Anderson’s result states that for a forward SDE of the form:

$$dx = f(x,t)dt + g(t)dB_t$$

The corresponding reverse SDE takes the form:

$$dx = \left(f(x,t) - g^2(t)\nabla_x(\log p_t(x))\right)dt + g(t)dB_t$$

where:

  • $f(x,t)$ is the drift term of the forward process
  • $g(t)$ is the diffusion coefficient
  • $\nabla_x(\log p_t(x))$ is the score function
  • The reverse process has the same diffusion term $g(t)dB_t$ as the forward process

This result shows that to reverse a diffusion process, we need to:

  1. Keep the same diffusion term
  2. Modify the drift term by subtracting the score matching term scaled by $g^2(t)$