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  Note: as indicated in the formulae, C(x, t) is constant. The intent is to make the contribution to the accumulated absolute gradients (see below) constant for all timesteps. As C(x, t) = L_0(x, t), this means L = 1. However, expressing L in this way overlooks the way the gradients are computed. I think the formulation with the expected value is less ambiguous.
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- from the DDPM paper (https://arxiv.org/abs/2006.11239):
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  $$
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  x_t = \sqrt{\bar{\alpha}_t} x + \sqrt{1 - \bar{\alpha}_t} \epsilon \\
 
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  Note: as indicated in the formulae, C(x, t) is constant. The intent is to make the contribution to the accumulated absolute gradients (see below) constant for all timesteps. As C(x, t) = L_0(x, t), this means L = 1. However, expressing L in this way overlooks the way the gradients are computed. I think the formulation with the expected value is less ambiguous.
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+ x_t and snr_t were taken from the DDPM paper (https://arxiv.org/abs/2006.11239):
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  $$
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  x_t = \sqrt{\bar{\alpha}_t} x + \sqrt{1 - \bar{\alpha}_t} \epsilon \\