Reverse-Time SDE

Theorem (Anderson 1982). If $x(t)$ solves the forward SDE

$$\mathrm{d}x=f(x,t)\mathrm{d}t+g(t)\mathrm{d}{W}_t$$

with marginal densities $p_t$, then the time-reversed process solves

$$\mathrm{d}x=[f(x,t)-g(t{)}^2{\nabla }_x\log p_t(x)]\mathrm{d}t+g(t)\mathrm{d}{\bar{W}}_t,$$

where time runs backwards from $T$ to $0$ ($\mathrm{d}t<0$) and $\bar{W}$ is a Brownian motion in reverse time. Starting from $x(T)\sim p_T$ and integrating to $t=0$ produces samples of $p_0=p_{\text{data}}$.

For the Variance-Preserving SDE ($f=-\frac{1}{2}\beta x$, $g=\sqrt{\beta }$) with the learned score $s_{\theta }=-{\epsilon }_{\theta }/\sqrt{1-\bar{\alpha }}$:

$$\mathrm{d}x=[-\frac{1}{2}\beta (t)x+\frac{\beta (t)}{\sqrt{1-\bar{\alpha }(t)}}{\epsilon }_{\theta }(x,t)]\mathrm{d}t+\sqrt{\beta (t)}\mathrm{d}{\bar{W}}_t.$$

The score term pulls the sample towards the data distribution. Because $\mathrm{d}t<0$, the backward step moves against the predicted noise.

It is discretised with the Euler-Maruyama Method or with DDPM’s exact Gaussian step (see DDPM Ancestral Sampling). The deterministic counterpart with the same marginals is the Probability Flow ODE.

References

1. B. D. O. Anderson (1982). Reverse-time diffusion equation models. Stoch. Proc. Appl. 12(3), 313–326. doi:10.1016/0304-4149(82)90051-5
2. Y. Song et al. (2021). Score-Based Generative Modeling through Stochastic Differential Equations. ICLR 2021. arXiv:2011.13456