, who's the father of whom) and Employed (unary, who is employed), a conjunctive query could be F a t h e r ( Mark , x ) ∧ E m p l o y e d ( x ) {\displaystyle Father({\text{Mark}},x)\wedge Employed(x)} . This query evaluates to true if there exists an individual x who is a child of Mark and employed. In other words, this query expresses the question: "does Mark have an employed child?"
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Graph Search
Facebook Graph Search was a semantic search engine that Facebook introduced in March 2013. It was designed to give answers to user natural language queries rather than a list of links. The name refers to the social graph nature of Facebook, which maps the relationships among users. The Graph Search feature combined the big data acquired from its over one billion users and external data into a search engine providing user-specific search results.
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Semantic Search
Semantic search denotes search with meaning, as distinguished from lexical search where the search engine looks for literal matches of the query words or variants of them, without understanding the overall meaning of the query. Semantic search seeks to improve search accuracy by understanding the searcher's intent and the contextual meaning of terms as they appear in the searchable dataspace, whether on the Web or within a closed system, to generate more relevant results. Content that ranks well in semantic search is well-written in a natural voice, focuses on the user's intent, and considers related topics that the user may look for in the future.Some authors regard semantic search as a set of techniques for retrieving knowledge from richly structured data sources like ontologies and XML as found on the Semantic Web. Such technologies enable the formal articulation of domain knowledge at a high level of expressiveness and could enable the user to specify their intent in more detail at query time.
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Entity integrity
The same format applies to the foreign key (abbreviated FK) because each FK matches a preexistent PK. Each of attributes being part of a PK (or of a FK) must have data values (such as numbers, letters or typographic symbols) but not data marks (also known as NULL marks in SQL world). Morphologically, a composite primary key is in a "steady state": If it is reduced, PK will lose its property of identifying every row of its relation but if it is extended, PK will be redundant. |
epfl-collab | Which of the following graph analysis techniques do you believe would be most appropriate to identify communities on a social graph? | ['Random Walks', 'Cliques', 'Association rules', 'Shortest Paths'] | B | null | Document 1:::
Social graph
The social graph is a graph that represents social relations between entities. In short, it is a model or representation of a social network, where the word graph has been taken from graph theory. The social graph has been referred to as "the global mapping of everybody and how they're related".The term was used as early as 1964, albeit in the context of isoglosses. Leo Apostel uses the term in the context here in 1978.
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Clique percolation method
The clique percolation method is a popular approach for analyzing the overlapping community structure of networks. The term network community (also called a module, cluster or cohesive group) has no widely accepted unique definition and it is usually defined as a group of nodes that are more densely connected to each other than to other nodes in the network. There are numerous alternative methods for detecting communities in networks, for example, the Girvan–Newman algorithm, hierarchical clustering and modularity maximization.
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Graph API
The social graph is a graph that represents social relations between entities. In short, it is a model or representation of a social network, where the word graph has been taken from graph theory. The social graph has been referred to as "the global mapping of everybody and how they're related".The term was used as early as 1964, albeit in the context of isoglosses. Leo Apostel uses the term in the context here in 1978.
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Clique percolation method
Clique percolation methods may be generalized by recording different amounts of overlap between the various k-cliques. This then defines a new type of graph, a clique graph, where each k-clique in the original graph is represented by a vertex in the new clique graph. The edges in the clique graph are used to record the strength of the overlap of cliques in the original graph. One may then apply any community detection method to this clique graph to identify the clusters in the original graph through the k-clique structure.
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Community structure
In the study of complex networks, a network is said to have community structure if the nodes of the network can be easily grouped into (potentially overlapping) sets of nodes such that each set of nodes is densely connected internally. In the particular case of non-overlapping community finding, this implies that the network divides naturally into groups of nodes with dense connections internally and sparser connections between groups. But overlapping communities are also allowed. The more general definition is based on the principle that pairs of nodes are more likely to be connected if they are both members of the same community(ies), and less likely to be connected if they do not share communities. A related but different problem is community search, where the goal is to find a community that a certain vertex belongs to. |
epfl-collab | Which of the following models for generating vector representations for text require to precompute the frequency of co-occurrence of words from the vocabulary in the document collection | ['Fasttext', 'LSI', 'CBOW', 'Glove'] | D | null | Document 1:::
Bag-of-words model in computer vision
In computer vision, the bag-of-words model (BoW model) sometimes called bag-of-visual-words model can be applied to image classification or retrieval, by treating image features as words. In document classification, a bag of words is a sparse vector of occurrence counts of words; that is, a sparse histogram over the vocabulary. In computer vision, a bag of visual words is a vector of occurrence counts of a vocabulary of local image features.
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Neural net language model
A language model is a probabilistic model of a natural language that can generate probabilities of a series of words, based on text corpora in one or multiple languages it was trained on. Large language models, as their most advanced form, are a combination of feedforward neural networks and transformers. They have superseded recurrent neural network-based models, which had previously superseded the pure statistical models, such as word n-gram language model. Language models are useful for a variety of tasks, including speech recognition (helping prevent predictions of low-probability (e.g. nonsense) sequences), machine translation, natural language generation (generating more human-like text), optical character recognition, handwriting recognition, grammar induction, information retrieval, and other.
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Random indexing
In Euclidean spaces, random projections are elucidated using the Johnson–Lindenstrauss lemma.The TopSig technique extends the random indexing model to produce bit vectors for comparison with the Hamming distance similarity function. It is used for improving the performance of information retrieval and document clustering. In a similar line of research, Random Manhattan Integer Indexing (RMII) is proposed for improving the performance of the methods that employ the Manhattan distance between text units. Many random indexing methods primarily generate similarity from co-occurrence of items in a corpus. Reflexive Random Indexing (RRI) generates similarity from co-occurrence and from shared occurrence with other items.
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Cosine similarity
For example, in information retrieval and text mining, each word is assigned a different coordinate and a document is represented by the vector of the numbers of occurrences of each word in the document. Cosine similarity then gives a useful measure of how similar two documents are likely to be, in terms of their subject matter, and independently of the length of the documents.The technique is also used to measure cohesion within clusters in the field of data mining.One advantage of cosine similarity is its low complexity, especially for sparse vectors: only the non-zero coordinates need to be considered. Other names for cosine similarity include Orchini similarity and Tucker coefficient of congruence; the Otsuka–Ochiai similarity (see below) is cosine similarity applied to binary data.
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Random indexing
Random indexing is a dimensionality reduction method and computational framework for distributional semantics, based on the insight that very-high-dimensional vector space model implementations are impractical, that models need not grow in dimensionality when new items (e.g. new terminology) are encountered, and that a high-dimensional model can be projected into a space of lower dimensionality without compromising L2 distance metrics if the resulting dimensions are chosen appropriately. This is the original point of the random projection approach to dimension reduction first formulated as the Johnson–Lindenstrauss lemma, and locality-sensitive hashing has some of the same starting points. Random indexing, as used in representation of language, originates from the work of Pentti Kanerva on sparse distributed memory, and can be described as an incremental formulation of a random projection.It can be also verified that random indexing is a random projection technique for the construction of Euclidean spaces—i.e. L2 normed vector spaces. |
epfl-collab | For which document classifier the training cost is low and inference is expensive? | ['for none', 'for kNN', 'for fasttext', 'for NB'] | B | null | Document 1:::
Document AI
Document AI or Document Intelligence is a technology that uses natural language processing (NLP) and machine learning (ML) to train computer models to simulate a human review of documents. NLP enables the computer system to grasp the relations between contents of documents, including the contextual nuances of the language within them, before extracting the information and insights contained in the documents. The technology can then categorize and organize the documents themselves.Document AI is used to process and parse forms, tables, receipts, invoices, tax forms, contracts, loan agreements, financial reports, etc. One of the applications of Document AI is in Real estate. A model that unifies existing approaches and leads to the development of a document classification standard and automated information extraction through AI will become instrumental for enhanced and innovative business models and products in real estate. The availability of up-to-date and complete documentation for transactions, such as real estate investments, can help reduce safety margins due to better estimated risks.
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Document AI
Document AI or Document Intelligence is a technology that uses natural language processing (NLP) and machine learning (ML) to train computer models to simulate a human review of documents. NLP enables the computer system to grasp the relations between contents of documents, including the contextual nuances of the language within them, before extracting the information and insights contained in the documents. The technology can then categorize and organize the documents themselves.Document AI is used to process and parse forms, tables, receipts, invoices, tax forms, contracts, loan agreements, financial reports, etc. One of the applications of Document AI is in real estate. A model that unifies existing approaches and leads to the development of a document classification standard and automated information extraction through AI will become instrumental for enhanced and innovative business models and products in real estate. The availability of up-to-date and complete documentation for transactions, such as real estate investments, can help reduce safety margins due to better estimated risks.
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Naive Bayes classifier
In statistics, naive Bayes classifiers are a family of simple "probabilistic classifiers" based on applying Bayes' theorem with strong (naive) independence assumptions between the features (see Bayes classifier). They are among the simplest Bayesian network models, but coupled with kernel density estimation, they can achieve high accuracy levels.Naive Bayes classifiers are highly scalable, requiring a number of parameters linear in the number of variables (features/predictors) in a learning problem. Maximum-likelihood training can be done by evaluating a closed-form expression,: 718 which takes linear time, rather than by expensive iterative approximation as used for many other types of classifiers. In the statistics literature, naive Bayes models are known under a variety of names, including simple Bayes and independence Bayes. All these names reference the use of Bayes' theorem in the classifier's decision rule, but naive Bayes is not (necessarily) a Bayesian method.
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Text categorization
Document classification or document categorization is a problem in library science, information science and computer science. The task is to assign a document to one or more classes or categories. This may be done "manually" (or "intellectually") or algorithmically. The intellectual classification of documents has mostly been the province of library science, while the algorithmic classification of documents is mainly in information science and computer science.
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Pythia (machine learning)
Pythia takes as input the damaged text, and is trained to return hypothesised restorations of ancient Greek inscriptions, working as an assistive aid for ancient historians. Its neural network architecture works at both the character- and word-level, thereby effectively handling long-term context information, and dealing efficiently with incomplete word representations. Pythia is applicable to any discipline dealing with ancient texts (philology, papyrology, codicology) and can work in any language (ancient or modern). == References == |
epfl-collab | In Ranked Retrieval, the result at position k is non-relevant and at k+1 is relevant. Which of the following is always true?
Hint: P@k and R@k are the precision and recall of the result set consisting of the k top-ranked documents. | ['P@k-1=P@k+1', 'R@k-1=R@k+1', 'R@k-1<R@k+1', 'P@k-1>P@k+1'] | C | null | Document 1:::
Precision and recall
In pattern recognition, information retrieval, object detection and classification (machine learning), precision and recall are performance metrics that apply to data retrieved from a collection, corpus or sample space. Precision (also called positive predictive value) is the fraction of relevant instances among the retrieved instances. Written as a formula: r e l e v a n t _ r e t r i e v e d _ i n s t a n c e s a l l _ r e t r i e v e d _ i n s t a n c e s {\displaystyle {\frac {relevant\_retrieved\_instances}{all\_{\mathbf {retrieved}}\_instances}}} . Recall (also known as sensitivity) is the fraction of relevant instances that were retrieved.
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Uncertain inference
Rather than retrieving a document that exactly matches the query we should rank the documents based on their plausibility in regards to that query. Since d and q are both generated by users, they are error prone; thus d → q {\displaystyle d\to q} is uncertain. This will affect the plausibility of a given query.
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Uncertain inference
Rather than retrieving a document that exactly matches the query we should rank the documents based on their plausibility in regards to that query. Since d and q are both generated by users, they are error prone; thus d → q {\displaystyle d\to q} is uncertain. This will affect the plausibility of a given query.
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Precision and recall
Written as a formula: r e l e v a n t _ r e t r i e v e d _ i n s t a n c e s a l l _ r e l e v a n t _ i n s t a n c e s {\displaystyle {\frac {relevant\_retrieved\_instances}{all\_{\mathbf {relevant}}\_instances}}} . Both precision and recall are therefore based on relevance.
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Average precision
Evaluation measures for an information retrieval (IR) system assess how well an index, search engine or database returns results from a collection of resources that satisfy a user's query. They are therefore fundamental to the success of information systems and digital platforms. The success of an IR system may be judged by a range of criteria including relevance, speed, user satisfaction, usability, efficiency and reliability. However, the most important factor in determining a system's effectiveness for users is the overall relevance of results retrieved in response to a query. |
epfl-collab | Regarding the Expectation-Maximization algorithm, which one of the following false? | ['It distinguishes experts from normal workers', 'The label with the highest probability is assigned as the new label', 'In E step the labels change, in M step the weights of the workers change', 'Assigning equal weights to workers initially decreases the convergence time'] | D | null | Document 1:::
EM algorithm
In statistics, an expectation–maximization (EM) algorithm is an iterative method to find (local) maximum likelihood or maximum a posteriori (MAP) estimates of parameters in statistical models, where the model depends on unobserved latent variables. The EM iteration alternates between performing an expectation (E) step, which creates a function for the expectation of the log-likelihood evaluated using the current estimate for the parameters, and a maximization (M) step, which computes parameters maximizing the expected log-likelihood found on the E step. These parameter-estimates are then used to determine the distribution of the latent variables in the next E step.
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Baum–Welch algorithm
In electrical engineering, statistical computing and bioinformatics, the Baum–Welch algorithm is a special case of the expectation–maximization algorithm used to find the unknown parameters of a hidden Markov model (HMM). It makes use of the forward-backward algorithm to compute the statistics for the expectation step.
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Maximum-entropy Markov model
In statistics, a maximum-entropy Markov model (MEMM), or conditional Markov model (CMM), is a graphical model for sequence labeling that combines features of hidden Markov models (HMMs) and maximum entropy (MaxEnt) models. An MEMM is a discriminative model that extends a standard maximum entropy classifier by assuming that the unknown values to be learnt are connected in a Markov chain rather than being conditionally independent of each other. MEMMs find applications in natural language processing, specifically in part-of-speech tagging and information extraction.
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Viterbi Algorithm
The Viterbi algorithm is a dynamic programming algorithm for obtaining the maximum a posteriori probability estimate of the most likely sequence of hidden states—called the Viterbi path—that results in a sequence of observed events, especially in the context of Markov information sources and hidden Markov models (HMM). The algorithm has found universal application in decoding the convolutional codes used in both CDMA and GSM digital cellular, dial-up modems, satellite, deep-space communications, and 802.11 wireless LANs. It is now also commonly used in speech recognition, speech synthesis, diarization, keyword spotting, computational linguistics, and bioinformatics. For example, in speech-to-text (speech recognition), the acoustic signal is treated as the observed sequence of events, and a string of text is considered to be the "hidden cause" of the acoustic signal. The Viterbi algorithm finds the most likely string of text given the acoustic signal.
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Maximum likelihood sequence estimation
Maximum likelihood sequence estimation (MLSE) is a mathematical algorithm to extract useful data out of a noisy data stream. |
epfl-collab | For an item that has not received any ratings, which method can make a prediction? | ['User-based collaborative RS', 'Content-based RS', 'Item-based collaborative RS', 'None of the above'] | B | null | Document 1:::
Naranjo algorithm
The Naranjo algorithm, Naranjo Scale, or Naranjo Nomogram is a questionnaire designed by Naranjo et al. for determining the likelihood of whether an ADR (adverse drug reaction) is actually due to the drug rather than the result of other factors. Probability is assigned via a score termed definite, probable, possible or doubtful. Values obtained from this algorithm are often used in peer reviews to verify the validity of author's conclusions regarding adverse drug reactions. It is often compared to the WHO-UMC system for standardized causality assessment for suspected adverse drug reactions (ADRs).
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Statistical association football predictions
Statistical Football prediction is a method used in sports betting, to predict the outcome of football matches by means of statistical tools. The goal of statistical match prediction is to outperform the predictions of bookmakers, who use them to set odds on the outcome of football matches. The most widely used statistical approach to prediction is ranking. Football ranking systems assign a rank to each team based on their past game results, so that the highest rank is assigned to the strongest team.
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Statistical association football predictions
The outcome of the match can be predicted by comparing the opponents’ ranks. Several different football ranking systems exist, for example some widely known are the FIFA World Rankings or the World Football Elo Ratings. There are three main drawbacks to football match predictions that are based on ranking systems: Ranks assigned to the teams do not differentiate between their attacking and defensive strengths.
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Predictive classification
Often the unknown event of interest is in the future, but predictive analytics can be applied to any type of unknown whether it be in the past, present or future. For example, identifying suspects after a crime has been committed, or credit card fraud as it occurs. The core of predictive analytics relies on capturing relationships between explanatory variables and the predicted variables from past occurrences, and exploiting them to predict the unknown outcome.
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Statistical association football predictions
Ranks are accumulated averages which do not account for skill changes in football teams. The main goal of a ranking system is not to predict the results of football games, but to sort the teams according to their average strength.Another approach to football prediction is known as rating systems. While ranking refers only to team order, rating systems assign to each team a continuously scaled strength indicator. Moreover, rating can be assigned not only to a team but to its attacking and defensive strengths, home field advantage or even to the skills of each team player (according to Stern ). |
epfl-collab | The SMART algorithm for query relevance feedback modifies? (Slide 11 Week 3) | ['The original document weight vectors', 'The keywords of the original user query', 'The result document weight vectors', 'The original query weight vectors'] | D | null | Document 1:::
Relevance (information retrieval)
In information science and information retrieval, relevance denotes how well a retrieved document or set of documents meets the information need of the user. Relevance may include concerns such as timeliness, authority or novelty of the result.
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Query optimization
Query optimization is a feature of many relational database management systems and other databases such as NoSQL and graph databases. The query optimizer attempts to determine the most efficient way to execute a given query by considering the possible query plans.Generally, the query optimizer cannot be accessed directly by users: once queries are submitted to the database server, and parsed by the parser, they are then passed to the query optimizer where optimization occurs. However, some database engines allow guiding the query optimizer with hints. A query is a request for information from a database.
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Average precision
Evaluation measures for an information retrieval (IR) system assess how well an index, search engine or database returns results from a collection of resources that satisfy a user's query. They are therefore fundamental to the success of information systems and digital platforms. The success of an IR system may be judged by a range of criteria including relevance, speed, user satisfaction, usability, efficiency and reliability. However, the most important factor in determining a system's effectiveness for users is the overall relevance of results retrieved in response to a query.
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Query expansion
Query expansion (QE) is the process of reformulating a given query to improve retrieval performance in information retrieval operations, particularly in the context of query understanding. In the context of search engines, query expansion involves evaluating a user's input (what words were typed into the search query area, and sometimes other types of data) and expanding the search query to match additional documents. Query expansion involves techniques such as: Finding synonyms of words, and searching for the synonyms as well Finding semantically related words (e.g. antonyms, meronyms, hyponyms, hypernyms) Finding all the various morphological forms of words by stemming each word in the search query Fixing spelling errors and automatically searching for the corrected form or suggesting it in the results Re-weighting the terms in the original queryQuery expansion is a methodology studied in the field of computer science, particularly within the realm of natural language processing and information retrieval.
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Query understanding
Query understanding is the process of inferring the intent of a search engine user by extracting semantic meaning from the searcher’s keywords. Query understanding methods generally take place before the search engine retrieves and ranks results. It is related to natural language processing but specifically focused on the understanding of search queries. Query understanding is at the heart of technologies like Amazon Alexa, Apple's Siri. Google Assistant, IBM's Watson, and Microsoft's Cortana. |
epfl-collab | Suppose that in a given FP Tree, an item in a leaf node N exists in every path. Which of the following is TRUE? | ['N co-occurs with its prefixes in every transaction', 'For every node P that is a parent of N in the FP tree, confidence (P->N) = 1', 'The item N exists in every candidate set', '{N}’s minimum possible support is equal to the number of paths'] | D | null | Document 1:::
Tree (automata theory)
If every node of a tree has finitely many successors, then it is called a finitely, otherwise an infinitely branching tree. A path π is a subset of T such that ε ∈ π and for every t ∈ T, either t is a leaf or there exists a unique c ∈ N {\displaystyle \mathbb {N} } such that t.c ∈ π. A path may be a finite or infinite set. If all paths of a tree are finite then the tree is called finite, otherwise infinite.
