generative

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 $\{\beta_t\}_{t=1}^{T}$:

NCSN — Noise-Conditional Score Network — is the score-based counterpart of DDPM. Rather than predicting noise inside a discrete Markov chain, it directly learns the (Stein) score function $s_\theta(x, \sigma) \approx \nabla_x \log p_\sigma(x)$, where $p_\sigma$ is the data distribution convolved with Gaussian noise of standard deviation $\sigma$.

The Variational Auto-Encoder defines a latent-variable generative model $p_\theta(x, z) = p_\theta(x \mid z)\, p(z)$ with a standard-normal prior $p(z) = \mathcal{N}(0, \mathbf{I})$ and a neural decoder $p_\theta(x \mid z)$. Because the marginal likelihood $p_\theta(x) = \int p_\theta(x \mid z)\, p(z)\, dz$ is intractable, training maximises the evidence lower bound (ELBO) instead: