# Sampling and Integration of Logconcave Functions by Algorithmic Diffusion

arXiv: https://arxiv.org/abs/2411.13462 · HTML: https://mlsanity.com/paper/2411.13462/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 37 references that are in the map.
- First posted on arXiv 20 Nov 2024.

## Builds on (most cited first, 7 of 7)

- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [Structured Logconcave Sampling with a Restricted Gaussian Oracle](https://mlsanity.com/paper/2010.03106/) (2020)
- [Improved analysis for a proximal algorithm for sampling](https://mlsanity.com/paper/2202.06386/) (2022)
- [Sampling with Riemannian Hamiltonian Monte Carlo in a Constrained Space](https://mlsanity.com/paper/2202.01908/) (2022)
- [In-and-Out: Algorithmic Diffusion for Sampling Convex Bodies](https://mlsanity.com/paper/2405.01425/) (2024)
- [Entropy contraction of the Gibbs sampler under log-concavity](https://mlsanity.com/paper/2410.00858/) (2024)
- [Reducing Isotropy and Volume to KLS: Faster Rounding and Volume Algorithms](https://mlsanity.com/paper/2008.02146/) (2020)

## Built on by (most cited first, 12 of 12)

- [Complexity Analysis of Normalizing Constant Estimation: from Jarzynski Equality to Annealed Importance Sampling and beyond](https://mlsanity.com/paper/2502.04575/) (2025)
- [Faster logconcave sampling from a cold start in high dimension](https://mlsanity.com/paper/2505.01937/) (2025)
- [Oracle-based Uniform Sampling from Convex Bodies](https://mlsanity.com/paper/2510.02983/) (2025)
- [A unified complexity bound for logconcave sampling](https://mlsanity.com/paper/2606.12694/) (2026)
- [Beyond the d2.5-mixing bound for Dikin walks on polytopes](https://mlsanity.com/paper/2607.13943/) (2026)
- [Spectral Gaps of Hit-and-Run and Coordinate Hit-and-Run](https://mlsanity.com/paper/2608.16878/) (2026)
- [A proximal gradient algorithm for composite log-concave sampling](https://mlsanity.com/paper/2605.12461/) (2026)
- [Functional Stochastic Localization](https://mlsanity.com/paper/2602.03999/) (2026)
- [The Geometry of Efficient Nonconvex Sampling](https://mlsanity.com/paper/2603.25622/) (2026)
- [Tight Bounds for Learning Polyhedra with a Margin](https://mlsanity.com/paper/2604.14614/) (2026)
- [On two proofs of d2 mixing of weighted Dikin walks](https://mlsanity.com/paper/2608.28566/) (2026)
- [Thin-shell stability of Gaussian cooling: logconcave sampling with sesteric complexity from a cold start](https://mlsanity.com/paper/2609.15884/) (2026)

## Abstract (arXiv)

We study the complexity of sampling, rounding, and integrating arbitrary logconcave functions. Our new approach provides the first complexity improvements in nearly two decades for general logconcave functions for all three problems, and matches the best-known complexities for the special case of uniform distributions on convex bodies. For the sampling problem, our output guarantees are significantly stronger than previously known, and lead to a streamlined analysis of statistical estimation based on dependent random samples.