Reverse SDE
Chapter 19: Stochastic Differential Equations — Time Reversal & Reverse-Time SDE Dynamics
From the book
Chapter 19: Stochastic Differential Equations. In the chapter mind map this icon labels Time Reversal & Reverse-Time SDE Dynamics. The discussion below is excerpted and lightly edited from § Reverse-Time SDE in Mathematics for AI and Machine Learning.
Under mild conditions (smoothness of drift and diffusion, non-degeneracy of diffusion), the reverse process satisfies (in the common state-independent diffusion case where $\mathbf{G}$ depends only on $t$, so $\mathbf{a}(t)=\mathbf{G}(t)\mathbf{G}(t)^\top$):
where $d\bar{\mathbf{W}}_t$ is a Brownian motion running backward in time.
Key observation: The reverse drift has an extra term $-\mathbf{G}(t)\mathbf{G}(t)^\top \nabla_{\mathbf{x}}\log p_t(\mathbf{X}_t)$ that depends on the score function. This term corrects for the asymmetry introduced by the forward diffusion.
What this drawing shows
What you see. Time-reversed SDE denoises a sample from noise (gray, right) back to data (blue, left); animation traces the state along the red reverse arrow opposite the black forward corruption path.
In the mind map. Chapter 19 — Time Reversal & Reverse-Time SDE Dynamics. See From the book above for definitions, figures, and worked examples.
Where to read next
Read the full definitions, figures, and worked examples in Chapter 19: Stochastic Differential Equations — see the mind-map node Time Reversal & Reverse-Time SDE Dynamics.