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Locally finite rooted tree
In graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by at most one path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees.A polytree (or directed tree or oriented tree or singly connected network) is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. The various kinds of data structures referred to as trees in computer science have underlying graphs that are trees in graph theory, although such data structures are generally rooted trees.
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Directed tree
A tree is an undirected graph G that satisfies any of the following equivalent conditions: G is connected and acyclic (contains no cycles). G is acyclic, and a simple cycle is formed if any edge is added to G. G is connected, but would become disconnected if any single edge is removed from G. G is connected and the 3-vertex complete graph K3 is not a minor of G. Any two vertices in G can be connected by a unique simple path.If G has finitely many vertices, say n of them, then the above statements are also equivalent to any of the following conditions: G is connected and has n − 1 edges. G is connected, and every subgraph of G includes at least one vertex with zero or one incident edges. (That is, G is connected and 1-degenerate.)
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Rooted tree
A labeled tree is a tree in which each vertex is given a unique label. The vertices of a labeled tree on n vertices (for nonnegative integers n) are typically given the labels 1, 2, …, n. A recursive tree is a labeled rooted tree where the vertex labels respect the tree order (i.e., if u < v for two vertices u and v, then the label of u is smaller than the label of v). In a rooted tree, the parent of a vertex v is the vertex connected to v on the path to the root; every vertex has a unique parent, except the root has no parent.
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Arborescence (graph theory)
In graph theory, an arborescence is a directed graph in which, for a vertex u (called the root) and any other vertex v, there is exactly one directed path from u to v. An arborescence is thus the directed-graph form of a rooted tree, understood here as an undirected graph.Equivalently, an arborescence is a directed, rooted tree in which all edges point away from the root; a number of other equivalent characterizations exist. Every arborescence is a directed acyclic graph (DAG), but not every DAG is an arborescence. An arborescence can equivalently be defined as a rooted digraph in which the path from the root to any other vertex is unique. |
epfl-collab | In Ranked Retrieval, the result at position k is non-relevant and at k+1 is relevant. Which of the following is always true?Hint: P@k and R@k are the precision and recall of the result set consisting of the k top ranked documents. | ['R@k-1=R@k+1', 'P@k-1>P@k+1', 'R@k-1<R@k+1', 'P@k-1=P@k+1'] | C | null | Document 1:::
Precision and recall
In pattern recognition, information retrieval, object detection and classification (machine learning), precision and recall are performance metrics that apply to data retrieved from a collection, corpus or sample space. Precision (also called positive predictive value) is the fraction of relevant instances among the retrieved instances. Written as a formula: r e l e v a n t _ r e t r i e v e d _ i n s t a n c e s a l l _ r e t r i e v e d _ i n s t a n c e s {\displaystyle {\frac {relevant\_retrieved\_instances}{all\_{\mathbf {retrieved}}\_instances}}} . Recall (also known as sensitivity) is the fraction of relevant instances that were retrieved.
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Uncertain inference
Rather than retrieving a document that exactly matches the query we should rank the documents based on their plausibility in regards to that query. Since d and q are both generated by users, they are error prone; thus d → q {\displaystyle d\to q} is uncertain. This will affect the plausibility of a given query.
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Uncertain inference
Rather than retrieving a document that exactly matches the query we should rank the documents based on their plausibility in regards to that query. Since d and q are both generated by users, they are error prone; thus d → q {\displaystyle d\to q} is uncertain. This will affect the plausibility of a given query.
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Precision and recall
Written as a formula: r e l e v a n t _ r e t r i e v e d _ i n s t a n c e s a l l _ r e l e v a n t _ i n s t a n c e s {\displaystyle {\frac {relevant\_retrieved\_instances}{all\_{\mathbf {relevant}}\_instances}}} . Both precision and recall are therefore based on relevance.
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Average precision
Evaluation measures for an information retrieval (IR) system assess how well an index, search engine or database returns results from a collection of resources that satisfy a user's query. They are therefore fundamental to the success of information systems and digital platforms. The success of an IR system may be judged by a range of criteria including relevance, speed, user satisfaction, usability, efficiency and reliability. However, the most important factor in determining a system's effectiveness for users is the overall relevance of results retrieved in response to a query. |
epfl-collab | Suppose that for points p, q, and t in metric space, the following hold:p is density-reachable from q t is density-reachable from qp is density-reachable from tWhich of the following statements is false? | ['p and q are density-connected', 't is a core point', 'p is a border point', 'q is a core point '] | C | null | Document 1:::
Discrete metric space
Formally, a metric space is an ordered pair (M, d) where M is a set and d is a metric on M, i.e., a functionsatisfying the following axioms for all points x , y , z ∈ M {\displaystyle x,y,z\in M} :The distance from a point to itself is zero: (Positivity) The distance between two distinct points is always positive: (Symmetry) The distance from x to y is always the same as the distance from y to x: The triangle inequality holds: This is a natural property of both physical and metaphorical notions of distance: you can arrive at z from x by taking a detour through y, but this will not make your journey any faster than the shortest path.If the metric d is unambiguous, one often refers by abuse of notation to "the metric space M". By taking all axioms except the second, one can show that distance is always non-negative:Therefore the second axiom can be weakened to If x ≠ y , then d ( x , y ) ≠ 0 {\textstyle {\text{If }}x\neq y{\text{, then }}d(x,y)\neq 0} and combined with the first to make d ( x , y ) = 0 ⟺ x = y {\textstyle d(x,y)=0\iff x=y} .
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Relation of norms and metrics
In fact, these three distances, while they have distinct properties, are similar in some ways. Informally, points that are close in one are close in the others, too. This observation can be quantified with the formula which holds for every pair of points p , q ∈ R 2 {\displaystyle p,q\in \mathbb {R} ^{2}} .
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Approach space
Given a metric space (X, d), or more generally, an extended pseudoquasimetric (which will be abbreviated ∞pq-metric here), one can define an induced map d: X × P(X) → by d(x, A) = inf{d(x, a): a ∈ A}. With this example in mind, a distance on X is defined to be a map X × P(X) → satisfying for all x in X and A, B ⊆ X, d(x, {x}) = 0, d(x, Ø) = ∞, d(x, A∪B) = min(d(x, A), d(x, B)), For all 0 ≤ ε ≤ ∞, d(x, A) ≤ d(x, A(ε)) + ε,where we define A(ε) = {x: d(x, A) ≤ ε}. (The "empty infimum is positive infinity" convention is like the nullary intersection is everything convention.) An approach space is defined to be a pair (X, d) where d is a distance function on X. Every approach space has a topology, given by treating A → A(0) as a Kuratowski closure operator. The appropriate maps between approach spaces are the contractions. A map f: (X, d) → (Y, e) is a contraction if e(f(x), f) ≤ d(x, A) for all x ∈ X and A ⊆ X.
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Density point
In mathematics, Lebesgue's density theorem states that for any Lebesgue measurable set A ⊂ R n {\displaystyle A\subset \mathbb {R} ^{n}} , the "density" of A is 0 or 1 at almost every point in R n {\displaystyle \mathbb {R} ^{n}} . Additionally, the "density" of A is 1 at almost every point in A. Intuitively, this means that the "edge" of A, the set of points in A whose "neighborhood" is partially in A and partially outside of A, is negligible. Let μ be the Lebesgue measure on the Euclidean space Rn and A be a Lebesgue measurable subset of Rn. Define the approximate density of A in a ε-neighborhood of a point x in Rn as d ε ( x ) = μ ( A ∩ B ε ( x ) ) μ ( B ε ( x ) ) {\displaystyle d_{\varepsilon }(x)={\frac {\mu (A\cap B_{\varepsilon }(x))}{\mu (B_{\varepsilon }(x))}}} where Bε denotes the closed ball of radius ε centered at x. Lebesgue's density theorem asserts that for almost every point x of A the density d ( x ) = lim ε → 0 d ε ( x ) {\displaystyle d(x)=\lim _{\varepsilon \to 0}d_{\varepsilon }(x)} exists and is equal to 0 or 1.
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Closeness (topology)
Given a metric space ( X , d ) {\displaystyle (X,d)} a point p {\displaystyle p} is called close or near to a set A {\displaystyle A} if d ( p , A ) = 0 {\displaystyle d(p,A)=0} ,where the distance between a point and a set is defined as d ( p , A ) := inf a ∈ A d ( p , a ) {\displaystyle d(p,A):=\inf _{a\in A}d(p,a)} where inf stands for infimum. Similarly a set B {\displaystyle B} is called close to a set A {\displaystyle A} if d ( B , A ) = 0 {\displaystyle d(B,A)=0} where d ( B , A ) := inf b ∈ B d ( b , A ) {\displaystyle d(B,A):=\inf _{b\in B}d(b,A)} . |
epfl-collab | If for the χ2 statistics for a binary feature, we obtain P(χ2 |DF = 1) < 0.05, this means: | ['That the class label correlates with the feature', 'No conclusion can be drawn', 'That the class labels depends on the feature', 'That the class label is independent of the feature'] | C | null | Document 1:::
5 sigma
In the case where X takes random values from a finite data set x1, x2, ..., xN, with each value having the same probability, the standard deviation is or, by using summation notation, If, instead of having equal probabilities, the values have different probabilities, let x1 have probability p1, x2 have probability p2, ..., xN have probability pN. In this case, the standard deviation will be
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Probit function
If we consider the familiar fact that the standard normal distribution places 95% of probability between −1.96 and 1.96, and is symmetric around zero, it follows that Φ ( − 1.96 ) = 0.025 = 1 − Φ ( 1.96 ) . {\displaystyle \Phi (-1.96)=0.025=1-\Phi (1.96).\,\!} The probit function gives the 'inverse' computation, generating a value of a standard normal random variable, associated with specified cumulative probability. Continuing the example, probit ( 0.025 ) = − 1.96 = − probit ( 0.975 ) {\displaystyle \operatorname {probit} (0.025)=-1.96=-\operatorname {probit} (0.975)} .In general, Φ ( probit ( p ) ) = p {\displaystyle \Phi (\operatorname {probit} (p))=p} and probit ( Φ ( z ) ) = z . {\displaystyle \operatorname {probit} (\Phi (z))=z.}
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Statistical power
In statistics, the power of a binary hypothesis test is the probability that the test correctly rejects the null hypothesis ( H 0 {\displaystyle H_{0}} ) when a specific alternative hypothesis ( H 1 {\displaystyle H_{1}} ) is true. It is commonly denoted by 1 − β {\displaystyle 1-\beta } , and represents the chances of a true positive detection conditional on the actual existence of an effect to detect. Statistical power ranges from 0 to 1, and as the power of a test increases, the probability β {\displaystyle \beta } of making a type II error by wrongly failing to reject the null hypothesis decreases.
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Dichotomous thinking
In statistics, dichotomous thinking or binary thinking is the process of seeing a discontinuity in the possible values that a p-value can take during null hypothesis significance testing: it is either above the significance threshold (usually 0.05) or below. When applying dichotomous thinking, a first p-value of 0.0499 will be interpreted the same as a p-value of 0.0001 (the null hypothesis is rejected) while a second p-value of 0.0501 will be interpreted the same as a p-value of 0.7 (the null hypothesis is accepted). The fact that first and second p-values are mathematically very close is thus completely disregarded and values of p are not considered as continuous but are interpreted dichotomously with respect to the significance threshold. A common measure of dichotomous thinking is the cliff effect. A reason to avoid dichotomous thinking is that p-values and other statistics naturally change from study to study due to random variation alone; decisions about refutation or support of a scientific hypothesis based on a result from a single study are therefore not reliable.Dichotomous thinking is very often associated with p-value reading but it can also happen with other statistical tools such as interval estimates.
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Sparse Distributed Memory
The mean of the binomial distribution is n/2, and the variance is n/4. This distribution function will be denoted by N(d). The normal distribution F with mean n/2 and standard deviation n / 2 {\displaystyle {\sqrt {n}}/2} is a good approximation to it: N(d) = Pr{d(x, y) ≤ d} ≅ F{(d − n / 2)/ n / 4 {\displaystyle {\sqrt {n/4}}} } Tendency to orthogonalityAn outstanding property of N is that most of it lies at approximately the mean (indifference) distance n/2 from a point (and its complement). In other words, most of the space is nearly orthogonal to any given point, and the larger n is, the more pronounced is this effect. |
epfl-collab | Which of the following is false regarding K-means and DBSCAN? | ['K-means takes the number of clusters as parameter, while DBSCAN does not take any parameter', 'K-means does not handle outliers, while DBSCAN does', 'Both are unsupervised', 'K-means does many iterations, while DBSCAN does not'] | A | null | Document 1:::
Determining the number of clusters in a data set
Determining the number of clusters in a data set, a quantity often labelled k as in the k-means algorithm, is a frequent problem in data clustering, and is a distinct issue from the process of actually solving the clustering problem. For a certain class of clustering algorithms (in particular k-means, k-medoids and expectation–maximization algorithm), there is a parameter commonly referred to as k that specifies the number of clusters to detect. Other algorithms such as DBSCAN and OPTICS algorithm do not require the specification of this parameter; hierarchical clustering avoids the problem altogether. The correct choice of k is often ambiguous, with interpretations depending on the shape and scale of the distribution of points in a data set and the desired clustering resolution of the user.
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K-means algorithm
k-means clustering is a method of vector quantization, originally from signal processing, that aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean (cluster centers or cluster centroid), serving as a prototype of the cluster. This results in a partitioning of the data space into Voronoi cells. k-means clustering minimizes within-cluster variances (squared Euclidean distances), but not regular Euclidean distances, which would be the more difficult Weber problem: the mean optimizes squared errors, whereas only the geometric median minimizes Euclidean distances.
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K-means algorithm
They both use cluster centers to model the data; however, k-means clustering tends to find clusters of comparable spatial extent, while the Gaussian mixture model allows clusters to have different shapes. The unsupervised k-means algorithm has a loose relationship to the k-nearest neighbor classifier, a popular supervised machine learning technique for classification that is often confused with k-means due to the name. Applying the 1-nearest neighbor classifier to the cluster centers obtained by k-means classifies new data into the existing clusters. This is known as nearest centroid classifier or Rocchio algorithm.
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K-means algorithm
For instance, better Euclidean solutions can be found using k-medians and k-medoids. The problem is computationally difficult (NP-hard); however, efficient heuristic algorithms converge quickly to a local optimum. These are usually similar to the expectation-maximization algorithm for mixtures of Gaussian distributions via an iterative refinement approach employed by both k-means and Gaussian mixture modeling.
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Determining the number of clusters in a data set
In addition, increasing k without penalty will always reduce the amount of error in the resulting clustering, to the extreme case of zero error if each data point is considered its own cluster (i.e., when k equals the number of data points, n). Intuitively then, the optimal choice of k will strike a balance between maximum compression of the data using a single cluster, and maximum accuracy by assigning each data point to its own cluster. If an appropriate value of k is not apparent from prior knowledge of the properties of the data set, it must be chosen somehow. There are several categories of methods for making this decision. |
epfl-collab | Which of the following is correct regarding community detection? | ['High modularity of a community indicates a large difference between the number of edges of the community and the number of edges of a null model', 'High betweenness of an edge indicates that the communities are well connected by that edge', 'The Girvan-Newman algorithm attempts to maximize the overall betweenness measure of a community graph', 'The Louvain algorithm attempts to minimize the overall modularity measure of a community graph'] | B | null | Document 1:::
Community structure
In the study of complex networks, a network is said to have community structure if the nodes of the network can be easily grouped into (potentially overlapping) sets of nodes such that each set of nodes is densely connected internally. In the particular case of non-overlapping community finding, this implies that the network divides naturally into groups of nodes with dense connections internally and sparser connections between groups. But overlapping communities are also allowed. The more general definition is based on the principle that pairs of nodes are more likely to be connected if they are both members of the same community(ies), and less likely to be connected if they do not share communities. A related but different problem is community search, where the goal is to find a community that a certain vertex belongs to.
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Clique percolation method
The clique percolation method is a popular approach for analyzing the overlapping community structure of networks. The term network community (also called a module, cluster or cohesive group) has no widely accepted unique definition and it is usually defined as a group of nodes that are more densely connected to each other than to other nodes in the network. There are numerous alternative methods for detecting communities in networks, for example, the Girvan–Newman algorithm, hierarchical clustering and modularity maximization.
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Clique percolation method
This definition allows overlaps between the communities in a natural way, as illustrated in Fig.1, showing four k-clique communities at k = 4. The communities are color-coded and the overlap between them is emphasized in red. The definition above is also local: if a certain sub-graph fulfills the criteria to be considered as a community, then it will remain a community independent of what happens to another part of the network far away.
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Modularity (networks)
Biological networks, including animal brains, exhibit a high degree of modularity. However, modularity maximization is not statistically consistent, and finds communities in its own null model, i.e. fully random graphs, and therefore it cannot be used to find statistically significant community structures in empirical networks. Furthermore, it has been shown that modularity suffers a resolution limit and, therefore, it is unable to detect small communities.
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Clique percolation method
The clique percolation method builds up the communities from k-cliques, which correspond to complete (fully connected) sub-graphs of k nodes. (E.g., a k-clique at k = 3 is equivalent to a triangle). Two k-cliques are considered adjacent if they share k − 1 nodes. A community is defined as the maximal union of k-cliques that can be reached from each other through a series of adjacent k-cliques. |
epfl-collab | When constructing a word embedding, negative samples are: | ['Only words that never appear as context word', 'Word - context word combinations that are not occurring in the document collection', 'All less frequent words that do not occur in the context of a given word', 'Context words that are not part of the vocabulary of the document collection'] | B | null | Document 1:::
Precision and recall
For classification tasks, the terms true positives, true negatives, false positives, and false negatives (see Type I and type II errors for definitions) compare the results of the classifier under test with trusted external judgments. The terms positive and negative refer to the classifier's prediction (sometimes known as the expectation), and the terms true and false refer to whether that prediction corresponds to the external judgment (sometimes known as the observation). Let us define an experiment from P positive instances and N negative instances for some condition. The four outcomes can be formulated in a 2×2 contingency table or confusion matrix, as follows: Precision and recall are then defined as: Recall in this context is also referred to as the true positive rate or sensitivity, and precision is also referred to as positive predictive value (PPV); other related measures used in classification include true negative rate and accuracy. True negative rate is also called specificity.
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Winnow (algorithm)
During training, Winnow is shown a sequence of positive and negative examples. From these it learns a decision hyperplane that can then be used to label novel examples as positive or negative. The algorithm can also be used in the online learning setting, where the learning and the classification phase are not clearly separated.
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Negative relationship
In statistics, there is a negative relationship or inverse relationship between two variables if higher values of one variable tend to be associated with lower values of the other. A negative relationship between two variables usually implies that the correlation between them is negative, or — what is in some contexts equivalent — that the slope in a corresponding graph is negative. A negative correlation between variables is also called anticorrelation or inverse correlation. Negative correlation can be seen geometrically when two normalized random vectors are viewed as points on a sphere, and the correlation between them is the cosine of the arc of separation of the points on the sphere.
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Strong and weak sampling
Strong and weak sampling are two sampling approach in Statistics, and are popular in computational cognitive science and language learning. In strong sampling, it is assumed that the data are intentionally generated as positive examples of a concept, while in weak sampling, it is assumed that the data are generated without any restrictions.
