The Geometry of Efficient Nonconvex Sampling

TL;DR

This paper introduces an efficient polynomial-time algorithm for uniform sampling from arbitrary compact bodies in high-dimensional space, generalizing previous methods for convex and star-shaped bodies under certain geometric conditions.

Contribution

It presents a novel sampling algorithm applicable to a broad class of nonconvex bodies, extending the scope of efficient sampling beyond convex and star-shaped sets.

Findings

Algorithm runs in polynomial time relative to dimension and geometric constants.

Generalizes known sampling methods for convex and star-shaped bodies.

Provides theoretical guarantees under isoperimetry and volume growth conditions.

Abstract

We present an efficient algorithm for uniformly sampling from an arbitrary compact body $\mathcal{X} \subset \mathbb{R}^n$ 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 $\mathcal{X}$ and the volume growth constant of the set $\mathcal{X}$.