Posterior Sampling by Combining Diffusion Models with Annealed Langevin Dynamics

TL;DR

This paper introduces a novel method combining diffusion models with annealed Langevin dynamics to enable efficient posterior sampling from log-concave distributions using only an $L^4$ score error bound.

Contribution

It proposes a new approach that achieves polynomial-time posterior sampling by integrating diffusion models with annealed Langevin dynamics under weaker score error assumptions.

Findings

Achieves posterior sampling with only an $L^4$ score error bound.

Combines diffusion models with annealed Langevin dynamics effectively.

Provides theoretical guarantees for sampling efficiency.

Abstract

Given a noisy linear measurement $y = Ax + \xi$ of a distribution $p(x)$, and a good approximation to the prior $p(x)$, when can we sample from the posterior $p(x \mid y)$? Posterior sampling provides an accurate and fair framework for tasks such as inpainting, deblurring, and MRI reconstruction, and several heuristics attempt to approximate it. Unfortunately, approximate posterior sampling is computationally intractable in general. To sidestep this hardness, we focus on (local or global) log-concave distributions $p(x)$. In this regime, Langevin dynamics yields posterior samples when the exact scores of $p(x)$ are available, but it is brittle to score--estimation error, requiring an MGF bound (sub-exponential error). By contrast, in the unconditional setting, diffusion models succeed with only an $L^2$ bound on the score error. We prove that combining diffusion models with an annealed variant of Langevin dynamics achieves conditional sampling in polynomial time using merely an $L^4$ bound on the score error.