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I have a question about forward process in DDPM. It is described as we sample our image from some distribution: $x_0\sim{q(x)}$ then in each time stamp $T$ we are applying gaussian noise $\epsilon\sim\mathcal{N}(0,1)$ and we can describe this as $q(x_{t}|x_{t-1}) =\mathcal{N}(x_{t},\sqrt{1-\beta_{t}}x_t,\beta_{t}I)$ ($\beta{t}$ is some scalar - not important yet).My question is how do we know that our new image at time step $T$ $x_{t}$ is from normal distribution? I know that we can add two normal variables by adding their mean and their variances (assumption: independent variables) but as far as I understand we don't have assumption about $x_0$ distribution, we don't know if is it normal distribution and we don't know mean and variance.

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  • $\begingroup$ a Gaussian random walk from any point will have a distribution of points that is Gaussian $\endgroup$
    – Alberto
    Commented Nov 21, 2023 at 13:15

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