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GloVe (machine learning)
GloVe, coined from Global Vectors, is a model for distributed word representation. The model is an unsupervised learning algorithm for obtaining vector representations for words. This is achieved by mapping words into a meaningful space where the distance between words is related to semantic similarity. Training is performed on aggregated global word-word co-occurrence statistics from a corpus, and the resulting representations showcase interesting linear substructures of the word vector space. It is developed as an open-source project at Stanford and was launched in 2014. As log-bilinear regression model for unsupervised learning of word representations, it combines the features of two model families, namely the global matrix factorization and local context window methods. |
epfl-collab | Which of the following statements about index merging (when constructing inverted files) is correct? | ['Index merging is used when the vocabulary does no longer fit into the main memory', 'The size of the final merged index file is O(nlog2(n)*M), where M is the size of the available memory', 'While merging two partial indices on disk, the inverted lists of a term are concatenated without sorting', 'While merging two partial indices on disk, the vocabularies are concatenated without sorting'] | C | null | Document 1:::
Inverted index
In computer science, an inverted index (also referred to as a postings list, postings file, or inverted file) is a database index storing a mapping from content, such as words or numbers, to its locations in a table, or in a document or a set of documents (named in contrast to a forward index, which maps from documents to content). The purpose of an inverted index is to allow fast full-text searches, at a cost of increased processing when a document is added to the database. The inverted file may be the database file itself, rather than its index. It is the most popular data structure used in document retrieval systems, used on a large scale for example in search engines.
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Inverted index
Additionally, several significant general-purpose mainframe-based database management systems have used inverted list architectures, including ADABAS, DATACOM/DB, and Model 204. There are two main variants of inverted indexes: A record-level inverted index (or inverted file index or just inverted file) contains a list of references to documents for each word. A word-level inverted index (or full inverted index or inverted list) additionally contains the positions of each word within a document. The latter form offers more functionality (like phrase searches), but needs more processing power and space to be created.
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External Sorting
External merge sort typically uses a hybrid sort-merge strategy. In the sorting phase, chunks of data small enough to fit in main memory are read, sorted, and written out to a temporary file. In the merge phase, the sorted subfiles are combined into a single larger file.
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Loop inversion
In computer science, loop inversion is a compiler optimization and loop transformation in which a while loop is replaced by an if block containing a do..while loop. When used correctly, it may improve performance due to instruction pipelining.
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Block merge sort
Block sort, or block merge sort, is a sorting algorithm combining at least two merge operations with an insertion sort to arrive at O(n log n) in-place stable sorting. It gets its name from the observation that merging two sorted lists, A and B, is equivalent to breaking A into evenly sized blocks, inserting each A block into B under special rules, and merging AB pairs. One practical algorithm for O(log n) in place merging was proposed by Pok-Son Kim and Arne Kutzner in 2008. |
epfl-collab | For his awesome research, Tugrulcan is going to use the PageRank with teleportation and HITS algorithm, not on a network of webpages but on the retweet network of Twitter! The retweet network is a directed graph, where nodes are users and an edge going out from a user A and to a user B means that "User A retweeted User B". Which one is false about a Twitter bot that retweeted other users frequently but got never retweeted by other users or by itself? | ['Its authority value will be equal to the hub value of a user who never retweets other users', 'It will have a non-zero hub value', 'It will have an authority value of zero', 'It will have a PageRank of zero'] | D | null | Document 1:::
PageRank
PageRank (PR) is an algorithm used by Google Search to rank web pages in their search engine results. It is named after both the term "web page" and co-founder Larry Page. PageRank is a way of measuring the importance of website pages. According to Google: PageRank works by counting the number and quality of links to a page to determine a rough estimate of how important the website is.
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Social graph
The social graph is a graph that represents social relations between entities. In short, it is a model or representation of a social network, where the word graph has been taken from graph theory. The social graph has been referred to as "the global mapping of everybody and how they're related".The term was used as early as 1964, albeit in the context of isoglosses. Leo Apostel uses the term in the context here in 1978.
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Web graph
The webgraph describes the directed links between pages of the World Wide Web. A graph, in general, consists of several vertices, some pairs connected by edges. In a directed graph, edges are directed lines or arcs. The webgraph is a directed graph, whose vertices correspond to the pages of the WWW, and a directed edge connects page X to page Y if there exists a hyperlink on page X, referring to page Y.
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Graph Search
In a presentation headed by Facebook CEO Mark Zuckerberg, it was announced that the Graph Search algorithm finds information from within a user's network of friends. Microsoft's Bing search engine provided additional results.
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Graph API
The social graph is a graph that represents social relations between entities. In short, it is a model or representation of a social network, where the word graph has been taken from graph theory. The social graph has been referred to as "the global mapping of everybody and how they're related".The term was used as early as 1964, albeit in the context of isoglosses. Leo Apostel uses the term in the context here in 1978. |
epfl-collab | Let $f_{\mathrm{MLP}}: \mathbb{R}^{d} \rightarrow \mathbb{R}$ be an $L$-hidden layer multi-layer perceptron (MLP) such that $$ f_{\mathrm{MLP}}(\mathbf{x})=\mathbf{w}^{\top} \sigma\left(\mathbf{W}_{L} \sigma\left(\mathbf{W}_{L-1} \ldots \sigma\left(\mathbf{W}_{1} \mathbf{x}\right)\right)\right) $$ with $\mathbf{w} \in \mathbb{R}^{M}, \mathbf{W}_{1} \in \mathbb{R}^{M \times d}$ and $\mathbf{W}_{\ell} \in \mathbb{R}^{M \times M}$ for $\ell=2, \ldots, L$, and $\sigma$ is an entry-wise activation function. Also, let $f_{\mathrm{CNN}}: \mathbb{R}^{d} \rightarrow \mathbb{R}$ be an $L^{\prime}$-hidden layer convolutional neural network (CNN) such that $$ f_{\mathrm{CNN}}(\mathbf{x})=\mathbf{w}^{\top} \sigma\left(\mathbf{w}_{L^{\prime}} \star \sigma\left(\mathbf{w}_{L^{\prime}-1} \star \ldots \sigma\left(\mathbf{w}_{1} \star \mathbf{x}\right)\right)\right) $$ with $\mathbf{w} \in \mathbb{R}^{d}, \mathbf{w}_{\ell} \in \mathbb{R}^{K}$ for $\ell=1, \ldots, L^{\prime}$ and $\star$ denoting the one-dimensional convolution operator with zero-padding, i.e., output of the convolution has the same dimensionality as the input. Let's assume $\sigma$ is a tanh activation function. Thus, by flipping the signs of all of the weights leading in and out of a hidden neuron, the input-output mapping function represented by the network is unchanged. Besides, interchanging the values of all of the weights (i.e., by permuting the ordering of the hidden neurons within the layer) also leaves the network input-output mapping function unchanged. Suppose that, given the training data, SGD can find a solution with zero training loss, and the (absolute value) weights of such solution are non-zero and all unique. Choose the largest lower bound on the number of solutions (with zero training loss) achievable by $f_{\mathrm{MLP}}$ with $L=1$ and $M$ hidden units on this dataset. | ['$M !$', '$2^M$', '$1$', '$M! 2^M$'] | D | null | Document 1:::
Multilayer perceptron
A multilayer perceptron (MLP) is a misnomer for a modern feedforward artificial neural network, consisting of fully connected neurons with a nonlinear kind of activation function, organized in at least three layers, notable for being able to distinguish data that is not linearly separable. It is a misnomer because the original perceptron used a Heaviside step function, instead of a nonlinear kind of activation function (used by modern networks). Modern feedforward networks are trained using the backpropagation method and are colloquially referred to as the "vanilla" neural networks.
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Tensor (machine learning)
Tensors may also be used to compute the layers of a fully connected neural network, where the tensor is applied to the entire layer instead of individual unit values. The output value of single layer unit y m {\displaystyle y_{m}} is the sum-product of its input units and the connection weights filtered through the activation function f {\displaystyle f}: y m = f ( ∑ n x n u m , n ) . {\displaystyle y_{m}=f\left(\sum _{n}x_{n}u_{m,n}\right).} The vectors x {\displaystyle x} and y {\displaystyle y} of output values can be expressed as a mode-1 tensors, while the hidden weights can be expressed as a mode-2 tensor.
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Tensor (machine learning)
Y = A , {\displaystyle {\mathcal {Y}}={\mathcal {A}},} where A , B {\displaystyle {\mathcal {A}},{\mathcal {B}}} and C {\displaystyle {\mathcal {C}}} are the inverse transform, data and kernel. The derivation is more complex when the filtering kernel also includes a non-linear activation function such as sigmoid or ReLU. The hidden weights of the convolution layer are the parameters to the filter. These can be reduced with a pooling layer which reduces the resolution (size) of the data, and can also be expressed as a tensor operation.
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Tensor (machine learning)
Tensors may be used as the unit values of neural networks which extend the concept of scalar, vector and matrix values to multiple dimensions. The output value of single layer unit y m {\displaystyle y_{m}} is the sum-product of its input units and the connection weights filtered through the activation function f {\displaystyle f}: y m = f ( ∑ n x n u m , n ) , {\displaystyle y_{m}=f\left(\sum _{n}x_{n}u_{m,n}\right),} where y m ∈ R . {\displaystyle y_{m}\in \mathbb {R} .} If each output element of y m {\displaystyle y_{m}} is a scalar, then we have the classical definition of an artificial neural network.
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Learning with errors
There exists a certain unknown linear function f: Z q n → Z q {\displaystyle f:\mathbb {Z} _{q}^{n}\rightarrow \mathbb {Z} _{q}} , and the input to the LWE problem is a sample of pairs ( x , y ) {\displaystyle (\mathbf {x} ,y)} , where x ∈ Z q n {\displaystyle \mathbf {x} \in \mathbb {Z} _{q}^{n}} and y ∈ Z q {\displaystyle y\in \mathbb {Z} _{q}} , so that with high probability y = f ( x ) {\displaystyle y=f(\mathbf {x} )} . Furthermore, the deviation from the equality is according to some known noise model. The problem calls for finding the function f {\displaystyle f} , or some close approximation thereof, with high probability. |
epfl-collab | Consider a linear regression problem with $N$ samples $\left\{\left(\boldsymbol{x}_{n}, y_{n}\right)\right\}_{n=1}^{N}$, where each input $\boldsymbol{x}_{n}$ is a $D$-dimensional vector $\{-1,+1\}^{D}$, and all output values are $y_{i} \in \mathbb{R}$. Which of the following statements is correct? | ['A linear regressor works very well if the data is linearly separable.', 'None of the above.', 'Linear regression always "works" very well for $N \\ll D$', 'Linear regression always "works" very well for $D \\ll N$'] | B | null | Document 1:::
Constrained least squares
Stochastic (linearly) constrained least squares: the elements of β {\displaystyle {\boldsymbol {\beta }}} must satisfy L β = d + ν {\displaystyle \mathbf {L} {\boldsymbol {\beta }}=\mathbf {d} +\mathbf {\nu } } , where ν {\displaystyle \mathbf {\nu } } is a vector of random variables such that E ( ν ) = 0 {\displaystyle \operatorname {E} (\mathbf {\nu } )=\mathbf {0} } and E ( ν ν T ) = τ 2 I {\displaystyle \operatorname {E} (\mathbf {\nu } \mathbf {\nu } ^{\rm {T}})=\tau ^{2}\mathbf {I} } . This effectively imposes a prior distribution for β {\displaystyle {\boldsymbol {\beta }}} and is therefore equivalent to Bayesian linear regression. Regularized least squares: the elements of β {\displaystyle {\boldsymbol {\beta }}} must satisfy ‖ L β − y ‖ ≤ α {\displaystyle \|\mathbf {L} {\boldsymbol {\beta }}-\mathbf {y} \|\leq \alpha } (choosing α {\displaystyle \alpha } in proportion to the noise standard deviation of y prevents over-fitting).
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Linear Regression
Most commonly, the conditional mean of the response given the values of the explanatory variables (or predictors) is assumed to be an affine function of those values; less commonly, the conditional median or some other quantile is used. Like all forms of regression analysis, linear regression focuses on the conditional probability distribution of the response given the values of the predictors, rather than on the joint probability distribution of all of these variables, which is the domain of multivariate analysis. Linear regression was the first type of regression analysis to be studied rigorously, and to be used extensively in practical applications.
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Mean absolute percentage error
Consider a standard regression setting in which the data are fully described by a random pair Z = ( X , Y ) {\displaystyle Z=(X,Y)} with values in R d × R {\displaystyle \mathbb {R} ^{d}\times \mathbb {R} } , and n i.i.d. copies ( X 1 , Y 1 ) , . . .
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Sufficient dimension reduction
In statistics, sufficient dimension reduction (SDR) is a paradigm for analyzing data that combines the ideas of dimension reduction with the concept of sufficiency. Dimension reduction has long been a primary goal of regression analysis. Given a response variable y and a p-dimensional predictor vector x {\displaystyle {\textbf {x}}} , regression analysis aims to study the distribution of y ∣ x {\displaystyle y\mid {\textbf {x}}} , the conditional distribution of y {\displaystyle y} given x {\displaystyle {\textbf {x}}} . A dimension reduction is a function R ( x ) {\displaystyle R({\textbf {x}})} that maps x {\displaystyle {\textbf {x}}} to a subset of R k {\displaystyle \mathbb {R} ^{k}} , k < p, thereby reducing the dimension of x {\displaystyle {\textbf {x}}} .
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Mean square deviation
If a vector of n {\displaystyle n} predictions is generated from a sample of n {\displaystyle n} data points on all variables, and Y {\displaystyle Y} is the vector of observed values of the variable being predicted, with Y ^ {\displaystyle {\hat {Y}}} being the predicted values (e.g. as from a least-squares fit), then the within-sample MSE of the predictor is computed as MSE = 1 n ∑ i = 1 n ( Y i − Y i ^ ) 2 . {\displaystyle \operatorname {MSE} ={\frac {1}{n}}\sum _{i=1}^{n}\left(Y_{i}-{\hat {Y_{i}}}\right)^{2}.} In other words, the MSE is the mean ( 1 n ∑ i = 1 n ) {\textstyle \left({\frac {1}{n}}\sum _{i=1}^{n}\right)} of the squares of the errors ( Y i − Y i ^ ) 2 {\textstyle \left(Y_{i}-{\hat {Y_{i}}}\right)^{2}} . This is an easily computable quantity for a particular sample (and hence is sample-dependent). |
epfl-collab | Let $\mathcal{R}_{p}(f, \varepsilon)$ be the $\ell_{p}$ adversarial risk of a classifier $f: \mathbb{R}^{d} \rightarrow\{ \pm 1\}$, i.e., $$ \mathcal{R}_{p}(f, \varepsilon)=\mathbb{E}_{(\mathbf{x}, y) \sim \mathcal{D}}\left[\max _{\tilde{\mathbf{x}}:\|\mathbf{x}-\tilde{\mathbf{x}}\|_{p} \leq \varepsilon} \mathbb{1}_{\{f(\tilde{\mathbf{x}}) \neq y\}}\right], $$ for $p=1,2, \infty$. Which of the following relationships between the adversarial risks is true? | ['$\\mathcal{R}_{\\infty}(f, \\varepsilon) \\leq \\mathcal{R}_{2}(f, \\varepsilon / d)$', '$\\mathcal{R}_{\\infty}(f, \\varepsilon) \\leq \\mathcal{R}_{2}(f, \\sqrt{d} \\varepsilon)$', '$\\mathcal{R}_{\\infty}(f, \\varepsilon) \\leq \\mathcal{R}_{1}(f, \\varepsilon)$', '$\\mathcal{R}_{2}(f, \\varepsilon) \\leq \\mathcal{R}_{1}(f, 2 \\varepsilon)$'] | B | null | Document 1:::
Loss functions for classification
In the case of binary classification, it is possible to simplify the calculation of expected risk from the integral specified above. Specifically, I = ∫ X × Y V ( f ( x → ) , y ) p ( x → , y ) d x → d y = ∫ X ∫ Y ϕ ( y f ( x → ) ) p ( y ∣ x → ) p ( x → ) d y d x → = ∫ X p ( x → ) d x → = ∫ X p ( x → ) d x → {\displaystyle {\begin{aligned}I&=\int _{{\mathcal {X}}\times {\mathcal {Y}}}V(f({\vec {x}}),y)\,p({\vec {x}},y)\,d{\vec {x}}\,dy\\&=\int _{\mathcal {X}}\int _{\mathcal {Y}}\phi (yf({\vec {x}}))\,p(y\mid {\vec {x}})\,p({\vec {x}})\,dy\,d{\vec {x}}\\&=\int _{\mathcal {X}}\,p({\vec {x}})\,d{\vec {x}}\\&=\int _{\mathcal {X}}\,p({\vec {x}})\,d{\vec {x}}\end{aligned}}} The second equality follows from the properties described above. The third equality follows from the fact that 1 and −1 are the only possible values for y {\displaystyle y} , and the fourth because p ( − 1 ∣ x ) = 1 − p ( 1 ∣ x ) {\displaystyle p(-1\mid x)=1-p(1\mid x)} .
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Sample-complexity bounds
Fix a loss function L: Y × Y → R ≥ 0 {\displaystyle {\mathcal {L}}\colon Y\times Y\to \mathbb {R} _{\geq 0}} , for example, the square loss L ( y , y ′ ) = ( y − y ′ ) 2 {\displaystyle {\mathcal {L}}(y,y')=(y-y')^{2}} , where h ( x ) = y ′ {\displaystyle h(x)=y'} . For a given distribution ρ {\displaystyle \rho } on X × Y {\displaystyle X\times Y} , the expected risk of a hypothesis (a function) h ∈ H {\displaystyle h\in {\mathcal {H}}} is E ( h ) := E ρ = ∫ X × Y L ( h ( x ) , y ) d ρ ( x , y ) {\displaystyle {\mathcal {E}}(h):=\mathbb {E} _{\rho }=\int _{X\times Y}{\mathcal {L}}(h(x),y)\,d\rho (x,y)} In our setting, we have h = A ( S n ) {\displaystyle h={\mathcal {A}}(S_{n})} , where A {\displaystyle {\mathcal {A}}} is a learning algorithm and S n = ( ( x 1 , y 1 ) , … , ( x n , y n ) ) ∼ ρ n {\displaystyle S_{n}=((x_{1},y_{1}),\ldots ,(x_{n},y_{n}))\sim \rho ^{n}} is a sequence of vectors which are all drawn independently from ρ {\displaystyle \rho } . Define the optimal riskSet h n = A ( S n ) {\displaystyle h_{n}={\mathcal {A}}(S_{n})} , for each n {\displaystyle n} .
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Sample complexity
Fix a loss function L: Y × Y → R ≥ 0 {\displaystyle {\mathcal {L}}\colon Y\times Y\to \mathbb {R} _{\geq 0}} , for example, the square loss L ( y , y ′ ) = ( y − y ′ ) 2 {\displaystyle {\mathcal {L}}(y,y')=(y-y')^{2}} , where h ( x ) = y ′ {\displaystyle h(x)=y'} . For a given distribution ρ {\displaystyle \rho } on X × Y {\displaystyle X\times Y} , the expected risk of a hypothesis (a function) h ∈ H {\displaystyle h\in {\mathcal {H}}} is E ( h ) := E ρ = ∫ X × Y L ( h ( x ) , y ) d ρ ( x , y ) {\displaystyle {\mathcal {E}}(h):=\mathbb {E} _{\rho }=\int _{X\times Y}{\mathcal {L}}(h(x),y)\,d\rho (x,y)} In our setting, we have h = A ( S n ) {\displaystyle h={\mathcal {A}}(S_{n})} , where A {\displaystyle {\mathcal {A}}} is a learning algorithm and S n = ( ( x 1 , y 1 ) , … , ( x n , y n ) ) ∼ ρ n {\displaystyle S_{n}=((x_{1},y_{1}),\ldots ,(x_{n},y_{n}))\sim \rho ^{n}} is a sequence of vectors which are all drawn independently from ρ {\displaystyle \rho } . Define the optimal riskSet h n = A ( S n ) {\displaystyle h_{n}={\mathcal {A}}(S_{n})} , for each n {\displaystyle n} .
