DDPM Reverse (Denoise) Step

Background

DDPM (Denoising Diffusion Probabilistic Models) sample by iteratively denoising: given the current noisy $x_t$ and the model's noise prediction $\hat\epsilon$, compute $x_{t-1}$ using the closed-form reverse update. Apply this $T$ to $1$ to sample from the model. This problem isolates one reverse step.

Problem statement

Implement ddpm_reverse_step(x_t, eps_theta, t, betas, seed=None) using:

$$x_{t-1} \;=\; \frac{1}{\sqrt{\alpha_t}}\Bigl(x_t - \frac{1-\alpha_t}{\sqrt{1-\bar\alpha_t}}\,\hat\epsilon\Bigr) + \sigma_t z,\quad z \sim \mathcal N(0, I)$$

where $\alpha_t = 1 - \beta_t$, $\bar\alpha_t = \prod_{s \le t}\alpha_s$, and $\sigma_t = \sqrt{\beta_t}$. At $t = 0$, no noise is added (deterministic).

Input

• x_t — np.ndarray of any shape, the current noisy sample.
• eps_theta — np.ndarray same shape, the model's noise prediction.
• t — int >= 0, the current diffusion step (in betas index).
• betas — 1-D array of $\beta_t$ values for $t = 0, 1, \dots, T-1$.
• seed — optional int for reproducibility.

Output

Returns np.ndarray of the same shape as x_t.

Examples

Example 1 — shape preserved

Input:  x_t=(3, 4) array, eps=(3, 4) array, betas=length-20, t=5
Output: (3, 4) array

Example 2 — t=0 is deterministic

Input:  any x, t=0
Output: identical across seeds (no noise term added)

Constraints

• t must be a valid index into betas.
• Use np.random.default_rng(seed) for the noise term so output is reproducible.
• At t = 0, return the mean term only.

Notes

• Variance scheduler. $\sigma_t = \sqrt{\beta_t}$ is the simpler choice; some implementations use $\tilde\beta_t = \frac{1-\bar\alpha_{t-1}}{1-\bar\alpha_t}\beta_t$. Either gives valid samples.
• Iterating. A full sampler calls this $T$ times starting from $x_T \sim \mathcal N(0, I)$.