# Chain of Log-Concave Markov Chains

arXiv: https://arxiv.org/abs/2305.19473 · HTML: https://mlsanity.com/paper/2305.19473/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 32 references that are in the map.
- First posted on arXiv 31 May 2023.

## Builds on (most cited first, 14 of 14)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Entropy-SGD: Biasing Gradient Descent Into Wide Valleys](https://mlsanity.com/paper/1611.01838/) (2016)
- [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014)
- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [Log-concave sampling: Metropolis-Hastings algorithms are fast](https://mlsanity.com/paper/1801.02309/) (2018)
- [The Randomized Midpoint Method for Log-Concave Sampling](https://mlsanity.com/paper/1909.05503/) (2019)
- [Neural Empirical Bayes](https://mlsanity.com/paper/1903.02334/) (2019)
- [High-Order Langevin Diffusion Yields an Accelerated MCMC Algorithm](https://mlsanity.com/paper/1908.10859/) (2019)
- [Fast Approximation of the Sliced-Wasserstein Distance Using Concentration of Random Projections](https://mlsanity.com/paper/2106.15427/) (2021)
- [Sqrt(d) Dimension Dependence of Langevin Monte Carlo](https://mlsanity.com/paper/2109.03839/) (2021)
- [Multimeasurement Generative Models](https://mlsanity.com/paper/2112.09822/) (2021)

## Built on by (most cited first, 14 of 14)

- [3D molecule generation by denoising voxel grids](https://mlsanity.com/paper/2306.07473/) (2023)
- [Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance](https://mlsanity.com/paper/2501.02298/) (2025)
- [Stochastic Localization via Iterative Posterior Sampling](https://mlsanity.com/paper/2402.10758/) (2024)
- [Convergence of Deterministic and Stochastic Diffusion-Model Samplers: A Simple Analysis in Wasserstein Distance](https://mlsanity.com/paper/2508.03210/) (2025)
- [Provable Convergence and Limitations of Geometric Tempering for Langevin Dynamics](https://mlsanity.com/paper/2410.09697/) (2024)
- [Wasserstein Bounds for generative diffusion models with Gaussian tail targets](https://mlsanity.com/paper/2412.11251/) (2024)
- [A Practical Diffusion Path for Sampling](https://mlsanity.com/paper/2406.14040/) (2024)
- [Sampling from multi-modal distributions with polynomial query complexity in fixed dimension via reverse diffusion](https://mlsanity.com/paper/2501.00565/) (2024)
- [Optimal Denoising in Score-Based Generative Models: The Role of Data Regularity](https://mlsanity.com/paper/2503.12966/) (2025)
- [Diffusion Path Samplers via Sequential Monte Carlo](https://mlsanity.com/paper/2601.21951/) (2026)
- [Unified all-atom molecule generation with neural fields](https://mlsanity.com/paper/2511.15906/) (2025)
- [Sampling by averaging: A multiscale approach to score estimation](https://mlsanity.com/paper/2508.15069/) (2025)
- [Sampling via Stochastic Interpolants by Langevin-based Velocity and Initialization Estimation in Flow ODEs](https://mlsanity.com/paper/2601.08527/) (2026)
- [Mixed Poisson diffusion models and application to posterior sampling](https://mlsanity.com/paper/2610.20218/) (2026)

## Abstract (arXiv)

We introduce a theoretical framework for sampling from unnormalized densities based on a smoothing scheme that uses an isotropic Gaussian kernel with a single fixed noise scale. We prove one can decompose sampling from a density (minimal assumptions made on the density) into a sequence of sampling from log-concave conditional densities via accumulation of noisy measurements with equal noise levels. Our construction is unique in that it keeps track of a history of samples, making it non-Markovian as a whole, but it is lightweight algorithmically as the history only shows up in the form of a running empirical mean of samples. Our sampling algorithm generalizes walk-jump sampling (Saremi & Hyv"arinen, 2019). The "walk" phase becomes a (non-Markovian) chain of (log-concave) Markov chains. The "jump" from the accumulated measurements is obtained by empirical Bayes. We study our sampling algorithm quantitatively using the 2-Wasserstein metric and compare it with various Langevin MCMC algorithms. We also report a remarkable capacity of our algorithm to "tunnel" between modes of a distribution.