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Loss functions for classification
Some of these surrogates are described below. In practice, the probability distribution p ( x → , y ) {\displaystyle p({\vec {x}},y)} is unknown. Consequently, utilizing a training set of n {\displaystyle n} independently and identically distributed sample points S = { ( x → 1 , y 1 ) , … , ( x → n , y n ) } {\displaystyle S=\{({\vec {x}}_{1},y_{1}),\dots ,({\vec {x}}_{n},y_{n})\}} drawn from the data sample space, one seeks to minimize empirical risk I S = 1 n ∑ i = 1 n V ( f ( x → i ) , y i ) {\displaystyle I_{S}={\frac {1}{n}}\sum _{i=1}^{n}V(f({\vec {x}}_{i}),y_{i})} as a proxy for expected risk. (See statistical learning theory for a more detailed description.)
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Bayes classifier
The probability of misclassification, or risk, of a classifier C is defined as R ( C ) = P { C ( X ) ≠ Y } . {\displaystyle {\mathcal {R}}(C)=\operatorname {P} \{C(X)\neq Y\}.} The Bayes classifier is C Bayes ( x ) = argmax r ∈ { 1 , 2 , … , K } P ( Y = r ∣ X = x ) . |
epfl-collab | We are given a data set $S=\left\{\left(\boldsymbol{x}_{n}, y_{n}\right)\right\}$ for a binary classification task where $\boldsymbol{x}_{n}$ in $\mathbb{R}^{D}$. We want to use a nearestneighbor classifier. In which of the following situations do we have a reasonable chance of success with this approach? [Ignore the issue of complexity.] | ['$ n=D^2, D \\rightarrow \\infty$', '$ n$ is fixed, $D \\rightarrow \\infty$', '$ n \\rightarrow \\infty, D \\ll \\ln (n)$', '$n \\rightarrow \\infty, D$ is fixed'] | D | null | Document 1:::
Nearest centroid classifier
In machine learning, a nearest centroid classifier or nearest prototype classifier is a classification model that assigns to observations the label of the class of training samples whose mean (centroid) is closest to the observation. When applied to text classification using word vectors containing tf*idf weights to represent documents, the nearest centroid classifier is known as the Rocchio classifier because of its similarity to the Rocchio algorithm for relevance feedback.An extended version of the nearest centroid classifier has found applications in the medical domain, specifically classification of tumors.
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Nearest neighbour function
{\displaystyle D_{o}(r)=1-P({N}(b(o,r))=1\mid o).} where P ( N ( b ( o , r ) ) = 1 ∣ o ) {\displaystyle \textstyle P({N}(b(o,r))=1\mid o)} denotes the conditional probability that there is one point of N {\displaystyle \textstyle {N}} located in b ( o , r ) {\displaystyle \textstyle b(o,r)} given there is a point of N {\displaystyle \textstyle {N}} located at o {\displaystyle \textstyle o} . The reference point need not be at the origin, and can be located at an arbitrary point x ∈ R d {\displaystyle \textstyle x\in {\textbf {R}}^{d}} . Given a point of N {\displaystyle \textstyle {N}} existing at x {\displaystyle \textstyle x} , then the nearest neighbor function, is defined as: D x ( r ) = 1 − P ( N ( b ( x , r ) ) = 1 ∣ x ) . {\displaystyle D_{x}(r)=1-P({N}(b(x,r))=1\mid x).}
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Nearest neighbor distance ratio
Nearest neighbor search (NNS), as a form of proximity search, is the optimization problem of finding the point in a given set that is closest (or most similar) to a given point. Closeness is typically expressed in terms of a dissimilarity function: the less similar the objects, the larger the function values. Formally, the nearest-neighbor (NN) search problem is defined as follows: given a set S of points in a space M and a query point q ∈ M, find the closest point in S to q. Donald Knuth in vol. 3 of The Art of Computer Programming (1973) called it the post-office problem, referring to an application of assigning to a residence the nearest post office.
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Nearest neighbour function
The nearest neighbor function, as opposed to the spherical contact distribution function, is defined in relation to some point of a point process already existing in some region of space. More precisely, for some point in the point process N {\displaystyle \textstyle {N}} , the nearest neighbor function is the probability distribution of the distance from that point to the nearest or closest neighboring point. To define this function for a point located in R d {\displaystyle \textstyle {\textbf {R}}^{d}} at, for example, the origin o {\displaystyle \textstyle o} , the d {\displaystyle \textstyle d} -dimensional ball b ( o , r ) {\displaystyle \textstyle b(o,r)} of radius r {\displaystyle \textstyle r} centered at the origin o is considered. Given a point of N {\displaystyle \textstyle {N}} existing at o {\displaystyle \textstyle o} , then the nearest neighbor function is defined as: D o ( r ) = 1 − P ( N ( b ( o , r ) ) = 1 ∣ o ) .
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K-nearest neighbors algorithm
In statistics, the k-nearest neighbors algorithm (k-NN) is a non-parametric supervised learning method first developed by Evelyn Fix and Joseph Hodges in 1951, and later expanded by Thomas Cover. It is used for classification and regression. In both cases, the input consists of the k closest training examples in a data set. The output depends on whether k-NN is used for classification or regression: In k-NN classification, the output is a class membership. |
epfl-collab | How does the bias-variance decomposition of a ridge regression estimator compare with that of the ordinary least-squares estimator in general? | ['Ridge has a smaller bias, and smaller variance.', 'Ridge has a larger bias, and larger variance.', 'Ridge has a larger bias, and smaller variance.', 'Ridge has a smaller bias, and larger variance.'] | C | null | Document 1:::
L2 regularization
It is particularly useful to mitigate the problem of multicollinearity in linear regression, which commonly occurs in models with large numbers of parameters. In general, the method provides improved efficiency in parameter estimation problems in exchange for a tolerable amount of bias (see bias–variance tradeoff).The theory was first introduced by Hoerl and Kennard in 1970 in their Technometrics papers “RIDGE regressions: biased estimation of nonorthogonal problems” and “RIDGE regressions: applications in nonorthogonal problems”. This was the result of ten years of research into the field of ridge analysis.Ridge regression was developed as a possible solution to the imprecision of least square estimators when linear regression models have some multicollinear (highly correlated) independent variables—by creating a ridge regression estimator (RR). This provides a more precise ridge parameters estimate, as its variance and mean square estimator are often smaller than the least square estimators previously derived.
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L2 regularization
Ridge regression is a method of estimating the coefficients of multiple-regression models in scenarios where the independent variables are highly correlated. It has been used in many fields including econometrics, chemistry, and engineering. Also known as Tikhonov regularization, named for Andrey Tikhonov, it is a method of regularization of ill-posed problems.
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Linear Regression
After developing such a model, if additional values of the explanatory variables are collected without an accompanying response value, the fitted model can be used to make a prediction of the response. If the goal is to explain variation in the response variable that can be attributed to variation in the explanatory variables, linear regression analysis can be applied to quantify the strength of the relationship between the response and the explanatory variables, and in particular to determine whether some explanatory variables may have no linear relationship with the response at all, or to identify which subsets of explanatory variables may contain redundant information about the response.Linear regression models are often fitted using the least squares approach, but they may also be fitted in other ways, such as by minimizing the "lack of fit" in some other norm (as with least absolute deviations regression), or by minimizing a penalized version of the least squares cost function as in ridge regression (L2-norm penalty) and lasso (L1-norm penalty). Conversely, the least squares approach can be used to fit models that are not linear models. Thus, although the terms "least squares" and "linear model" are closely linked, they are not synonymous.
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Unequal variance
The existence of heteroscedasticity is a major concern in regression analysis and the analysis of variance, as it invalidates statistical tests of significance that assume that the modelling errors all have the same variance. While the ordinary least squares estimator is still unbiased in the presence of heteroscedasticity, it is inefficient and inference based on the assumption of homoskedasticity is misleading. In that case, generalized least squares (GLS) was frequently used in the past. Nowadays, standard practice in econometrics is to include Heteroskedasticity-consistent standard errors instead of using GLS, as GLS can exhibit strong bias in small samples if the actual Skedastic function is unknown.Because heteroscedasticity concerns expectations of the second moment of the errors, its presence is referred to as misspecification of the second order.The econometrician Robert Engle was awarded the 2003 Nobel Memorial Prize for Economics for his studies on regression analysis in the presence of heteroscedasticity, which led to his formulation of the autoregressive conditional heteroscedasticity (ARCH) modeling technique.
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Best linear unbiased prediction
However, the equations for the "fixed" effects and for the random effects are different. In practice, it is often the case that the parameters associated with the random effect(s) term(s) are unknown; these parameters are the variances of the random effects and residuals. Typically the parameters are estimated and plugged into the predictor, leading to the empirical best linear unbiased predictor (EBLUP). Notice that by simply plugging in the estimated parameter into the predictor, additional variability is unaccounted for, leading to overly optimistic prediction variances for the EBLUP.Best linear unbiased predictions are similar to empirical Bayes estimates of random effects in linear mixed models, except that in the latter case, where weights depend on unknown values of components of variance, these unknown variances are replaced by sample-based estimates. |
epfl-collab | You are given two distributions over $\mathbb{R}$ : Uniform on the interval $[a, b]$ and Gaussian with mean $\mu$ and variance $\sigma^{2}$. Their respective probability density functions are $$ p_{\mathcal{U}}(y \mid a, b):=\left\{\begin{array}{ll} \frac{1}{b-a}, & \text { for } a \leq y \leq b, \\ 0 & \text { otherwise } \end{array} \quad p_{\mathcal{G}}\left(y \mid \mu, \sigma^{2}\right):=\frac{1}{\sqrt{2 \pi \sigma^{2}}} \exp \left(-\frac{(y-\mu)^{2}}{2 \sigma^{2}}\right)\right. $$ Which one(s) belong to the exponential family? | ['Both of them.', 'Only Gaussian.', 'None of them.', 'Only Uniform.'] | B | null | Document 1:::
Natural parameters
In probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculate expectations, covariances using differentiation based on some useful algebraic properties, as well as for generality, as exponential families are in a sense very natural sets of distributions to consider. The term exponential class is sometimes used in place of "exponential family", or the older term Koopman–Darmois family. The terms "distribution" and "family" are often used loosely: specifically, an exponential family is a set of distributions, where the specific distribution varies with the parameter; however, a parametric family of distributions is often referred to as "a distribution" (like "the normal distribution", meaning "the family of normal distributions"), and the set of all exponential families is sometimes loosely referred to as "the" exponential family.
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Uniform probability distribution
Therefore, the distribution is often abbreviated U ( a , b ) , {\displaystyle U(a,b),} where U {\displaystyle U} stands for uniform distribution. The difference between the bounds defines the interval length; all intervals of the same length on the distribution's support are equally probable. It is the maximum entropy probability distribution for a random variable X {\displaystyle X} under no constraint other than that it is contained in the distribution's support.
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Rectangular distribution
The probability density function of the continuous uniform distribution is: f ( x ) = { 1 b − a for a ≤ x ≤ b , 0 for x < a or x > b . {\displaystyle f(x)={\begin{cases}{\frac {1}{b-a}}&{\text{for }}a\leq x\leq b,\\0&{\text{for }}x b.\end{cases}}} The values of f ( x ) {\displaystyle f(x)} at the two boundaries a {\displaystyle a} and b {\displaystyle b} are usually unimportant, because they do not alter the value of ∫ c d f ( x ) d x {\textstyle \int _{c}^{d}f(x)dx} over any interval , {\displaystyle ,} nor of ∫ a b x f ( x ) d x , {\textstyle \int _{a}^{b}xf(x)dx,} nor of any higher moment. Sometimes they are chosen to be zero, and sometimes chosen to be 1 b − a . {\displaystyle {\tfrac {1}{b-a}}.}
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Normally distributed
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f ( x ) = 1 σ 2 π e − 1 2 ( x − μ σ ) 2 {\displaystyle f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}} The parameter μ {\displaystyle \mu } is the mean or expectation of the distribution (and also its median and mode), while the parameter σ {\displaystyle \sigma } is its standard deviation. The variance of the distribution is σ 2 {\displaystyle \sigma ^{2}} . A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate.
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Gamma distribution
In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution. There are two equivalent parameterizations in common use: With a shape parameter k {\displaystyle k} and a scale parameter θ {\displaystyle \theta } . With a shape parameter α = k {\displaystyle \alpha =k} and an inverse scale parameter β = 1 / θ {\displaystyle \beta =1/\theta } , called a rate parameter.In each of these forms, both parameters are positive real numbers. The gamma distribution is the maximum entropy probability distribution (both with respect to a uniform base measure and a 1 / x {\displaystyle 1/x} base measure) for a random variable X {\displaystyle X} for which E = kθ = α/β is fixed and greater than zero, and E = ψ(k) + ln(θ) = ψ(α) − ln(β) is fixed (ψ is the digamma function). |
epfl-collab | Church booleans are a representation of booleans in the lambda calculus. The Church encoding of true and false are functions of two parameters: Church encoding of tru: t => f => t Church encoding of fls: t => f => f What should replace ??? so that the following function computes not(b and c)? b => c => b ??? (not b) | ['(not b)', 'fls', '(not c)', 'tru'] | C | null | Document 1:::
Simply typed λ-calculus
The simply typed lambda calculus ( λ → {\displaystyle \lambda ^{\to }} ), a form of type theory, is a typed interpretation of the lambda calculus with only one type constructor ( → {\displaystyle \to } ) that builds function types. It is the canonical and simplest example of a typed lambda calculus. The simply typed lambda calculus was originally introduced by Alonzo Church in 1940 as an attempt to avoid paradoxical use of the untyped lambda calculus.The term simple type is also used to refer extensions of the simply typed lambda calculus such as products, coproducts or natural numbers (System T) or even full recursion (like PCF). In contrast, systems which introduce polymorphic types (like System F) or dependent types (like the Logical Framework) are not considered simply typed. The simple types, except for full recursion, are still considered simple because the Church encodings of such structures can be done using only → {\displaystyle \to } and suitable type variables, while polymorphism and dependency cannot.
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Boolean-valued function
A Boolean-valued function (sometimes called a predicate or a proposition) is a function of the type f: X → B, where X is an arbitrary set and where B is a Boolean domain, i.e. a generic two-element set, (for example B = {0, 1}), whose elements are interpreted as logical values, for example, 0 = false and 1 = true, i.e., a single bit of information. In the formal sciences, mathematics, mathematical logic, statistics, and their applied disciplines, a Boolean-valued function may also be referred to as a characteristic function, indicator function, predicate, or proposition. In all of these uses, it is understood that the various terms refer to a mathematical object and not the corresponding semiotic sign or syntactic expression. In formal semantic theories of truth, a truth predicate is a predicate on the sentences of a formal language, interpreted for logic, that formalizes the intuitive concept that is normally expressed by saying that a sentence is true. A truth predicate may have additional domains beyond the formal language domain, if that is what is required to determine a final truth value.
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Boolean data type
In computer science, the Boolean (sometimes shortened to Bool) is a data type that has one of two possible values (usually denoted true and false) which is intended to represent the two truth values of logic and Boolean algebra. It is named after George Boole, who first defined an algebraic system of logic in the mid 19th century. The Boolean data type is primarily associated with conditional statements, which allow different actions by changing control flow depending on whether a programmer-specified Boolean condition evaluates to true or false. It is a special case of a more general logical data type—logic does not always need to be Boolean (see probabilistic logic).
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K combinator
Moreover, in lambda calculus, notions such as '3' and ' ∗ {\displaystyle *} ' can be represented without any need for externally defined primitive operators or constants. It is possible to identify terms in lambda calculus, which, when suitably interpreted, behave like the number 3 and like the multiplication operator, q.v. Church encoding.
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Truth table
For a relation to be a function, the special requirement is that each element of the domain of the function must be mapped to one and only one member of the codomain. Thus, the function f itself can be listed as: f = {((0, 0), f0), ((0, 1), f1), ((1, 0), f2), ((1, 1), f3)}, where f0, f1, f2, and f3 are each Boolean, 0 or 1, values as members of the codomain {0, 1}, as the outputs corresponding to the member of the domain, respectively. |
epfl-collab | To which expression is the following for-loop translated? for x <- xs if x > 5; y <- ys yield x + y | ['xs.flatMap(x => ys.map(y => x + y)).withFilter(x => x > 5)', 'xs.withFilter(x => x > 5).flatMap(x => ys.map(y => x + y))', 'xs.withFilter(x => x > 5).map(x => ys.flatMap(y => x + y))', 'xs.map(x => ys.flatMap(y => x + y)).withFilter(x => x > 5)'] | B | null | Document 1:::
Iterative for loop
In computer science a for-loop or for loop is a control flow statement for specifying iteration. Specifically, a for loop functions by running a section of code repeatedly until a certain condition has been satisfied. For-loops have two parts: a header and a body. The header defines the iteration and the body is the code that is executed once per iteration.
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Iterative for loop
For-loops can be thought of as shorthands for while-loops which increment and test a loop variable. Various keywords are used to indicate the usage of a for loop: descendants of ALGOL use "for", while descendants of Fortran use "do". There are other possibilities, for example COBOL which uses PERFORM VARYING.
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Iterative for loop
It is the direct translation of the earlier German für and was used in Superplan (1949–1951) by Heinz Rutishauser. Rutishauser was involved in defining ALGOL 58 and ALGOL 60. The loop body is executed "for" the given values of the loop variable. This is more explicit in ALGOL versions of the for statement where a list of possible values and increments can be specified. In Fortran and PL/I, the keyword DO is used for the same thing and it is called a do-loop; this is different from a do-while loop.
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Predicative programming
Loop proofs are greatly simplified. For example, if x {\displaystyle x} is an integer variable, to prove that while x {\displaystyle x} >0 do x {\displaystyle x} := x {\displaystyle x} –1 od refines, or implements the specification x {\displaystyle x} ≥0 ⇒ x ′ {\displaystyle x'} =0, prove if x {\displaystyle x} >0 then x {\displaystyle x} := x {\displaystyle x} –1; ( x {\displaystyle x} ≥0 ⇒ x ′ {\displaystyle x'} =0) else o k {\displaystyle ok} fi ⇒ ( x {\displaystyle x} ≥0 ⇒ x ′ {\displaystyle x'} =0) where o k {\displaystyle ok} = ( x ′ {\displaystyle x'} = x {\displaystyle x} ) is the empty, or do-nothing command.
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Iterative for loop
The header often declares an explicit loop counter or loop variable. This allows the body to know which iteration is being executed. For-loops are typically used when the number of iterations is known before entering the loop. |
epfl-collab | Why is natural language processing difficult?Select all that apply.You will get a penalty for wrong answers. | ['High dimensionality and sparseness of data', 'Lack of linguistic competence and resources', 'Impact of power laws', 'Subjectivity of annotators'] | B | null | Document 1:::
Natural language understanding
Natural-language understanding (NLU) or natural-language interpretation (NLI) is a subtopic of natural-language processing in artificial intelligence that deals with machine reading comprehension. Natural-language understanding is considered an AI-hard problem.There is considerable commercial interest in the field because of its application to automated reasoning, machine translation, question answering, news-gathering, text categorization, voice-activation, archiving, and large-scale content analysis.
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Natural language generation
NLG may be viewed as complementary to natural-language understanding (NLU): whereas in natural-language understanding, the system needs to disambiguate the input sentence to produce the machine representation language, in NLG the system needs to make decisions about how to put a representation into words. The practical considerations in building NLU vs. NLG systems are not symmetrical. NLU needs to deal with ambiguous or erroneous user input, whereas the ideas the system wants to express through NLG are generally known precisely.
