John's Walk

TL;DR

This paper introduces an affine-invariant random walk using John's ellipsoids for efficient uniform sampling from convex bodies, with proven polynomial mixing times from various starting points.

Contribution

It proposes a novel affine-invariant walk based on John's ellipsoids and provides rigorous polynomial mixing time bounds for sampling from convex bodies.

Findings

Mixing in O(n^7) steps from a warm start

Polynomial mixing bounds from any fixed point

Uses maximum volume inscribed ellipsoids for proposals

Abstract

We present an affine-invariant random walk for drawing uniform random samples from a convex body $\mathcal{K} \subset \mathbb{R}^n$ that uses maximum volume inscribed ellipsoids, known as John's ellipsoids, for the proposal distribution. Our algorithm makes steps using uniform sampling from the John's ellipsoid of the symmetrization of $\mathcal{K}$ at the current point. We show that from a warm start, the random walk mixes in $\widetilde{O}(n^7)$ steps where the log factors depend only on constants associated with the warm start and desired total variation distance to uniformity. We also prove polynomial mixing bounds starting from any fixed point $x$ such that for any chord $pq$ of $\mathcal{K}$ containing $x$, $\left|\log \frac{|p-x|}{|q-x|}\right|$ is bounded above by a polynomial in $n$.