A noise-corrected Langevin algorithm and sampling by half-denoising

TL;DR

This paper introduces a noise-corrected Langevin algorithm that effectively samples from distributions using noisy score functions, requiring only a single noise level, and offers an intuitive half-denoising approach.

Contribution

The authors develop a novel noise-corrected Langevin algorithm that removes bias from noisy data score functions with minimal noise level requirements.

Findings

The algorithm accurately samples from target distributions using noisy score functions.

It only requires knowledge of the noisy score at one noise level.

The method includes a simple half-denoising interpretation.

Abstract

The Langevin algorithm is a classic method for sampling from a given pdf in a real space. In its basic version, it only requires knowledge of the gradient of the log-density, also called the score function. However, in deep learning, it is often easier to learn the so-called "noisy-data score function", i.e. the gradient of the log-density of noisy data, more precisely when Gaussian noise is added to the data. Such an estimate is biased and complicates the use of the Langevin method. Here, we propose a noise-corrected version of the Langevin algorithm, where the bias due to noisy data is removed, at least regarding first-order terms. Unlike diffusion models, our algorithm only needs to know the noisy score function for one single noise level. We further propose a simple special case which has an interesting intuitive interpretation of iteratively adding noise the data and then attempting to remove half of that noise.