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Natural language generation
Psycholinguists prefer the term language production for this process, which can also be described in mathematical terms, or modeled in a computer for psychological research. NLG systems can also be compared to translators of artificial computer languages, such as decompilers or transpilers, which also produce human-readable code generated from an intermediate representation. Human languages tend to be considerably more complex and allow for much more ambiguity and variety of expression than programming languages, which makes NLG more challenging.
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Natural language programming
Natural-language programming (NLP) is an ontology-assisted way of programming in terms of natural-language sentences, e.g. English. A structured document with Content, sections and subsections for explanations of sentences forms a NLP document, which is actually a computer program. Natural language programming is not to be mixed up with natural language interfacing or voice control where a program is first written and then communicated with through natural language using an interface added on. In NLP the functionality of a program is organised only for the definition of the meaning of sentences.
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Natural Language Toolkit
The Natural Language Toolkit, or more commonly NLTK, is a suite of libraries and programs for symbolic and statistical natural language processing (NLP) for English written in the Python programming language. It supports classification, tokenization, stemming, tagging, parsing, and semantic reasoning functionalities. It was developed by Steven Bird and Edward Loper in the Department of Computer and Information Science at the University of Pennsylvania. |
epfl-collab | A query \(q\) has been submitted to two distinct Information Retrieval engines operating on the same document collection containing 1'000 documents, with 50 documents being truly relevant for \(q\).The following result lists have been produced by the two IR engines, \(S_1\) and \(S_2\) respectively:
\(S_1\text{:}\)
\(d_1\)
\(d_2\text{ (*)}\)
\(d_3\text{ (*)}\)
\(d_4\)
\(d_5\text{ (*)}\)
\(S_2\text{:}\)
\(d^\prime_1\text{ (*)}\)
\(d^\prime_2\text{ (*)}\)
\(d^\prime_3\)
\(d^\prime_4\)
\(d^\prime_5\)
In these result lists, the stars \(\text{(*)}\) identify the truly relevant documents.
By convention, we consider that any non retrieved document has been retrieved at rank 6.If Average Precision is used as evaluation metric, which of the two IR engines is performing better for the query \(q\)? | ['\\(S_2\\)', 'This evaluation metric cannot be computed.', 'Both engines perform equally.', '\\(S_1\\)'] | A | null | Document 1:::
Average precision
Evaluation measures for an information retrieval (IR) system assess how well an index, search engine or database returns results from a collection of resources that satisfy a user's query. They are therefore fundamental to the success of information systems and digital platforms. The success of an IR system may be judged by a range of criteria including relevance, speed, user satisfaction, usability, efficiency and reliability. However, the most important factor in determining a system's effectiveness for users is the overall relevance of results retrieved in response to a query.
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Average precision
Evaluation measures may be categorised in various ways including offline or online, user-based or system-based and include methods such as observed user behaviour, test collections, precision and recall, and scores from prepared benchmark test sets. Evaluation for an information retrieval system should also include a validation of the measures used, i.e. an assessment of how well they measure what they are intended to measure and how well the system fits its intended use case. Measures are generally used in two settings: online experimentation, which assesses users' interactions with the search system, and offline evaluation, which measures the effectiveness of an information retrieval system on a static offline collection.
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Uncertain inference
Rather than retrieving a document that exactly matches the query we should rank the documents based on their plausibility in regards to that query. Since d and q are both generated by users, they are error prone; thus d → q {\displaystyle d\to q} is uncertain. This will affect the plausibility of a given query.
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Uncertain inference
Rather than retrieving a document that exactly matches the query we should rank the documents based on their plausibility in regards to that query. Since d and q are both generated by users, they are error prone; thus d → q {\displaystyle d\to q} is uncertain. This will affect the plausibility of a given query.
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Precision and recall
In pattern recognition, information retrieval, object detection and classification (machine learning), precision and recall are performance metrics that apply to data retrieved from a collection, corpus or sample space. Precision (also called positive predictive value) is the fraction of relevant instances among the retrieved instances. Written as a formula: r e l e v a n t _ r e t r i e v e d _ i n s t a n c e s a l l _ r e t r i e v e d _ i n s t a n c e s {\displaystyle {\frac {relevant\_retrieved\_instances}{all\_{\mathbf {retrieved}}\_instances}}} . Recall (also known as sensitivity) is the fraction of relevant instances that were retrieved. |
epfl-collab | A major specificity of natural languages is that they are inherently implicit and ambiguous. How should this be taken into account in the NLP perspective?
(penalty for wrong ticks) | ['by interacting with human experts to formulate precise interpretation rules for linguistic entities', 'by teaching humans to talk and write in a way that reduces implicitness and ambiguity', 'by increasing the amount of a priori knowledge that NLP systems are able to exploit', 'by designing NLP algorithms and data structures able to efficiently cope with very ambiguous representations'] | D | null | Document 1:::
Natural language generation
NLG may be viewed as complementary to natural-language understanding (NLU): whereas in natural-language understanding, the system needs to disambiguate the input sentence to produce the machine representation language, in NLG the system needs to make decisions about how to put a representation into words. The practical considerations in building NLU vs. NLG systems are not symmetrical. NLU needs to deal with ambiguous or erroneous user input, whereas the ideas the system wants to express through NLG are generally known precisely.
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Logic translation
Many of the difficulties of the process are caused by vague or ambiguous expressions in natural language. For example, the English word "is" can mean that something exists, that it is identical to something else, or that it has a certain property. This contrasts with the precise nature of formal logic, which avoids such ambiguities.
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Logic translation
Many of the difficulties of the process are caused by vague or ambiguous expressions in natural language. For example, the English word "is" can mean that something exists, that it is identical to something else, or that it has a certain property. This contrasts with the precise nature of formal logic, which avoids such ambiguities.
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Natural language understanding
Natural-language understanding (NLU) or natural-language interpretation (NLI) is a subtopic of natural-language processing in artificial intelligence that deals with machine reading comprehension. Natural-language understanding is considered an AI-hard problem.There is considerable commercial interest in the field because of its application to automated reasoning, machine translation, question answering, news-gathering, text categorization, voice-activation, archiving, and large-scale content analysis.
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Natural language programming
Natural-language programming (NLP) is an ontology-assisted way of programming in terms of natural-language sentences, e.g. English. A structured document with Content, sections and subsections for explanations of sentences forms a NLP document, which is actually a computer program. Natural language programming is not to be mixed up with natural language interfacing or voice control where a program is first written and then communicated with through natural language using an interface added on. In NLP the functionality of a program is organised only for the definition of the meaning of sentences. |
epfl-collab | Consider 3 regular expressions \(A\), \(B\), and \(C\), such that:the sets of strings recognized by each of the regular expressions is non empty;the set of strings recognized by \(B\) is included in the set of strings recognized by \(A\);some strings are recognized simultaneously by \(A\) and by \(C\); andno string is recognized simultaneously by \(B\) and \(C\).Which of the following statements are true?(where, for a regular expression \(X\), \((X)\) denotes the transducer which associates every string recognized by \(X\) to itself)(Penalty for wrong ticks) | ['Any string recognized by \\(B\\) is (at least) associated to itself by the transducer \\(A\\otimes B\\)', '\\((A\\otimes B)\\circ (C)\\) recognizes a non empty set of string associations', '\\((B\\otimes A)\\circ (C)\\) recognizes a non empty set of string associations', 'Any string recognized by \\(A\\) but not by \\(B\\) is a left string in an association recognized by the transducer \\((A)(C)\\)'] | A | null | Document 1:::
Comparison of regular expression engines
This is a comparison of regular expression engines.
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Regular Expression
A regular expression (shortened as regex or regexp; sometimes referred to as rational expression) is a sequence of characters that specifies a match pattern in text. Usually such patterns are used by string-searching algorithms for "find" or "find and replace" operations on strings, or for input validation. Regular expression techniques are developed in theoretical computer science and formal language theory. The concept of regular expressions began in the 1950s, when the American mathematician Stephen Cole Kleene formalized the concept of a regular language.
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Regular Expression
They came into common use with Unix text-processing utilities. Different syntaxes for writing regular expressions have existed since the 1980s, one being the POSIX standard and another, widely used, being the Perl syntax. Regular expressions are used in search engines, in search and replace dialogs of word processors and text editors, in text processing utilities such as sed and AWK, and in lexical analysis. Regular expressions are supported in many programming languages.
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Thompson's construction algorithm
From a theoretical point of view, this algorithm is a part of the proof that they both accept exactly the same languages, that is, the regular languages. An NFA can be made deterministic by the powerset construction and then be minimized to get an optimal automaton corresponding to the given regular expression. However, an NFA may also be interpreted directly. To decide whether two given regular expressions describe the same language, each can be converted into an equivalent minimal deterministic finite automaton via Thompson's construction, powerset construction, and DFA minimization. If, and only if, the resulting automata agree up to renaming of states, the regular expressions' languages agree.
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Pattern Recognition
Pattern recognition is a more general problem that encompasses other types of output as well. Other examples are regression, which assigns a real-valued output to each input; sequence labeling, which assigns a class to each member of a sequence of values (for example, part of speech tagging, which assigns a part of speech to each word in an input sentence); and parsing, which assigns a parse tree to an input sentence, describing the syntactic structure of the sentence.Pattern recognition algorithms generally aim to provide a reasonable answer for all possible inputs and to perform "most likely" matching of the inputs, taking into account their statistical variation. This is opposed to pattern matching algorithms, which look for exact matches in the input with pre-existing patterns. A common example of a pattern-matching algorithm is regular expression matching, which looks for patterns of a given sort in textual data and is included in the search capabilities of many text editors and word processors. |
epfl-collab | Why is natural language processing difficult?
Select all that apply.A penalty will be applied for wrong answers. | ['Impact of power laws', 'Lack of linguistic competence and resources', 'Subjectivity of annotators', 'High dimensionality and sparseness of data'] | B | null | Document 1:::
Natural language understanding
Natural-language understanding (NLU) or natural-language interpretation (NLI) is a subtopic of natural-language processing in artificial intelligence that deals with machine reading comprehension. Natural-language understanding is considered an AI-hard problem.There is considerable commercial interest in the field because of its application to automated reasoning, machine translation, question answering, news-gathering, text categorization, voice-activation, archiving, and large-scale content analysis.
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Natural language programming
Natural-language programming (NLP) is an ontology-assisted way of programming in terms of natural-language sentences, e.g. English. A structured document with Content, sections and subsections for explanations of sentences forms a NLP document, which is actually a computer program. Natural language programming is not to be mixed up with natural language interfacing or voice control where a program is first written and then communicated with through natural language using an interface added on. In NLP the functionality of a program is organised only for the definition of the meaning of sentences.
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Natural Language Toolkit
The Natural Language Toolkit, or more commonly NLTK, is a suite of libraries and programs for symbolic and statistical natural language processing (NLP) for English written in the Python programming language. It supports classification, tokenization, stemming, tagging, parsing, and semantic reasoning functionalities. It was developed by Steven Bird and Edward Loper in the Department of Computer and Information Science at the University of Pennsylvania.
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Natural language generation
NLG may be viewed as complementary to natural-language understanding (NLU): whereas in natural-language understanding, the system needs to disambiguate the input sentence to produce the machine representation language, in NLG the system needs to make decisions about how to put a representation into words. The practical considerations in building NLU vs. NLG systems are not symmetrical. NLU needs to deal with ambiguous or erroneous user input, whereas the ideas the system wants to express through NLG are generally known precisely.
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Decision problems
The field of computational complexity categorizes decidable decision problems by how difficult they are to solve. "Difficult", in this sense, is described in terms of the computational resources needed by the most efficient algorithm for a certain problem. The field of recursion theory, meanwhile, categorizes undecidable decision problems by Turing degree, which is a measure of the noncomputability inherent in any solution. |
epfl-collab | A multiset is an unordered collection where elements can appear multiple times. We will represent a multiset of Char elements as a function from Char to Int: the function returns 0 for any Char argument that is not in the multiset, and the (positive) number of times it appears otherwise: type Multiset = Char => Int The filter operation on a multiset m returns the subset of m for which p holds. What should replace ??? so that the filter function is correct? def filter(m: Multiset, p: Char => Boolean): Multiset = ??? | ['x => if !m(x) then p(x) else 0', 'x => if m(x) then p(x) else 0', 'x => m(x) && p(x)', 'x => if p(x) then m(x) else 0'] | D | null | Document 1:::
Set (abstract data type)
Static sets allow only query operations on their elements — such as checking whether a given value is in the set, or enumerating the values in some arbitrary order. Other variants, called dynamic or mutable sets, allow also the insertion and deletion of elements from the set. A multiset is a special kind of set in which an element can appear multiple times in the set.
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Filter (higher-order function)
In functional programming, filter is a higher-order function that processes a data structure (usually a list) in some order to produce a new data structure containing exactly those elements of the original data structure for which a given predicate returns the boolean value true.
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Multibrot set
In mathematics, a Multibrot set is the set of values in the complex plane whose absolute value remains below some finite value throughout iterations by a member of the general monic univariate polynomial family of recursions. The name is a portmanteau of multiple and Mandelbrot set. The same can be applied to the Julia set, this being called Multijulia set.
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Filter (mathematics)
In mathematics, a filter or order filter is a special subset of a partially ordered set (poset), describing "large" or "eventual" elements. Filters appear in order and lattice theory, but also topology, whence they originate. The notion dual to a filter is an order ideal. Special cases of filters include ultrafilters, which are filters that cannot be enlarged, and describe nonconstructive techniques in mathematical logic.
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Dedekind number
A Boolean function is a function that takes as input n Boolean variables (that is, values that can be either false or true, or equivalently binary values that can be either 0 or 1), and produces as output another Boolean variable. It is monotonic if, for every combination of inputs, switching one of the inputs from false to true can only cause the output to switch from false to true and not from true to false. The Dedekind number M(n) is the number of different monotonic Boolean functions on n variables.An antichain of sets (also known as a Sperner family) is a family of sets, none of which is contained in any other set. If V is a set of n Boolean variables, an antichain A of subsets of V defines a monotone Boolean function f, where the value of f is true for a given set of inputs if some subset of the true inputs to f belongs to A and false otherwise. |
epfl-collab | The little Fermat theorem states that for a prime $n$ and any $b\in \mathbb{Z}_n ^\star$ we have\dots | ['$b^{n-1}\\mod n = 1$.', '$b^{n-1}\\mod n = n$.', '$b^{n-1}\\mod n = b$.', '$b^{n}\\mod n = 1$.'] | A | null | Document 1:::
Cubic residue symbol
An analogue of Fermat's little theorem is true in Z {\displaystyle \mathbb {Z} }: if α {\displaystyle \alpha } is not divisible by a prime π {\displaystyle \pi } , α N ( π ) − 1 ≡ 1 mod π . {\displaystyle \alpha ^{N(\pi )-1}\equiv 1{\bmod {\pi }}.} Now assume that N ( π ) ≠ 3 {\displaystyle N(\pi )\neq 3} so that N ( π ) ≡ 1 mod 3 . {\displaystyle N(\pi )\equiv 1{\bmod {3}}.}
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Fermat pseudoprime
Fermat's little theorem states that if p is prime and a is coprime to p, then ap−1 − 1 is divisible by p. For an integer a > 1, if a composite integer x divides ax−1 − 1, then x is called a Fermat pseudoprime to base a.: Def. 3.32 In other words, a composite integer is a Fermat pseudoprime to base a if it successfully passes the Fermat primality test for the base a. The false statement that all numbers that pass the Fermat primality test for base 2, are prime, is called the Chinese hypothesis. The smallest base-2 Fermat pseudoprime is 341. It is not a prime, since it equals 11·31, but it satisfies Fermat's little theorem: 2340 ≡ 1 (mod 341) and thus passes the Fermat primality test for the base 2.
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Fermat pseudoprime
In number theory, the Fermat pseudoprimes make up the most important class of pseudoprimes that come from Fermat's little theorem.
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Fermat's theorem
The works of the 17th-century mathematician Pierre de Fermat engendered many theorems. Fermat's theorem may refer to one of the following theorems: Fermat's Last Theorem, about integer solutions to an + bn = cn Fermat's little theorem, a property of prime numbers Fermat's theorem on sums of two squares, about primes expressible as a sum of squares Fermat's theorem (stationary points), about local maxima and minima of differentiable functions Fermat's principle, about the path taken by a ray of light Fermat polygonal number theorem, about expressing integers as a sum of polygonal numbers Fermat's right triangle theorem, about squares not being expressible as the difference of two fourth powers
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Carmichael number
They are infinite in number. They constitute the comparatively rare instances where the strict converse of Fermat's Little Theorem does not hold. This fact precludes the use of that theorem as an absolute test of primality.The Carmichael numbers form the subset K1 of the Knödel numbers. |
epfl-collab | The number of permutations on a set of $n$ elements | ['is always greater than $2^n$', 'is approximately $n(\\log n - 1)$', 'can be approximated using the Stirling formula', 'is independent of the size of the set'] | C | null | Document 1:::
Circular notation
In computer science, they are used for analyzing sorting algorithms; in quantum physics, for describing states of particles; and in biology, for describing RNA sequences. The number of permutations of n distinct objects is n factorial, usually written as n!, which means the product of all positive integers less than or equal to n. Technically, a permutation of a set S is defined as a bijection from S to itself. That is, it is a function from S to S for which every element occurs exactly once as an image value.
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Circular notation
As properties of permutations do not depend on the nature of the set elements, it is often the permutations of the set { 1 , 2 , … , n } {\displaystyle \{1,2,\ldots ,n\}} that are considered for studying permutations. In elementary combinatorics, the k-permutations, or partial permutations, are the ordered arrangements of k distinct elements selected from a set. When k is equal to the size of the set, these are the permutations of the set.
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Enumerative combinatorics
Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations. More generally, given an infinite collection of finite sets Si indexed by the natural numbers, enumerative combinatorics seeks to describe a counting function which counts the number of objects in Sn for each n. Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a relatively simple combinatorial description. The twelvefold way provides a unified framework for counting permutations, combinations and partitions.
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Factorial function
For instance, the number of ways to choose all n {\displaystyle n} elements from a set of n {\displaystyle n} is ( n n ) = n ! n !
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Symmetric group
( n {\displaystyle n} factorial) such permutation operations, the order (number of elements) of the symmetric group S n {\displaystyle \mathrm {S} _{n}} is n ! {\displaystyle n!} . |
epfl-collab | Select \emph{incorrect} statement. Complexity analysis of an attack consideres | ['probability of success.', 'memory complexity.', 'time complexity.', 'difficulty to understand a corresponding journal paper.'] | D | null | Document 1:::
Algorithmic complexity attack
An algorithmic complexity attack (ACA) is a form of attack in which the system is attacked by an exhaustion resource to take advantage of worst-case performance.
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Brute force attack
In cryptography, a brute-force attack consists of an attacker submitting many passwords or passphrases with the hope of eventually guessing correctly. The attacker systematically checks all possible passwords and passphrases until the correct one is found. Alternatively, the attacker can attempt to guess the key which is typically created from the password using a key derivation function. This is known as an exhaustive key search.
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Padding oracle attack
In cryptography, a padding oracle attack is an attack which uses the padding validation of a cryptographic message to decrypt the ciphertext. In cryptography, variable-length plaintext messages often have to be padded (expanded) to be compatible with the underlying cryptographic primitive. The attack relies on having a "padding oracle" who freely responds to queries about whether a message is correctly padded or not. Padding oracle attacks are mostly associated with CBC mode decryption used within block ciphers. Padding modes for asymmetric algorithms such as OAEP may also be vulnerable to padding oracle attacks.
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Inference attack
An Inference Attack is a data mining technique performed by analyzing data in order to illegitimately gain knowledge about a subject or database. A subject's sensitive information can be considered as leaked if an adversary can infer its real value with a high confidence. This is an example of breached information security. An Inference attack occurs when a user is able to infer from trivial information more robust information about a database without directly accessing it.
