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The Geometry of Efficient Nonconvex Sampling

Santosh S. Vempala, Andre Wibisono

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 $\mathcal{X}$.

The Geometry of Efficient Nonconvex Sampling

Abstract

We present an efficient algorithm for uniformly sampling from an arbitrary compact body 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 and the volume growth constant of the set .

Paper Structure

This paper contains 51 sections, 18 theorems, 123 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Isoperimetry.
  3. Volume growth condition.
  4. Algorithm: In-and-Out
  5. Main result: Iteration complexity of In-and-Out with a warm start
  6. Preliminaries
  7. Notation and definitions.
  8. Geometry of the support set
  9. Isoperimetry: Poincaré inequality
  10. Volume growth condition
  11. Volume growth for convex bodies
  12. Volume growth for star-shaped bodies
  13. Volume growth under union
  14. Volume growth under set exclusion
  15. Rényi divergence and other statistical divergences
  16. ...and 36 more sections

Key Result

Theorem 1

Let $\pi \propto \mathbf{1}_\mathcal{X}$ where $\mathcal{X} \subset \mathbb{R}^n$ is a compact body, $n \ge 2$. Assume: Let $q \in [2,\infty)$, $\varepsilon \in (0, \frac{1}{2})$, and $M \in [1,\infty)$ be arbitrary. Suppose $x_0 \sim \rho_0$ is $M$-warm with respect to $\pi$. Then with a suitable choice of parameters (see Eq:TPIDef for $T$, Eq:hDef for $h$, Eq:NDef for $N$), with probability at

Figures (3)

  • Figure 1: Examples of bodies $\mathcal{X} \subset \mathbb{R}^n$ that current theory covers ((a) convex and (b) star-shaped); and examples of $\mathcal{X}$ that current theory does not cover ((c) with a hole and (d) not star-shaped).
  • Figure 2: A body with (a) poor isoperimetry (large Poincaré constant), and (b) good isoperimetry (small Poincaré constant) but large volume growth rate.
  • Figure 3: Difference of containment guarantee in the (a) convex case and (b) nonconvex case.

Theorems & Definitions (38)

  • Theorem 1
  • Definition 1
  • Remark 1
  • Remark 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • ...and 28 more