
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 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 at the current point. We show that from a warm start, the random walk mixes in 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 such that for any chord of containing , is bounded above by a polynomial in .
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Taxonomy
TopicsPoint processes and geometric inequalities · Markov Chains and Monte Carlo Methods · Computational Geometry and Mesh Generation