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Brute force attack
A brute-force attack is a cryptanalytic attack that can, in theory, be used to attempt to decrypt any encrypted data (except for data encrypted in an information-theoretically secure manner). Such an attack might be used when it is not possible to take advantage of other weaknesses in an encryption system (if any exist) that would make the task easier. When password-guessing, this method is very fast when used to check all short passwords, but for longer passwords other methods such as the dictionary attack are used because a brute-force search takes too long. |
epfl-collab | Which one of these is \emph{not} a stream cipher? | ['A5/1', 'IDEA', 'RC4', 'E0'] | B | null | Document 1:::
SSS (cipher)
In cryptography, SSS is a stream cypher algorithm developed by Gregory Rose, Philip Hawkes, Michael Paddon, and Miriam Wiggers de Vries. It includes a message authentication code feature. It has been submitted to the eSTREAM Project of the eCRYPT network. It has not selected for focus nor for consideration during Phase 2; it has been 'archived'. == References ==
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KCipher-2
KCipher-2 is a stream cipher jointly developed by Kyushu University and Japanese telecommunications company KDDI. It is standardized as ISO/IEC 18033–4, and is on the list of recommended ciphers published by the Japanese Cryptography Research and Evaluation Committees (CRYPTREC). It has a key length of 128 bits, and can encrypt and decrypt around seven to ten times faster than the Advanced Encryption Standard (AES) algorithm.
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HC-256
HC-256 is a stream cipher designed to provide bulk encryption in software at high speeds while permitting strong confidence in its security. A 128-bit variant was submitted as an eSTREAM cipher candidate and has been selected as one of the four final contestants in the software profile.The algorithm is designed by Hongjun Wu, and was first published in 2004. It is not patented.
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ABC (cipher)
In cryptography, ABC is a stream cypher algorithm developed by Vladimir Anashin, Andrey Bogdanov, Ilya Kizhvatov, and Sandeep Kumar. It has been submitted to the eSTREAM Project of the eCRYPT network. == References ==
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CryptMT
In cryptography, CryptMT is a stream cipher algorithm which internally uses the Mersenne twister. It was developed by Makoto Matsumoto, Mariko Hagita, Takuji Nishimura and Mutsuo Saito and is patented. It has been submitted to the eSTREAM project of the eCRYPT network. In that submission to eSTREAM, the authors also included another cipher named Fubuki, which also uses the Mersenne twister. |
epfl-collab | Tick the \emph{correct} assertion regarding GSM. | ['The integrity of GSM messages is well protected.', 'In GSM, the phone is authenticated to the network.', 'GSM uses the GSME cipher to encrypt messages.', 'In GSM, the communication is always encrypted.'] | B | null | Document 1:::
Timing advance
In the GSM cellular mobile phone standard, timing advance (TA) value corresponds to the length of time a signal takes to reach the base station from a mobile phone. GSM uses TDMA technology in the radio interface to share a single frequency between several users, assigning sequential timeslots to the individual users sharing a frequency. Each user transmits periodically for less than one-eighth of the time within one of the eight timeslots. Since the users are at various distances from the base station and radio waves travel at the finite speed of light, the precise arrival-time within the slot can be used by the base station to determine the distance to the mobile phone.
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GSM frequency ranges
GSM frequency bands or frequency ranges are the cellular frequencies designated by the ITU for the operation of GSM mobile phones and other mobile devices.
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Enhanced Full Rate
Enhanced Full Rate or EFR or GSM-EFR or GSM 06.60 is a speech coding standard that was developed in order to improve the quality of GSM. Enhanced Full Rate was developed by Nokia and the Université de Sherbrooke (Canada). In 1995, ETSI selected the Enhanced Full Rate voice codec as the industry standard codec for GSM/DCS.
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EDGE Evolution
Enhanced Data rates for GSM Evolution (EDGE), also known as Enhanced GPRS (EGPRS), IMT Single Carrier (IMT-SC), or Enhanced Data rates for Global Evolution, is a digital mobile phone technology that allows improved data transmission rates as a backward-compatible extension of GSM. EDGE is considered a pre-3G radio technology and is part of ITU's 3G definition. EDGE was deployed on GSM networks beginning in 2003 – initially by Cingular (now AT&T) in the United States.EDGE is standardized also by 3GPP as part of the GSM family.
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Enhanced Observed Time Difference
However, GSM and 3G networks are not necessarily synchronised, so further information is needed. The E-OTD standard provides a method for pseudo-synchronisation. A Location Measurement Unit (LMU) can be used to estimate the transmission time offset between two base stations. |
epfl-collab | Tick the \emph{wrong} assertion concerning 3G. | ['3G uses f8 for encryption.', 'In 3G, there is a counter to protect against replay attacks.', 'In 3G, the network is authenticated to the phone.', 'The integrity of 3G messages is well protected.'] | C | null | Document 1:::
3G network
3G is the third generation of wireless mobile telecommunications technology. It is the upgrade over 2G, 2.5G, GPRS and 2.75G Enhanced Data Rates for GSM Evolution networks, offering faster data transfer, and better voice quality. This network was superseded by 4G, and later on by 5G. This network is based on a set of standards used for mobile devices and mobile telecommunications use services and networks that comply with the International Mobile Telecommunications-2000 (IMT-2000) specifications by the International Telecommunication Union.
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3GPP
The 3rd Generation Partnership Project (3GPP) is an umbrella term for a number of standards organizations which develop protocols for mobile telecommunications. Its best known work is the development and maintenance of: GSM and related 2G and 2.5G standards, including GPRS and EDGE UMTS and related 3G standards, including HSPA and HSPA+ LTE and related 4G standards, including LTE Advanced and LTE Advanced Pro 5G NR and related 5G standards, including 5G-Advanced An evolved IP Multimedia Subsystem (IMS) developed in an access independent manner3GPP is a consortium with seven national or regional telecommunication standards organizations as primary members ("organizational partners") and a variety of other organizations as associate members ("market representation partners"). The 3GPP organizes its work into three different streams: Radio Access Networks, Services and Systems Aspects, and Core Network and Terminals.The project was established in December 1998 with the goal of developing a specification for a 3G mobile phone system based on the 2G GSM system, within the scope of the International Telecommunication Union's International Mobile Telecommunications-2000, hence the name 3GPP. It should not be confused with 3rd Generation Partnership Project 2 (3GPP2), which developed a competing 3G system, CDMA2000.The 3GPP administrative support team (known as the "Mobile Competence Centre") is located at the European Telecommunications Standards Institute headquarters in the Sophia Antipolis technology park in France.
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Femtocell
A Home eNodeB (HeNB) is an LTE 4G femtocell. Theoretically the range of a standard base station may be up to 35 kilometres (22 mi), and in practice could be 5–10 km (3–6 mi), a microcell is less than two kilometers (1 mile) wide, a picocell is 200 meters (yards) or less, and a femtocell is in the order of 10 meters (yards), although AT&T calls its product, with a range of 40 feet (12 m), a "microcell". AT&T uses "AT&T 3G MicroCell" as a trademark and not necessarily the "microcell" technology, however.
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3G network
3G finds application in wireless voice telephony, mobile Internet access, fixed wireless Internet access, video calls and mobile TV.3G telecommunication networks support services that provide an information transfer rate of at least 144 kbit/s. Later 3G releases, often denoted 3.5G and 3.75G, also provide mobile broadband access of several Mbit/s to smartphones and mobile modems in laptop computers. This ensures it can be applied to wireless voice calls, mobile Internet access, fixed wireless Internet access, video calls and mobile TV technologies.
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3G network
A new generation of cellular standards has appeared approximately every tenth year since 1G systems were introduced in 1979 and the early to mid-1980s. Each generation is characterized by new frequency bands, higher data rates and non–backward-compatible transmission technology. The first commercial 3G networks were introduced in mid-2001. |
epfl-collab | Tick the \textbf{false} statement. | ['In WEP, authentication is done with the pre-shared keys.', 'Due to memory limitations, dummy devices can share the same key with everyone.', 'Cryptographic primitives used in Bluetooth are provably secure.', 'The security of Bluetooth 2.0 pairing is based on PIN.'] | C | null | Document 1:::
False (logic)
In logic, false or untrue is the state of possessing negative truth value and is a nullary logical connective. In a truth-functional system of propositional logic, it is one of two postulated truth values, along with its negation, truth. Usual notations of the false are 0 (especially in Boolean logic and computer science), O (in prefix notation, Opq), and the up tack symbol ⊥ {\displaystyle \bot } .Another approach is used for several formal theories (e.g., intuitionistic propositional calculus), where a propositional constant (i.e. a nullary connective), ⊥ {\displaystyle \bot } , is introduced, the truth value of which being always false in the sense above. It can be treated as an absurd proposition, and is often called absurdity.
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Boolean expression
In computer science, a Boolean expression is an expression used in programming languages that produces a Boolean value when evaluated. A Boolean value is either true or false. A Boolean expression may be composed of a combination of the Boolean constants true or false, Boolean-typed variables, Boolean-valued operators, and Boolean-valued functions.Boolean expressions correspond to propositional formulas in logic and are a special case of Boolean circuits.
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Boolean flag
A Boolean flag, truth bit or truth flag in computer science is a Boolean value represented as one or more bits, which encodes a state variable with two possible values.
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Boolean domain
The initial object in the category of bounded lattices is a Boolean domain. In computer science, a Boolean variable is a variable that takes values in some Boolean domain. Some programming languages feature reserved words or symbols for the elements of the Boolean domain, for example false and true. However, many programming languages do not have a Boolean datatype in the strict sense. In C or BASIC, for example, falsity is represented by the number 0 and truth is represented by the number 1 or −1, and all variables that can take these values can also take any other numerical values.
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Boolean-valued function
A Boolean-valued function (sometimes called a predicate or a proposition) is a function of the type f: X → B, where X is an arbitrary set and where B is a Boolean domain, i.e. a generic two-element set, (for example B = {0, 1}), whose elements are interpreted as logical values, for example, 0 = false and 1 = true, i.e., a single bit of information. In the formal sciences, mathematics, mathematical logic, statistics, and their applied disciplines, a Boolean-valued function may also be referred to as a characteristic function, indicator function, predicate, or proposition. In all of these uses, it is understood that the various terms refer to a mathematical object and not the corresponding semiotic sign or syntactic expression. In formal semantic theories of truth, a truth predicate is a predicate on the sentences of a formal language, interpreted for logic, that formalizes the intuitive concept that is normally expressed by saying that a sentence is true. A truth predicate may have additional domains beyond the formal language domain, if that is what is required to determine a final truth value. |
epfl-collab | Why do block ciphers use modes of operation? | ['to use keys of any size.', 'to be provably secure.', 'to encrypt messages of any size.', 'it is necessary for the decryption to work.'] | C | null | Document 1:::
Cipher-block chaining
In cryptography, a block cipher mode of operation is an algorithm that uses a block cipher to provide information security such as confidentiality or authenticity. A block cipher by itself is only suitable for the secure cryptographic transformation (encryption or decryption) of one fixed-length group of bits called a block. A mode of operation describes how to repeatedly apply a cipher's single-block operation to securely transform amounts of data larger than a block.Most modes require a unique binary sequence, often called an initialization vector (IV), for each encryption operation. The IV has to be non-repeating and, for some modes, random as well.
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Block cipher
Even a secure block cipher is suitable for the encryption of only a single block of data at a time, using a fixed key. A multitude of modes of operation have been designed to allow their repeated use in a secure way to achieve the security goals of confidentiality and authenticity. However, block ciphers may also feature as building blocks in other cryptographic protocols, such as universal hash functions and pseudorandom number generators.
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Cipher-block chaining
There are, however, modes that do not require padding because they effectively use a block cipher as a stream cipher. Historically, encryption modes have been studied extensively in regard to their error propagation properties under various scenarios of data modification. Later development regarded integrity protection as an entirely separate cryptographic goal. Some modern modes of operation combine confidentiality and authenticity in an efficient way, and are known as authenticated encryption modes.
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Cipher-block chaining
The initialization vector is used to ensure distinct ciphertexts are produced even when the same plaintext is encrypted multiple times independently with the same key. Block ciphers may be capable of operating on more than one block size, but during transformation the block size is always fixed. Block cipher modes operate on whole blocks and require that the last part of the data be padded to a full block if it is smaller than the current block size.
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EAX mode
EAX mode (encrypt-then-authenticate-then-translate) is a mode of operation for cryptographic block ciphers. It is an Authenticated Encryption with Associated Data (AEAD) algorithm designed to simultaneously provide both authentication and privacy of the message (authenticated encryption) with a two-pass scheme, one pass for achieving privacy and one for authenticity for each block. EAX mode was submitted on October 3, 2003 to the attention of NIST in order to replace CCM as standard AEAD mode of operation, since CCM mode lacks some desirable attributes of EAX and is more complex. |
epfl-collab | If we pick independent random numbers in $\{1, 2, \dots, N\}$ with uniform distribution, $\theta \sqrt{N}$ times, we get at least one number twice with probability\dots | ['$e^{\\theta ^2}$', '$1-e^{-\\theta ^2 /2}$', '$1-e^{\\theta ^2}$', '$e^{-\\theta ^2 /2}$'] | B | null | Document 1:::
Discrete uniform random variable
In probability theory and statistics, the discrete uniform distribution is a symmetric probability distribution wherein a finite number of values are equally likely to be observed; every one of n values has equal probability 1/n. Another way of saying "discrete uniform distribution" would be "a known, finite number of outcomes equally likely to happen". A simple example of the discrete uniform distribution is throwing a fair die. The possible values are 1, 2, 3, 4, 5, 6, and each time the die is thrown the probability of a given score is 1/6.
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Random number sampling
Non-uniform random variate generation or pseudo-random number sampling is the numerical practice of generating pseudo-random numbers (PRN) that follow a given probability distribution. Methods are typically based on the availability of a uniformly distributed PRN generator. Computational algorithms are then used to manipulate a single random variate, X, or often several such variates, into a new random variate Y such that these values have the required distribution. The first methods were developed for Monte-Carlo simulations in the Manhattan project, published by John von Neumann in the early 1950s.
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Random number
In mathematics and statistics, a random number is either Pseudo-random or a number generated for, or part of, a set exhibiting statistical randomness.
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Uniform probability distribution
For n ≥ 2 , {\displaystyle n\geq 2,} the n {\displaystyle n} -th cumulant of the continuous uniform distribution on the interval {\displaystyle } is B n n , {\displaystyle {\tfrac {B_{n}}{n}},} where B n {\displaystyle B_{n}} is the n {\displaystyle n} -th Bernoulli number.
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RP (complexity)
If the correct answer is YES and the algorithm is run n times with the result of each run statistically independent of the others, then it will return YES at least once with probability at least 1 − 2−n. So if the algorithm is run 100 times, then the chance of it giving the wrong answer every time is lower than the chance that cosmic rays corrupted the memory of the computer running the algorithm. In this sense, if a source of random numbers is available, most algorithms in RP are highly practical. The fraction 1/2 in the definition is arbitrary. The set RP will contain exactly the same problems, even if the 1/2 is replaced by any constant nonzero probability less than 1; here constant means independent of the input to the algorithm. |
epfl-collab | In practice, what is the typical size of an RSA modulus? | ['256 bits', '64 bits', '1024 bits', '8192 bits'] | C | null | Document 1:::
Factoring integers
The researchers estimated that a 1024-bit RSA modulus would take about 500 times as long.Not all numbers of a given length are equally hard to factor. The hardest instances of these problems (for currently known techniques) are semiprimes, the product of two prime numbers. When they are both large, for instance more than two thousand bits long, randomly chosen, and about the same size (but not too close, for example, to avoid efficient factorization by Fermat's factorization method), even the fastest prime factorization algorithms on the fastest computers can take enough time to make the search impractical; that is, as the number of digits of the integer being factored increases, the number of operations required to perform the factorization on any computer increases drastically. Many cryptographic protocols are based on the difficulty of factoring large composite integers or a related problem—for example, the RSA problem. An algorithm that efficiently factors an arbitrary integer would render RSA-based public-key cryptography insecure.
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RSA Cryptosystem
An RSA user creates and publishes a public key based on two large prime numbers, along with an auxiliary value. The prime numbers are kept secret. Messages can be encrypted by anyone, via the public key, but can only be decoded by someone who knows the prime numbers.The security of RSA relies on the practical difficulty of factoring the product of two large prime numbers, the "factoring problem".
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Modular exponentiation
Modular exponentiation is exponentiation performed over a modulus. It is useful in computer science, especially in the field of public-key cryptography, where it is used in both Diffie-Hellman Key Exchange and RSA public/private keys. Modular exponentiation is the remainder when an integer b (the base) is raised to the power e (the exponent), and divided by a positive integer m (the modulus); that is, c = be mod m. From the definition of division, it follows that 0 ≤ c < m. For example, given b = 5, e = 3 and m = 13, dividing 53 = 125 by 13 leaves a remainder of c = 8. Modular exponentiation can be performed with a negative exponent e by finding the modular multiplicative inverse d of b modulo m using the extended Euclidean algorithm.
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RSA Factoring Challenge
The RSA Factoring Challenge was a challenge put forward by RSA Laboratories on March 18, 1991 to encourage research into computational number theory and the practical difficulty of factoring large integers and cracking RSA keys used in cryptography. They published a list of semiprimes (numbers with exactly two prime factors) known as the RSA numbers, with a cash prize for the successful factorization of some of them. The smallest of them, a 100-decimal digit number called RSA-100 was factored by April 1, 1991.
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Montgomery modular multiplication
The need to convert a and b into Montgomery form and their product out of Montgomery form means that computing a single product by Montgomery multiplication is slower than the conventional or Barrett reduction algorithms. However, when performing many multiplications in a row, as in modular exponentiation, intermediate results can be left in Montgomery form. Then the initial and final conversions become a negligible fraction of the overall computation. Many important cryptosystems such as RSA and Diffie–Hellman key exchange are based on arithmetic operations modulo a large odd number, and for these cryptosystems, computations using Montgomery multiplication with R a power of two are faster than the available alternatives. |
epfl-collab | The one-time pad is\dots | ['A perfectly binding commitment scheme.', 'A computationally (but not statistically) binding commitment scheme.', 'Not a commitment scheme.', 'A statistically (but not perfectly) binding commitment scheme.'] | C | null | Document 1:::
Blinding (cryptography)
The one-time pad (OTP) is an application of blinding to the secure communication problem, by its very nature. Alice would like to send a message to Bob secretly, however all of their communication can be read by Oscar. Therefore, Alice sends the message after blinding it with a secret key or OTP that she shares with Bob.
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Dot-decimal notation
Dot-decimal notation is a presentation format for numerical data. It consists of a string of decimal numbers, using the full stop (dot) as a separation character.A common use of dot-decimal notation is in information technology where it is a method of writing numbers in octet-grouped base-10 (decimal) numbers. In computer networking, Internet Protocol Version 4 (IPv4) addresses are commonly written using the quad-dotted notation of four decimal integers, ranging from 0 to 255 each.
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Numerical keypad
A numeric keypad, number pad, numpad, or ten key, is the palm-sized, usually-17-key section of a standard computer keyboard, usually on the far right. It provides calculator-style efficiency for entering numbers. The idea of a 10-key number pad cluster was originally introduced by Tadao Kashio, the developer of Casio electronic calculators.The numpad's keys are digits 0 to 9, + (addition), - (subtraction), * (multiplication) and / (division) symbols, .
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Talk:Fibonacci sequence
With lots. Of. Periods.
