mlsanityData to 14 Oct 2026

Chain of Log-Concave Markov Chains

arXiv 2305.19473 ↗First posted 31 May 2023stat.ML cs.LG stat.COcited by 14

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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.

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

14 of 32 references.

References · 14

  1. 2020Denoising Diffusion Probabilistic Modelscited by 21,076
  2. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,058
  3. 2022Elucidating the Design Space of Diffusion-Based Generative Modelscited by 3,045
  4. 2016Entropy-SGD: Biasing Gradient Descent Into Wide Valleyscited by 359
  5. 2014Theoretical guarantees for approximate sampling from smooth and log-concave densitiescited by 208
  6. 2015Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithmcited by 197
  7. 2017Underdamped Langevin MCMC: A non-asymptotic analysiscited by 133
  8. 2018Log-concave sampling: Metropolis-Hastings algorithms are fastcited by 118
  9. 2019The Randomized Midpoint Method for Log-Concave Samplingcited by 78
  10. 2019Neural Empirical Bayescited by 46
  11. 2019High-Order Langevin Diffusion Yields an Accelerated MCMC Algorithmcited by 36
  12. 2021Fast Approximation of the Sliced-Wasserstein Distance Using Concentration of Random Projectionscited by 33
2 more
  1. 2021Sqrt(d) Dimension Dependence of Langevin Monte Carlocited by 25
  2. 2021Multimeasurement Generative Modelscited by 8

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Citing papers per year: 2023: 1; 2024: 5; 2025: 5; 2026: 3.

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  1. 20233D molecule generation by denoising voxel gridscited by 40
  2. 2025Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distancecited by 29
  3. 2024Stochastic Localization via Iterative Posterior Samplingcited by 28
  4. 2025Convergence of Deterministic and Stochastic Diffusion-Model Samplers: A Simple Analysis in Wasserstein Distancecited by 21
  5. 2024Provable Convergence and Limitations of Geometric Tempering for Langevin Dynamicscited by 10
  6. 2024Wasserstein Bounds for generative diffusion models with Gaussian tail targetscited by 10
  7. 2024A Practical Diffusion Path for Samplingcited by 5
  8. 2024Sampling from multi-modal distributions with polynomial query complexity in fixed dimension via reverse diffusioncited by 4
  9. 2025Optimal Denoising in Score-Based Generative Models: The Role of Data Regularitycited by 4
  10. 2026Diffusion Path Samplers via Sequential Monte Carlocited by 4
  11. 2025Unified all-atom molecule generation with neural fieldscited by 3
  12. 2025Sampling by averaging: A multiscale approach to score estimationcited by 1
2 more
  1. 2026Sampling via Stochastic Interpolants by Langevin-based Velocity and Initialization Estimation in Flow ODEscited by 0
  2. 2026Mixed Poisson diffusion models and application to posterior samplingcited by 0