# The Geometry of Efficient Nonconvex Sampling

arXiv: https://arxiv.org/abs/2603.25622 · HTML: https://mlsanity.com/paper/2603.25622/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 37 references that are in the map.
- First posted on arXiv 26 Mar 2026.

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

- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [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)
- [Improved dimension dependence of a proximal algorithm for sampling](https://mlsanity.com/paper/2302.10081/) (2023)
- [A Proximal Algorithm for Sampling from Non-smooth Potentials](https://mlsanity.com/paper/2110.04597/) (2021)
- [Efficient Convex Optimization with Membership Oracles](https://mlsanity.com/paper/1706.07357/) (2017)
- [A Proximal Algorithm for Sampling](https://mlsanity.com/paper/2202.13975/) (2022)
- [In-and-Out: Algorithmic Diffusion for Sampling Convex Bodies](https://mlsanity.com/paper/2405.01425/) (2024)
- [Sampling and Integration of Logconcave Functions by Algorithmic Diffusion](https://mlsanity.com/paper/2411.13462/) (2024)
- [Reducing Isotropy and Volume to KLS: Faster Rounding and Volume Algorithms](https://mlsanity.com/paper/2008.02146/) (2020)
- [Faster logconcave sampling from a cold start in high dimension](https://mlsanity.com/paper/2505.01937/) (2025)
- [Hit-and-Run for Sampling and Planning in Non-Convex Spaces](https://mlsanity.com/paper/1610.08865/) (2016)
- [Oracle-based Uniform Sampling from Convex Bodies](https://mlsanity.com/paper/2510.02983/) (2025)

## Abstract (arXiv)

We present an efficient algorithm for uniformly sampling from an arbitrary compact body X subset ℝn from a warm start under isoperimetry and a natural volume growth condition. Our result provides a substantial common generalization of known results for convex bodies and star-shaped bodies. The complexity of the algorithm is polynomial in the dimension, the Poincar'e constant of the uniform distribution on X and the volume growth constant of the set X.