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OT1 encoding
OT1 (aka TeX text) is a 7-bit TeX encoding developed by Donald E. Knuth. |
epfl-collab | Tick the \textbf{false} statement. | ['If a point is singular on an Elliptic curve, we can draw a tangent to this point.', 'The identity element of $E_{a,b}$ is the point at infinity.', 'Elliptic curve cryptography is useful in public-key cryptography.', '$P=(x_p,y_p)$ and $Q=(x_p,-y_p)$ are the inverse of each other on an Elliptic curve of equation $y^2=x^3+ax+b$.'] | A | null | Document 1:::
False (logic)
In logic, false or untrue is the state of possessing negative truth value and is a nullary logical connective. In a truth-functional system of propositional logic, it is one of two postulated truth values, along with its negation, truth. Usual notations of the false are 0 (especially in Boolean logic and computer science), O (in prefix notation, Opq), and the up tack symbol ⊥ {\displaystyle \bot } .Another approach is used for several formal theories (e.g., intuitionistic propositional calculus), where a propositional constant (i.e. a nullary connective), ⊥ {\displaystyle \bot } , is introduced, the truth value of which being always false in the sense above. It can be treated as an absurd proposition, and is often called absurdity.
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Boolean expression
In computer science, a Boolean expression is an expression used in programming languages that produces a Boolean value when evaluated. A Boolean value is either true or false. A Boolean expression may be composed of a combination of the Boolean constants true or false, Boolean-typed variables, Boolean-valued operators, and Boolean-valued functions.Boolean expressions correspond to propositional formulas in logic and are a special case of Boolean circuits.
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Boolean flag
A Boolean flag, truth bit or truth flag in computer science is a Boolean value represented as one or more bits, which encodes a state variable with two possible values.
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Boolean domain
The initial object in the category of bounded lattices is a Boolean domain. In computer science, a Boolean variable is a variable that takes values in some Boolean domain. Some programming languages feature reserved words or symbols for the elements of the Boolean domain, for example false and true. However, many programming languages do not have a Boolean datatype in the strict sense. In C or BASIC, for example, falsity is represented by the number 0 and truth is represented by the number 1 or −1, and all variables that can take these values can also take any other numerical values.
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Boolean-valued function
A Boolean-valued function (sometimes called a predicate or a proposition) is a function of the type f: X → B, where X is an arbitrary set and where B is a Boolean domain, i.e. a generic two-element set, (for example B = {0, 1}), whose elements are interpreted as logical values, for example, 0 = false and 1 = true, i.e., a single bit of information. In the formal sciences, mathematics, mathematical logic, statistics, and their applied disciplines, a Boolean-valued function may also be referred to as a characteristic function, indicator function, predicate, or proposition. In all of these uses, it is understood that the various terms refer to a mathematical object and not the corresponding semiotic sign or syntactic expression. In formal semantic theories of truth, a truth predicate is a predicate on the sentences of a formal language, interpreted for logic, that formalizes the intuitive concept that is normally expressed by saying that a sentence is true. A truth predicate may have additional domains beyond the formal language domain, if that is what is required to determine a final truth value. |
epfl-collab | Diffie-Hellman refers to \ldots | ['a signature scheme.', 'the inventors of the RSA cryptosystem.', 'a public-key cryptosystem.', 'a key-agreement protocol.'] | D | null | Document 1:::
Diffie–Hellman problem
The Diffie–Hellman problem (DHP) is a mathematical problem first proposed by Whitfield Diffie and Martin Hellman in the context of cryptography. The motivation for this problem is that many security systems use one-way functions: mathematical operations that are fast to compute, but hard to reverse. For example, they enable encrypting a message, but reversing the encryption is difficult. If solving the DHP were easy, these systems would be easily broken.
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Elliptic-curve Diffie-Hellman
Elliptic-curve Diffie–Hellman (ECDH) is a key agreement protocol that allows two parties, each having an elliptic-curve public–private key pair, to establish a shared secret over an insecure channel. This shared secret may be directly used as a key, or to derive another key. The key, or the derived key, can then be used to encrypt subsequent communications using a symmetric-key cipher. It is a variant of the Diffie–Hellman protocol using elliptic-curve cryptography.
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Computational Diffie–Hellman assumption
The computational Diffie–Hellman (CDH) assumption is a computational hardness assumption about the Diffie–Hellman problem. The CDH assumption involves the problem of computing the discrete logarithm in cyclic groups. The CDH problem illustrates the attack of an eavesdropper in the Diffie–Hellman key exchange protocol to obtain the exchanged secret key.
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Computational Diffie–Hellman assumption
Consider a cyclic group G of order q. The CDH assumption states that, given ( g , g a , g b ) {\displaystyle (g,g^{a},g^{b})\,} for a randomly chosen generator g and random a , b ∈ { 0 , … , q − 1 } , {\displaystyle a,b\in \{0,\ldots ,q-1\},\,} it is computationally intractable to compute the value g a b . {\displaystyle g^{ab}.\,}
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Decision Linear assumption
The Decision Linear (DLIN) assumption is a computational hardness assumption used in elliptic curve cryptography. In particular, the DLIN assumption is useful in settings where the decisional Diffie–Hellman assumption does not hold (as is often the case in pairing-based cryptography). The Decision Linear assumption was introduced by Boneh, Boyen, and Shacham.Informally the DLIN assumption states that given ( u , v , h , u x , v y ) {\displaystyle (u,\,v,\,h,\,u^{x},\,v^{y})} , with u , v , h {\displaystyle u,\,v,\,h} random group elements and x , y {\displaystyle x,\,y} random exponents, it is hard to distinguish h x + y {\displaystyle h^{x+y}} from an independent random group element η {\displaystyle \eta } . |
epfl-collab | Consider the Rabin cryptosystem using a modulus $N=pq$ where $p$ and $q$ are both $\ell$-bit primes. What is the tightest complexity of the encryption algorithm? | ['$O(\\ell^2)$', '$O(\\ell)$', '$O(\\ell^4)$', '$O(\\ell^3)$'] | A | null | Document 1:::
Trapdoor permutation
As of 2004, the best known trapdoor function (family) candidates are the RSA and Rabin families of functions. Both are written as exponentiation modulo a composite number, and both are related to the problem of prime factorization. Functions related to the hardness of the discrete logarithm problem (either modulo a prime or in a group defined over an elliptic curve) are not known to be trapdoor functions, because there is no known "trapdoor" information about the group that enables the efficient computation of discrete logarithms. A trapdoor in cryptography has the very specific aforementioned meaning and is not to be confused with a backdoor (these are frequently used interchangeably, which is incorrect). A backdoor is a deliberate mechanism that is added to a cryptographic algorithm (e.g., a key pair generation algorithm, digital signing algorithm, etc.) or operating system, for example, that permits one or more unauthorized parties to bypass or subvert the security of the system in some fashion.
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Rabin fingerprint
The Rabin fingerprinting scheme is a method for implementing fingerprints using polynomials over a finite field. It was proposed by Michael O. Rabin.
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Benaloh cryptosystem
To decrypt a ciphertext c ∈ Z n ∗ {\displaystyle c\in \mathbb {Z} _{n}^{*}}: Compute a = c ϕ / r mod n {\displaystyle a=c^{\phi /r}\mod n} Output m = log x ( a ) {\displaystyle m=\log _{x}(a)} , i.e., find m such that x m ≡ a mod n {\displaystyle x^{m}\equiv a\mod n} To understand decryption, first notice that for any m ∈ Z r {\displaystyle m\in \mathbb {Z} _{r}} and u ∈ Z n ∗ {\displaystyle u\in \mathbb {Z} _{n}^{*}} we have: a = ( c ) ϕ / r ≡ ( y m u r ) ϕ / r ≡ ( y m ) ϕ / r ( u r ) ϕ / r ≡ ( y ϕ / r ) m ( u ) ϕ ≡ ( x ) m ( u ) 0 ≡ x m mod n {\displaystyle a=(c)^{\phi /r}\equiv (y^{m}u^{r})^{\phi /r}\equiv (y^{m})^{\phi /r}(u^{r})^{\phi /r}\equiv (y^{\phi /r})^{m}(u)^{\phi }\equiv (x)^{m}(u)^{0}\equiv x^{m}\mod n} To recover m from a, we take the discrete log of a base x. If r is small, we can recover m by an exhaustive search, i.e. checking if x i ≡ a mod n {\displaystyle x^{i}\equiv a\mod n} for all 0 … ( r − 1 ) {\displaystyle 0\dots (r-1)} . For larger values of r, the Baby-step giant-step algorithm can be used to recover m in O ( r ) {\displaystyle O({\sqrt {r}})} time and space.
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Factoring integers
The researchers estimated that a 1024-bit RSA modulus would take about 500 times as long.Not all numbers of a given length are equally hard to factor. The hardest instances of these problems (for currently known techniques) are semiprimes, the product of two prime numbers. When they are both large, for instance more than two thousand bits long, randomly chosen, and about the same size (but not too close, for example, to avoid efficient factorization by Fermat's factorization method), even the fastest prime factorization algorithms on the fastest computers can take enough time to make the search impractical; that is, as the number of digits of the integer being factored increases, the number of operations required to perform the factorization on any computer increases drastically. Many cryptographic protocols are based on the difficulty of factoring large composite integers or a related problem—for example, the RSA problem. An algorithm that efficiently factors an arbitrary integer would render RSA-based public-key cryptography insecure.
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Benaloh cryptosystem
The security of this scheme rests on the Higher residuosity problem, specifically, given z,r and n where the factorization of n is unknown, it is computationally infeasible to determine whether z is an rth residue mod n, i.e. if there exists an x such that z ≡ x r mod n {\displaystyle z\equiv x^{r}\mod n} . == References == |
epfl-collab | Select the \emph{incorrect} statement. | ['Plain RSA encryption is deterministic.', 'The non-deterministic encryption can encrypt one plaintext into many ciphertexts.', 'The non-deterministic encryption always provides perfect secrecy.', 'ElGamal encryption is non-deterministic.'] | C | null | Document 1:::
Uncertain inference
Rather than retrieving a document that exactly matches the query we should rank the documents based on their plausibility in regards to that query. Since d and q are both generated by users, they are error prone; thus d → q {\displaystyle d\to q} is uncertain. This will affect the plausibility of a given query.
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Uncertain inference
Rather than retrieving a document that exactly matches the query we should rank the documents based on their plausibility in regards to that query. Since d and q are both generated by users, they are error prone; thus d → q {\displaystyle d\to q} is uncertain. This will affect the plausibility of a given query.
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Transposition error
A transcription error is a specific type of data entry error that is commonly made by human operators or by optical character recognition (OCR) programs. Human transcription errors are commonly the result of typographical mistakes; putting one's fingers in the wrong place while touch typing is the easiest way to make this error. Electronic transcription errors occur when the scan of some printed matter is compromised or in an unusual font – for example, if the paper is crumpled, or the ink is smudged, the OCR may make transcription errors when reading.
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Ordered field
The element − 1 {\displaystyle -1} is not in P . {\displaystyle P.}
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False (logic)
In logic, false or untrue is the state of possessing negative truth value and is a nullary logical connective. In a truth-functional system of propositional logic, it is one of two postulated truth values, along with its negation, truth. Usual notations of the false are 0 (especially in Boolean logic and computer science), O (in prefix notation, Opq), and the up tack symbol ⊥ {\displaystyle \bot } .Another approach is used for several formal theories (e.g., intuitionistic propositional calculus), where a propositional constant (i.e. a nullary connective), ⊥ {\displaystyle \bot } , is introduced, the truth value of which being always false in the sense above. It can be treated as an absurd proposition, and is often called absurdity. |
epfl-collab | Which mode of operation is similar to a stream cipher? | ['OFB', 'ECB', 'CFB', 'CBC'] | A | null | Document 1:::
Cipher-block chaining
In cryptography, a block cipher mode of operation is an algorithm that uses a block cipher to provide information security such as confidentiality or authenticity. A block cipher by itself is only suitable for the secure cryptographic transformation (encryption or decryption) of one fixed-length group of bits called a block. A mode of operation describes how to repeatedly apply a cipher's single-block operation to securely transform amounts of data larger than a block.Most modes require a unique binary sequence, often called an initialization vector (IV), for each encryption operation. The IV has to be non-repeating and, for some modes, random as well.
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Cipher-block chaining
There are, however, modes that do not require padding because they effectively use a block cipher as a stream cipher. Historically, encryption modes have been studied extensively in regard to their error propagation properties under various scenarios of data modification. Later development regarded integrity protection as an entirely separate cryptographic goal. Some modern modes of operation combine confidentiality and authenticity in an efficient way, and are known as authenticated encryption modes.
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Cipher
Without knowledge of the key, it should be extremely difficult, if not impossible, to decrypt the resulting ciphertext into readable plaintext. Most modern ciphers can be categorized in several ways By whether they work on blocks of symbols usually of a fixed size (block ciphers), or on a continuous stream of symbols (stream ciphers). By whether the same key is used for both encryption and decryption (symmetric key algorithms), or if a different key is used for each (asymmetric key algorithms).
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LEX (cipher)
LEX is a stream cipher based on the round transformation of AES. LEX provides the same key agility and short message block performance as AES while handling longer messages faster than AES. In addition, it has the same hardware and software flexibility as AES, and hardware implementations of LEX can share resources with AES implementations.
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QUAD (cipher)
In cryptography, the QUAD cipher is a stream cipher which was designed with provable security arguments in mind. |
epfl-collab | Select the \emph{incorrect} statement. | ['The Discrete Logarithm can be solved in polynomial time on a quantum computer.', 'The Discrete Logarithm is hard to compute for the additive group $\\mathbf{Z}_{n}$.', 'The Computational Diffie-Hellman problem reduces to the Discrete Logarithm problem.', 'The ElGamal cryptosystem is based on the Discrete Logarithm problem.'] | B | null | Document 1:::
Uncertain inference
Rather than retrieving a document that exactly matches the query we should rank the documents based on their plausibility in regards to that query. Since d and q are both generated by users, they are error prone; thus d → q {\displaystyle d\to q} is uncertain. This will affect the plausibility of a given query.
Document 2:::
Uncertain inference
Rather than retrieving a document that exactly matches the query we should rank the documents based on their plausibility in regards to that query. Since d and q are both generated by users, they are error prone; thus d → q {\displaystyle d\to q} is uncertain. This will affect the plausibility of a given query.
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Transposition error
A transcription error is a specific type of data entry error that is commonly made by human operators or by optical character recognition (OCR) programs. Human transcription errors are commonly the result of typographical mistakes; putting one's fingers in the wrong place while touch typing is the easiest way to make this error. Electronic transcription errors occur when the scan of some printed matter is compromised or in an unusual font – for example, if the paper is crumpled, or the ink is smudged, the OCR may make transcription errors when reading.
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Ordered field
The element − 1 {\displaystyle -1} is not in P . {\displaystyle P.}
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False (logic)
In logic, false or untrue is the state of possessing negative truth value and is a nullary logical connective. In a truth-functional system of propositional logic, it is one of two postulated truth values, along with its negation, truth. Usual notations of the false are 0 (especially in Boolean logic and computer science), O (in prefix notation, Opq), and the up tack symbol ⊥ {\displaystyle \bot } .Another approach is used for several formal theories (e.g., intuitionistic propositional calculus), where a propositional constant (i.e. a nullary connective), ⊥ {\displaystyle \bot } , is introduced, the truth value of which being always false in the sense above. It can be treated as an absurd proposition, and is often called absurdity. |
epfl-collab | In Bluetooth, the link key $K_{link}$ is ... | ['used to generate an epheremal key $K_{init}$.', 'the input to the pairing protocol.', 'used to authenticate devices.', 'not used to generate the encryption key.'] | C | null | Document 1:::
Bluetooth Basic Rate/Enhanced Data Rate
Bluetooth is a short-range wireless technology standard that is used for exchanging data between fixed and mobile devices over short distances and building personal area networks (PANs). In the most widely used mode, transmission power is limited to 2.5 milliwatts, giving it a very short range of up to 10 metres (33 ft). It employs UHF radio waves in the ISM bands, from 2.402 GHz to 2.48 GHz.
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Bluetooth Basic Rate/Enhanced Data Rate
It is mainly used as an alternative to wire connections, to exchange files between nearby portable devices and connect cell phones and music players with wireless headphones. Bluetooth is managed by the Bluetooth Special Interest Group (SIG), which has more than 35,000 member companies in the areas of telecommunication, computing, networking, and consumer electronics. The IEEE standardized Bluetooth as IEEE 802.15.1, but no longer maintains the standard.
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Bluetooth Basic Rate/Enhanced Data Rate
The Bluetooth SIG oversees development of the specification, manages the qualification program, and protects the trademarks. A manufacturer must meet Bluetooth SIG standards to market it as a Bluetooth device. A network of patents apply to the technology, which are licensed to individual qualifying devices. As of 2021, 4.7 billion Bluetooth integrated circuit chips are shipped annually.
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Piconet
A piconet is an ad hoc network that links a wireless user group of devices using Bluetooth technology protocols. A piconet consists of two or more devices occupying the same physical channel (synchronized to a common clock and hopping sequence). It allows one master device to interconnect with up to seven active slave devices. Up to 255 further slave devices can be inactive, or parked, which the master device can bring into active status at any time, but an active station must go into parked first. Some examples of piconets include a cell phone connected to a computer, a laptop and a Bluetooth-enabled digital camera, or several PDAs that are connected to each other.
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Fast Pair
The Google Fast Pair Service, or simply Fast Pair, is Google's proprietary standard for quickly pairing Bluetooth devices when they come in close proximity for the first time using Bluetooth Low Energy (BLE). It was announced in October 2017 and initially designed for connecting audio devices such as speakers, headphones and car kits with the Android operating system. In 2018, Google added support for ChromeOS devices, and in 2019, Google announced that Fast Pair connections could now be synced with other Android devices with the same Google Account. |
epfl-collab | Let $n=pq$ where $p$ and $q$ are prime numbers. We have: | ['$\\varphi (n) = n-1$', '$\\varphi (n) = p + q$', '$\\varphi (n) = (p-1) (q-1)$', '$\\varphi (n) = pq$'] | C | null | Document 1:::
Pythagorean prime
A Pythagorean prime is a prime number of the form 4 n + 1 {\displaystyle 4n+1} . Pythagorean primes are exactly the odd prime numbers that are the sum of two squares; this characterization is Fermat's theorem on sums of two squares. Equivalently, by the Pythagorean theorem, they are the odd prime numbers p {\displaystyle p} for which p {\displaystyle {\sqrt {p}}} is the length of the hypotenuse of a right triangle with integer legs, and they are also the prime numbers p {\displaystyle p} for which p {\displaystyle p} itself is the hypotenuse of a primitive Pythagorean triangle. For instance, the number 5 is a Pythagorean prime; 5 {\displaystyle {\sqrt {5}}} is the hypotenuse of a right triangle with legs 1 and 2, and 5 itself is the hypotenuse of a right triangle with legs 3 and 4.
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Proth prime
A Proth number takes the form N = k 2 n + 1 {\displaystyle N=k2^{n}+1} where k and n are positive integers, k {\displaystyle k} is odd and 2 n > k {\displaystyle 2^{n}>k} . A Proth prime is a Proth number that is prime. Without the condition that 2 n > k {\displaystyle 2^{n}>k} , all odd integers larger than 1 would be Proth numbers.
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Partition number
Furthermore p(n) = 0 when n is negative. The first few values of the partition function, starting with p(0) = 1, are: Some exact values of p(n) for larger values of n include: As of June 2022, the largest known prime number among the values of p(n) is p(1289844341), with 40,000 decimal digits. Until March 2022, this was also the largest prime that has been proved using elliptic curve primality proving.
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Blum integer
In mathematics, a natural number n is a Blum integer if n = p × q is a semiprime for which p and q are distinct prime numbers congruent to 3 mod 4. That is, p and q must be of the form 4t + 3, for some integer t. Integers of this form are referred to as Blum primes. This means that the factors of a Blum integer are Gaussian primes with no imaginary part. The first few Blum integers are 21, 33, 57, 69, 77, 93, 129, 133, 141, 161, 177, 201, 209, 213, 217, 237, 249, 253, 301, 309, 321, 329, 341, 381, 393, 413, 417, 437, 453, 469, 473, 489, 497, ... (sequence A016105 in the OEIS)The integers were named for computer scientist Manuel Blum.
