DDPM Reverse Sampling

The problem, from first principles

To generate an image, start from pure noise x_T and denoise one step at a time, from T down to 1. A DDPM reverse step takes the current noisy image x_t, asks the network for its noise estimate, and computes the mean of a Gaussian over the slightly less noisy x_{t-1}. The mean subtracts the predicted noise, scaled by a factor that depends on the schedule: mu = (x_t - beta_t / sqrt(1 - ab_t) * eps) / sqrt(alpha_t) where alpha_t = 1 - beta_t. Then a bit of fresh Gaussian noise with standard deviation sqrt(beta_t) is added back, except at the very last step where the output should be the clean sample. That injected noise makes DDPM sampling stochastic.

From theory to code

Implement ddpm_step.

Constraints

• ddpm_step(x_t, eps_pred, t, betas, alpha_bars, z) uses 0-based t (index into betas and alpha_bars), with alpha_t = 1 - betas[t].
• The mean is mu = (x_t - betas[t] / sqrt(1 - alpha_bars[t]) * eps_pred) / sqrt(alpha_t).
• If t > 0 return mu + sqrt(betas[t]) * z. If t == 0 return mu with no noise added.
• z is standard Gaussian noise of the same shape as x_t, passed in so the step is deterministic.

Hints