Expectation_X0

[ryancll]

Thx for the amazing work!

When you calculating the Expectation_X0 and Tr_Cov_d for Analytic-Init, did you assume the pixel value for X0 follow a normal distribution?

[zhuhz22]

Hi @ryancll ,
We make no assumptions about the distribution of the true data X_0. We assume that the prior distribution in the generation process is a normal distribution, but not a standard Gaussian. To minimize the KL divergence between this prior distribution and the true distribution, we derive the analytical form of its mean and variance.

[ryancll]

@zhuhz22 Thank you for your response. The assumption makes sense to me.
While, under this assumption, I'm not sure whether the effect of Analytic-Init would decrease as the latent resolution increases. According to central limit theorem (CLT), I would guess the higher the data resolution, the less likely the prior distribution is noraml. I wonder if there is a more general form of this analytical process that works well under any form of prior distribution.

[zhuhz22]

Hi @ryancll ,thank you for your suggestions.Mathematically,we may have a more complex analytic form of the prior,and the more complex it is,the smaller the minimal KL divergence will be. However,the more complex the prior distribution is, the greater the computational time complexity is, and the lower the computational efficiency is.For example,if we adpot a normal distribution $N(\mu,\Sigma)$,where $\Sigma$ isn't a scalar matrix,it could be tough to sample from this prior distribution.