# A unified complexity bound for logconcave sampling

arXiv: https://arxiv.org/abs/2606.12694 · HTML: https://mlsanity.com/paper/2606.12694/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 18 references that are in the map.
- First posted on arXiv 10 Jun 2026.

## Builds on (most cited first, 8 of 8)

- [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)
- [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)
- [Sampling and Integration of Logconcave Functions by Algorithmic Diffusion](https://mlsanity.com/paper/2411.13462/) (2024)
- [Faster logconcave sampling from a cold start in high dimension](https://mlsanity.com/paper/2505.01937/) (2025)
- [The entropic barrier is n-self-concordant](https://mlsanity.com/paper/2112.10947/) (2021)
- [Zeroth-order Logconcave Sampling](https://mlsanity.com/paper/2507.18021/) (2025)

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

- [Spectral Gaps of Hit-and-Run and Coordinate Hit-and-Run](https://mlsanity.com/paper/2608.16878/) (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 give a simple, unified, and nearly tight bound for sampling arbitrary logconcave distributions from a warm start using the In-and-Out algorithm along with exponential lifting. The main new ingredient in the analysis is an improved bound on the Poincar'e constant of a lifted distribution. As a consequence, the resulting convergence rate is nearly tight for both constrained settings (e.g., Gaussian restricted to a convex body) and well-conditioned settings (e.g., strongly logconcave and smooth densities).