DDPM Ancestral Sampling

For the discrete DDPM Forward Process, the reverse step conditioned on a clean image is Gaussian with closed-form parameters:

$$q(x_{t-1}\mid x_t,x_0)=\mathcal{N}({\tilde{\mu }}_t(x_t,x_0),{\tilde{\beta }}_{t}I),$$ $${\tilde{\mu }}_t=\frac{\sqrt{{\alpha }_t}(1-{\bar{\alpha }}_{t-1})}{1-{\bar{\alpha }}_t}{x}_t+\frac{\sqrt{{\bar{\alpha }}_{t-1}}{\beta }_t}{1-{\bar{\alpha }}_t}{x}_0,{\tilde{\beta }}_t=\frac{1-{\bar{\alpha }}_{t-1}}{1-{\bar{\alpha }}_t}{\beta }_t.$$

Ancestral sampling. $x_0$ is unknown during generation, so it is replaced by the denoised estimate from Tweedie’s Formula, ${\hat{x}}_0=(x_t-\sqrt{1-{\bar{\alpha }}_t}{\epsilon }_{\theta }(x_t,t))/\sqrt{{\bar{\alpha }}_t}$. Then:

x_T ← sample N(0, I)
for t = T, …, 1:
    ε̂  ← ε_θ(x_t, t)
    x̂0 ← (x_t − sqrt(1−ᾱ_t) ε̂) / sqrt(ᾱ_t)
    μ  ← sqrt(α_t)(1−ᾱ_{t−1})/(1−ᾱ_t) · x_t + sqrt(ᾱ_{t−1}) β_t/(1−ᾱ_t) · x̂0
    x_{t−1} ← μ + sqrt(β̃_t) · z,   z ~ N(0, I)   (z = 0 at t = 1)
return x_0

Substituting ${\hat{x}}_0$ gives the equivalent DDPM form $\mu =\frac{1}{\sqrt{{\alpha }_t}}(x_t-\frac{{\beta }_t}{\sqrt{1-{\bar{\alpha }}_t}}{\epsilon }_{\theta })$. For small ${\beta }_t$ this step agrees with an Euler–Maruyama step of the Reverse-Time SDE to first order.

This step is the backbone that Diffusion Posterior Sampling augments with a likelihood gradient.

References

1. J. Ho, A. Jain, P. Abbeel (2020). Denoising Diffusion Probabilistic Models. NeurIPS 33. arXiv:2006.11239