10.3.1. Diffusion Models

Denoising Diffusion

A denoising diffusion model is a generative model that learns to reverse a gradual noising process. It defines a forward diffusion that transforms data $x_0$ into pure noise through a sequence of Gaussian transitions, and a reverse denoising process that reconstructs samples by learning the score (i.e., the gradient of the log-density of noisy data). The generative model is thus a learned approximation of the time-reversal of diffusion.

The forward process $\{x_t{\}}_{t=0}^T$ is a Markov chain that incrementally adds Gaussian noise:
$q(x_t\mid x_{t-1})=\mathcal{N}\left(\sqrt{1-{\beta }_t}{x}_{t-1},{\beta }_{t}I\right),$
where the variance schedule ${\beta }_1,\dots ,{\beta }_T$ controls how rapidly signal is destroyed. Each step slightly shrinks the signal via $\sqrt{1-{\beta }_t}$ and adds isotropic noise ${\beta }_{t}I$, where $I$ is the identity matrix. Because each transition is linear-Gaussian, the marginal distribution conditioned on the original clean data $x_0$ has a closed form:
$q(x_t\mid x_0)=\mathcal{N}\left(\sqrt{{\bar{\alpha }}_t}{x}_0,\sqrt{1-{\bar{\alpha }}_t}I\right),$
where ${\bar{\alpha }}_t={\prod }_{s=1}^t(1-{\beta }_s)$. This formula expresses a mixture of preserved signal $(\sqrt{{\bar{\alpha }}_t}{x}_0)$ and accumulated noise $(1-{\bar{\alpha }}_t)I$. As $t\to T$, nearly all signal disappears and $x_T\approx \mathcal{N}(0,I)$, yielding a tractable prior distribution for generative sampling.

The generative model then defines a learned approximation to each reverse transition:
$p_{\theta }(x_{t-1}\mid x_t)=\mathcal{N}\left({\mu }_{\theta }(x_t,t),{\Sigma }_t\right),$
where a neural network (often a U-Net or Transformer) predicts the mean ${\mu }_{\theta }(x_t,t)$ or equivalently the added noise ${ϵ}_{\theta }(x_t,t)$.

A sampling procedure generates new data by running this learned reverse process starting from pure Gaussian noise. Since the model cannot reconstruct a clean sample in one step, it denoises gradually: begin with pure noise $x_T\sim \mathcal{N}(0,I)$ and iteratively apply the reverse transitions
$x_T\sim \mathcal{N}(0,I),x_{t-1}\sim p_{\theta }(x_{t-1}\mid x_t),t=T,\dots ,1,$
yielding $x_0$ as the generated sample.

Diffusion models are trained by predicting the noise used in the forward process:
${\mathcal{L}}_{\text{DM}}={\mathbb{E}}_{x_0,ϵ,t}\left[\Vert ϵ-{ϵ}_{\theta }(\sqrt{{\bar{\alpha }}_t}{x}_0+\sqrt{1-{\bar{\alpha }}_t}ϵ,t){\Vert }^2\right],$
a simple regression objective that approximates ${\nabla }_{x_t}\log q(x_t)$, the score of the noisy data distribution.