The Geometry of Efficient Nonconvex Sampling

2026-03-26

AI Summary

This work addresses the problem of efficiently sampling uniformly from arbitrary compact non-convex sets that satisfy isoperimetric and natural volume growth conditions. To this end, the authors propose a novel Markov chain Monte Carlo algorithm that extends efficient uniform sampling beyond convex and star-shaped domains to a significantly broader class of non-convex geometries. By integrating isoperimetric inequalities, Poincaré constants, and volume growth analysis, the method achieves polynomial-time complexity under a warm start, with runtime depending polynomially on the ambient dimension, the Poincaré constant, and the volume growth constant. This result substantially broadens the theoretical scope of existing sampling guarantees, offering the first provably efficient approach for such general non-convex settings.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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 Poincaré constant of the uniform distribution on $\mathcal{X}$ and the volume growth constant of the set $\mathcal{X}$.

Problem

• nonconvex sampling
• uniform sampling
• isoperimetry
• volume growth
• compact body

Innovation

• nonconvex sampling
• isoperimetry
• volume growth
• Poincaré constant
• polynomial complexity