DDPM

DDPM — Denoising Diffusion Probabilistic Models — defines a latent variable generative model through a pair of Markov chains. The forward (noising) chain progressively corrupts data $x_0 \sim q(x_0)$ over $T$ steps with a fixed Gaussian schedule $\{\beta_t\}_{t=1}^{T}$:

With $\alpha_t = 1 - \beta_t$ and $\bar{\alpha}_t = \prod_{s=1}^{t} \alpha_s$, the marginal admits a closed form that supports cheap one-shot sampling at any step:

The reverse (denoising) chain is parameterised by a U-Net $\epsilon_\theta(x_t, t)$ trained to predict the injected noise. Ho et al. show that the variational ELBO

decomposes into per-timestep KL divergences which, after a reweighting, simplify to the deceptively elegant noise-prediction objective

Sampling iterates $t = T, T-1, \dots, 1$ via $x_{t-1} = \tfrac{1}{\sqrt{\alpha_t}} \big(x_t - \tfrac{\beta_t}{\sqrt{1 - \bar{\alpha}_t}}\, \epsilon_\theta(x_t, t)\big) + \sigma_t z$. The U-Net itself follows the original architecture (Ronneberger 2015) augmented with sinusoidal timestep embeddings, self-attention at low-resolution feature maps (default $16 \times 16$), and GroupNorm-equipped residual blocks — the now-canonical recipe inherited by ADM, Imagen, Stable Diffusion, and beyond.