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Proth number
A Proth number is a natural number N of the form N = k × 2 n + 1 {\displaystyle N=k\times 2^{n}+1} where k and n are positive integers, k is odd and 2 n > k {\displaystyle 2^{n}>k} . A Proth prime is a Proth number that is prime. They are named after the French mathematician François Proth. The first few Proth primes are 3, 5, 13, 17, 41, 97, 113, 193, 241, 257, 353, 449, 577, 641, 673, 769, 929, 1153, 1217, 1409, 1601, 2113, 2689, 2753, 3137, 3329, 3457, 4481, 4993, 6529, 7297, 7681, 7937, 9473, 9601, 9857 (OEIS: A080076).It is still an open question whether an infinite number of Proth primes exist. It was shown in 2022 that the reciprocal sum of Proth primes converges to a real number near 0.747392479, substantially less than the value of 1.093322456 for the reciprocal sum of Proth numbers.The primality of Proth numbers can be tested more easily than many other numbers of similar magnitude. |
epfl-collab | Which of the following elements belongs to $\mathbb{Z}_{78}^*$? | ['35', '65', '46', '21'] | A | null | Document 1:::
Abell 78
Abell 78 is a planetary nebula located in the constellation of Cygnus. It has a fainter halo consisting mostly hydrogen, and an inner elliptical ring that is mostly made of helium. The central star of the planetary nebula has a spectral type of , similar to that of a carbon-rich Wolf–Rayet star.
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Zinc finger protein 780a
Zinc finger protein 780A is a protein that in humans is encoded by the ZNF780A gene. == References ==
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Z* theorem
In mathematics, George Glauberman's Z* theorem is stated as follows: Z* theorem: Let G be a finite group, with O(G) being its maximal normal subgroup of odd order. If T is a Sylow 2-subgroup of G containing an involution not conjugate in G to any other element of T, then the involution lies in Z*(G), which is the inverse image in G of the center of G/O(G). This generalizes the Brauer–Suzuki theorem (and the proof uses the Brauer–Suzuki theorem to deal with some small cases).
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Ring of all algebraic integers
In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients are integers. The set of all algebraic integers A is closed under addition, subtraction and multiplication and therefore is a commutative subring of the complex numbers. The ring of integers of a number field K, denoted by OK, is the intersection of K and A: it can also be characterised as the maximal order of the field K. Each algebraic integer belongs to the ring of integers of some number field. A number α is an algebraic integer if and only if the ring Z {\displaystyle \mathbb {Z} } is finitely generated as an abelian group, which is to say, as a Z {\displaystyle \mathbb {Z} } -module.
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Kepler-78
Kepler-78 (formerly known as KIC 8435766) is a 12th magnitude star 407 light-years (125 parsecs) away in the constellation Cygnus. Initially classified as an eclipsing binary with orbital period 0.710015 days, it was later re-classified as a single star with significant interaction between star magnetosphere and close-in planet. The radius of the star is of about 74% of the Sun, and the effective temperature is about 5100 K. |
epfl-collab | Tick the \textbf{false} statement. Moore's Law ... | ['implies that the heat generated by transistors of CPU doubles every 18 months.', 'assumes the number of transistors per CPU increases exponentially fast with time.', 'is partly a reason why some existing cryptosystems are insecure.', 'was stated by the founder of Intel.'] | A | null | Document 1:::
Computational power
Moore's law is the observation that the number of transistors in an integrated circuit (IC) doubles about every two years. Moore's law is an observation and projection of a historical trend. Rather than a law of physics, it is an empirical relationship linked to gains from experience in production. The observation is named after Gordon Moore, the co-founder of Fairchild Semiconductor and Intel (and former CEO of the latter), who in 1965 posited a doubling every year in the number of components per integrated circuit, and projected this rate of growth would continue for at least another decade.
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Computational power
Industry experts have not reached a consensus on exactly when Moore's law will cease to apply. Microprocessor architects report that semiconductor advancement has slowed industry-wide since around 2010, slightly below the pace predicted by Moore's law. In September 2022 Nvidia CEO Jensen Huang considered Moore's law dead, while Intel CEO Pat Gelsinger was of the opposite view.
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Computational power
Advancements in digital electronics, such as the reduction in quality-adjusted microprocessor prices, the increase in memory capacity (RAM and flash), the improvement of sensors, and even the number and size of pixels in digital cameras, are strongly linked to Moore's law. These ongoing changes in digital electronics have been a driving force of technological and social change, productivity, and economic growth.
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Moore's second law
Rock's law or Moore's second law, named for Arthur Rock or Gordon Moore, says that the cost of a semiconductor chip fabrication plant doubles every four years. As of 2015, the price had already reached about 14 billion US dollars.Rock's law can be seen as the economic flip side to Moore's (first) law – that the number of transistors in a dense integrated circuit doubles every two years. The latter is a direct consequence of the ongoing growth of the capital-intensive semiconductor industry— innovative and popular products mean more profits, meaning more capital available to invest in ever higher levels of large-scale integration, which in turn leads to the creation of even more innovative products.The semiconductor industry has always been extremely capital-intensive, with ever-dropping manufacturing unit costs. Thus, the ultimate limits to growth of the industry will constrain the maximum amount of capital that can be invested in new products; at some point, Rock's Law will collide with Moore's Law.It has been suggested that fabrication plant costs have not increased as quickly as predicted by Rock's law – indeed plateauing in the late 1990s – and also that the fabrication plant cost per transistor (which has shown a pronounced downward trend) may be more relevant as a constraint on Moore's Law.
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Moore Machine
In the theory of computation, a Moore machine is a finite-state machine whose current output values are determined only by its current state. This is in contrast to a Mealy machine, whose output values are determined both by its current state and by the values of its inputs. Like other finite state machines, in Moore machines, the input typically influences the next state. Thus the input may indirectly influence subsequent outputs, but not the current or immediate output. The Moore machine is named after Edward F. Moore, who presented the concept in a 1956 paper, “Gedanken-experiments on Sequential Machines.” |
epfl-collab | The elements of $\mathbf{Z}_{14}^*$ are | ['$\\{ 1, 2, 3, 9, 11 \\}$', '$\\{ 1, 3, 5, 9, 11, 13\\}$', '$\\{ 0, 1, 3, 5, 9, 11, 13\\}$', '$\\{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13\\}$'] | B | null | Document 1:::
Z* theorem
In mathematics, George Glauberman's Z* theorem is stated as follows: Z* theorem: Let G be a finite group, with O(G) being its maximal normal subgroup of odd order. If T is a Sylow 2-subgroup of G containing an involution not conjugate in G to any other element of T, then the involution lies in Z*(G), which is the inverse image in G of the center of G/O(G). This generalizes the Brauer–Suzuki theorem (and the proof uses the Brauer–Suzuki theorem to deal with some small cases).
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14 Trianguli
14 Tri is slightly metal-deficient with = −0.16, and spins modestly with a projected rotational velocity of 2.1 km/s. This is a single-lined spectroscopic binary that completes an eccentric orbit within 17 years. The secondary star has not been detected visually or in the spectrum and is expected to be a low-mass red dwarf or white dwarf. 14 Tri may be part of the Wolf 630 moving group.
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14 Trianguli
14 Trianguli (14 Tri), also known as HD 15656, is a spectroscopic binary located in the northern constellation Triangulum. It has an apparent magnitude of 5.14, making it faintly visible to the naked eye in ideal conditions. Gaia DR3 parallax measurements place the system 433 light years away, and it is currently approaching the Solar System with a heliocentric radial velocity of −37 km/s. At its current distance, 14 Tri's brightness is diminished by 0.21 magnitude due to interstellar dust.
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Trivalent group 14 radicals
A trivalent group 14 radical (also known as a trivalent tetrel radical) is a molecule that contains a group 14 element (E = C, Si, Ge, Sn, Pb) with three bonds and a free radical, having the general formula of R3E•. Such compounds can be categorized into three different types, depending on the structure (or equivalently the orbital in which the unpaired electron resides) and the energetic barrier to inversion. A molecule that remains rigidly in a pyramidal structure has an electron in a sp3 orbital is denoted as Type A. A structure that is pyramidal, but flexible, is denoted as Type B. And a planar structure with an electron that typically would reside in a pure p orbital is denoted as Type C. The structure of such molecules has been determined by probing the nature of the orbital that the unpaired electron resides in using spectroscopy, as well as directly with X-ray methods. Trivalent tetrel radicals tend to be synthesized from their tetravalent counterparts (i.e. R3EY where Y is a species that will dissociate).
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L1014
L1014 is a dark nebula in the Cygnus constellation. It may be among the most centrally condensed small dark clouds known, perhaps indicative of the earliest stages of star formation processes. This cloud harbors at its core a very young low-mass star named L1014 IRS; some astronomers have suggested that this object may be a brown dwarf or even a rogue planet at the earliest stage of its lifetime. |
epfl-collab | Tick the \textbf{false} statement. | ['RSA can be accelerated by using CRT (Chinese Remainder Theorem).', 'The CRT states $\\mathbb{Z}_{mn} \\equiv \\mathbb{Z}_{m} \\cup \\mathbb{Z}_{n}$.', 'The CRT implies $\\varphi(mn)=\\varphi(m)\\varphi(n)$ for $\\mathsf{gcd}(m,n)=1$.', 'An isomorphism is defined as a bijective homomorphism.'] | B | null | Document 1:::
False (logic)
In logic, false or untrue is the state of possessing negative truth value and is a nullary logical connective. In a truth-functional system of propositional logic, it is one of two postulated truth values, along with its negation, truth. Usual notations of the false are 0 (especially in Boolean logic and computer science), O (in prefix notation, Opq), and the up tack symbol ⊥ {\displaystyle \bot } .Another approach is used for several formal theories (e.g., intuitionistic propositional calculus), where a propositional constant (i.e. a nullary connective), ⊥ {\displaystyle \bot } , is introduced, the truth value of which being always false in the sense above. It can be treated as an absurd proposition, and is often called absurdity.
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Boolean expression
In computer science, a Boolean expression is an expression used in programming languages that produces a Boolean value when evaluated. A Boolean value is either true or false. A Boolean expression may be composed of a combination of the Boolean constants true or false, Boolean-typed variables, Boolean-valued operators, and Boolean-valued functions.Boolean expressions correspond to propositional formulas in logic and are a special case of Boolean circuits.
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Boolean flag
A Boolean flag, truth bit or truth flag in computer science is a Boolean value represented as one or more bits, which encodes a state variable with two possible values.
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Boolean domain
The initial object in the category of bounded lattices is a Boolean domain. In computer science, a Boolean variable is a variable that takes values in some Boolean domain. Some programming languages feature reserved words or symbols for the elements of the Boolean domain, for example false and true. However, many programming languages do not have a Boolean datatype in the strict sense. In C or BASIC, for example, falsity is represented by the number 0 and truth is represented by the number 1 or −1, and all variables that can take these values can also take any other numerical values.
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Boolean-valued function
A Boolean-valued function (sometimes called a predicate or a proposition) is a function of the type f: X → B, where X is an arbitrary set and where B is a Boolean domain, i.e. a generic two-element set, (for example B = {0, 1}), whose elements are interpreted as logical values, for example, 0 = false and 1 = true, i.e., a single bit of information. In the formal sciences, mathematics, mathematical logic, statistics, and their applied disciplines, a Boolean-valued function may also be referred to as a characteristic function, indicator function, predicate, or proposition. In all of these uses, it is understood that the various terms refer to a mathematical object and not the corresponding semiotic sign or syntactic expression. In formal semantic theories of truth, a truth predicate is a predicate on the sentences of a formal language, interpreted for logic, that formalizes the intuitive concept that is normally expressed by saying that a sentence is true. A truth predicate may have additional domains beyond the formal language domain, if that is what is required to determine a final truth value. |
epfl-collab | What is the advantage of using a salt in a password authentication protocol? | ['It protects against online attacks.', 'It avoids single-target exhaustive search attacks from the database.', 'It avoids multi-target bruteforce attacks from the database.', 'It makes the protocol more spicy.'] | C | null | Document 1:::
Salt (cryptography)
In cryptography, a salt is random data fed as an additional input to a one-way function that hashes data, a password or passphrase. Salting helps defend against attacks that use precomputed tables (e.g. rainbow tables), by vastly growing the size of table needed for a successful attack. It also helps protect passwords that occur multiple times in a database, as a new salt is used for each password instance.
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Salt (cryptography)
Additionally, salting does not place any burden on users. The way salting is typically done is that a new salt is randomly generated for each password. The salt and the password (or its version after key stretching) are concatenated and fed to a cryptographic hash function, and the output hash value is then stored with the salt in a database. The salt need not be encrypted, because knowing the salt would not help the attacker.Salting is broadly used in cybersecurity, from Unix system credentials to Internet security. Salts are related to cryptographic nonces.
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Rainbow tables
A common defense against this attack is to compute the hashes using a key derivation function that adds a "salt" to each password before hashing it, with different passwords receiving different salts, which are stored in plain text along with the hash. Rainbow tables are a practical example of a space–time tradeoff: they use less computer processing time and more storage than a brute-force attack which calculates a hash on every attempt, but more processing time and less storage than a simple table that stores the hash of every possible password. Rainbow tables were invented by Philippe Oechslin as an application of an earlier, simpler algorithm by Martin Hellman.
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Crypt (C)
crypt is a POSIX C library function. It is typically used to compute the hash of user account passwords. The function outputs a text string which also encodes the salt (usually the first two characters are the salt itself and the rest is the hashed result), and identifies the hash algorithm used (defaulting to the "traditional" one explained below). This output string forms a password record, which is usually stored in a text file. More formally, crypt provides cryptographic key derivation functions for password validation and storage on Unix systems.
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Pass the hash
In computer security, pass the hash is a hacking technique that allows an attacker to authenticate to a remote server or service by using the underlying NTLM or LanMan hash of a user's password, instead of requiring the associated plaintext password as is normally the case. It replaces the need for stealing the plaintext password to gain access with stealing the hash. The attack exploits an implementation weakness in the authentication protocol, where password hashes remain static from session to session until the password is next changed. This technique can be performed against any server or service accepting LM or NTLM authentication, whether it runs on a machine with Windows, Unix, or any other operating system. |
epfl-collab | Select \emph{incorrect} statement. The birthday paradox | ['implies that majority of people is born at full moon.', 'implies that in a list of $\\Theta\\sqrt{N}$ random numbers from $\\mathbb{Z}_N$ we have at least one number twice with probability $1- e^{-{\\Theta^2\\over 2}}$.', 'can be used to find collisions in hash function.', 'implies that in class of $23$ students we have two student with same birthday with approximately $50\\%$ probability.'] | A | null | Document 1:::
Artificial precision
In numerical mathematics, artificial precision is a source of error that occurs when a numerical value or semantic is expressed with more precision than was initially provided from measurement or user input. For example, a person enters their birthday as the date 1984-01-01 but it is stored in a database as 1984-01-01T00:00:00Z which introduces the artificial precision of the hour, minute, and second they were born, and may even affect the date, depending on the user's actual place of birth. This is also an example of false precision, which is artificial precision specifically of numerical quantities or measures.
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Kleene–Rosser paradox
In mathematics, the Kleene–Rosser paradox is a paradox that shows that certain systems of formal logic are inconsistent, in particular the version of Haskell Curry's combinatory logic introduced in 1930, and Alonzo Church's original lambda calculus, introduced in 1932–1933, both originally intended as systems of formal logic. The paradox was exhibited by Stephen Kleene and J. B. Rosser in 1935.
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False (logic)
In logic, false or untrue is the state of possessing negative truth value and is a nullary logical connective. In a truth-functional system of propositional logic, it is one of two postulated truth values, along with its negation, truth. Usual notations of the false are 0 (especially in Boolean logic and computer science), O (in prefix notation, Opq), and the up tack symbol ⊥ {\displaystyle \bot } .Another approach is used for several formal theories (e.g., intuitionistic propositional calculus), where a propositional constant (i.e. a nullary connective), ⊥ {\displaystyle \bot } , is introduced, the truth value of which being always false in the sense above. It can be treated as an absurd proposition, and is often called absurdity.
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Temperature paradox
Therefore, ninety is rising. (invalid conclusion)Despite its obvious invalidity, this argument would be valid in most formalizations based on traditional extensional systems of logic.
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Leap year problem
The leap year problem (also known as the leap year bug or the leap day bug) is a problem for both digital (computer-related) and non-digital documentation and data storage situations which results from errors in the calculation of which years are leap years, or from manipulating dates without regard to the difference between leap years and common years. |
epfl-collab | Which scheme is the most secure? | ['DES.', 'Three-key triple DES.', 'Double DES.', 'Two-key triple DES.'] | B | null | Document 1:::
Asymptotic security
In cryptography, concrete security or exact security is a practice-oriented approach that aims to give more precise estimates of the computational complexities of adversarial tasks than polynomial equivalence would allow. It quantifies the security of a cryptosystem by bounding the probability of success for an adversary running for a fixed amount of time. Security proofs with precise analyses are referred to as concrete.Traditionally, provable security is asymptotic: it classifies the hardness of computational problems using polynomial-time reducibility. Secure schemes are defined to be those in which the advantage of any computationally bounded adversary is negligible.
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Indistinguishability (cryptography)
A cryptosystem is considered secure in terms of indistinguishability if no adversary, given an encryption of a message randomly chosen from a two-element message space determined by the adversary, can identify the message choice with probability significantly better than that of random guessing (1⁄2). If any adversary can succeed in distinguishing the chosen ciphertext with a probability significantly greater than 1⁄2, then this adversary is considered to have an "advantage" in distinguishing the ciphertext, and the scheme is not considered secure in terms of indistinguishability. This definition encompasses the notion that in a secure scheme, the adversary should learn no information from seeing a ciphertext. Therefore, the adversary should be able to do no better than if it guessed randomly.
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Cryptosystem
In cryptography, a cryptosystem is a suite of cryptographic algorithms needed to implement a particular security service, such as confidentiality (encryption).Typically, a cryptosystem consists of three algorithms: one for key generation, one for encryption, and one for decryption. The term cipher (sometimes cypher) is often used to refer to a pair of algorithms, one for encryption and one for decryption. Therefore, the term cryptosystem is most often used when the key generation algorithm is important. For this reason, the term cryptosystem is commonly used to refer to public key techniques; however both "cipher" and "cryptosystem" are used for symmetric key techniques.
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Hybrid cryptosystem
This is addressed by hybrid systems by using a combination of both.A hybrid cryptosystem can be constructed using any two separate cryptosystems: a key encapsulation mechanism, which is a public-key cryptosystem a data encapsulation scheme, which is a symmetric-key cryptosystemThe hybrid cryptosystem is itself a public-key system, whose public and private keys are the same as in the key encapsulation scheme.Note that for very long messages the bulk of the work in encryption/decryption is done by the more efficient symmetric-key scheme, while the inefficient public-key scheme is used only to encrypt/decrypt a short key value.All practical implementations of public key cryptography today employ the use of a hybrid system. Examples include the TLS protocol and the SSH protocol, that use a public-key mechanism for key exchange (such as Diffie-Hellman) and a symmetric-key mechanism for data encapsulation (such as AES). The OpenPGP file format and the PKCS#7 file format are other examples.
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Probabilistic signature scheme
Probabilistic Signature Scheme (PSS) is a cryptographic signature scheme designed by Mihir Bellare and Phillip Rogaway.RSA-PSS is an adaptation of their work and is standardized as part of PKCS#1 v2.1. In general, RSA-PSS should be used as a replacement for RSA-PKCS#1 v1.5. |
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