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

arXiv: https://arxiv.org/abs/2410.05837 · HTML: https://mlsanity.com/paper/2410.05837/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 26 references that are in the map.
- First posted on arXiv 8 Oct 2024.

## Builds on (most cited first, 15 of 15)

- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Diffusion Models in Vision: A Survey](https://mlsanity.com/paper/2209.04747/) (2022)
- [How to Train Your Energy-Based Models](https://mlsanity.com/paper/2101.03288/) (2021)
- [Score-Based Generative Modeling with Critically-Damped Langevin Diffusion](https://mlsanity.com/paper/2112.07068/) (2021)
- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [A Complete Recipe for Stochastic Gradient MCMC](https://mlsanity.com/paper/1506.04696/) (2015)
- [Generalized Denoising Auto-Encoders as Generative Models](https://mlsanity.com/paper/1305.6663/) (2013)
- [Neural Empirical Bayes](https://mlsanity.com/paper/1903.02334/) (2019)
- [3D molecule generation by denoising voxel grids](https://mlsanity.com/paper/2306.07473/) (2023)
- [Protein Discovery with Discrete Walk-Jump Sampling](https://mlsanity.com/paper/2306.12360/) (2023)
- [Beyond Log-concavity: Provable Guarantees for Sampling Multi-modal Distributions using Simulated Tempering Langevin Monte Carlo](https://mlsanity.com/paper/1710.02736/) (2017)
- [Provable Benefit of Annealed Langevin Monte Carlo for Non-log-concave Sampling](https://mlsanity.com/paper/2407.16936/) (2024)
- [Heavy-tailed denoising score matching](https://mlsanity.com/paper/2112.09788/) (2021)
- [Universal Smoothed Score Functions for Generative Modeling](https://mlsanity.com/paper/2303.11669/) (2023)

## Built on by (most cited first, 3 of 3)

- [Optimal Denoising in Score-Based Generative Models: The Role of Data Regularity](https://mlsanity.com/paper/2503.12966/) (2025)
- [Divergence-Free Diffusion Models for Incompressible Fluid Flows](https://mlsanity.com/paper/2601.19368/) (2026)
- [Generative Shape Reconstruction with Geometry-Guided Langevin Dynamics](https://mlsanity.com/paper/2603.27016/) (2026)

## Abstract (arXiv)

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.