Random Walks with Tweedie: A Unified View of Score-Based Diffusion Models

TL;DR

The paper addresses unifying score-based diffusion models by deriving a simple, textbook-backed theory that connects MMSE denoisers to the score via Tweedie’s formula, enabling score-based sampling with only denoiser-based estimators. It introduces training and sampling templates that reproduce and subsume NCSN, DDPM, and SGM-SDE within a common framework, and shows how conditional sampling can be performed without likelihood approximations. The key contributions include linking denoisers to the score, formulating Langevin-style sampling with a potential derived from the target distribution, and enabling a noise-schedule driven sequence of sampling steps that converge to the true distribution. The approach provides a practical, flexible, and accessible perspective for signal-processing researchers, with code available for unconditional and conditional sampling.

Abstract

We present a concise derivation for several influential score-based diffusion models that relies on only a few textbook results. Diffusion models have recently emerged as powerful tools for generating realistic, synthetic signals -- particularly natural images -- and often play a role in state-of-the-art algorithms for inverse problems in image processing. While these algorithms are often surprisingly simple, the theory behind them is not, and multiple complex theoretical justifications exist in the literature. Here, we provide a simple and largely self-contained theoretical justification for score-based diffusion models that is targeted towards the signal processing community. This approach leads to generic algorithmic templates for training and generating samples with diffusion models. We show that several influential diffusion models correspond to particular choices within these templates and demonstrate that alternative, more straightforward algorithmic choices can provide comparable results. This approach has the added benefit of enabling conditional sampling without any likelihood approximation.