2608.16878Aug 17, 202615 pages. AI disclosure included

Spectral Gaps of Hit-and-Run and Coordinate Hit-and-Run

Yunbum Kook, Santosh S. Vempala

Data Structures and Algorithms Machine Learning Probability Statistics Theory

Abstract

For any convex body $\mathcal{K}\subset\mathbb{R}^{n}$ containing a unit ball, the spectral gap of Hit-and-Run is $Ω(1/(n^2 C_{\mathsf{PI}}))$

Briefing

What this paper is actually doing

Sign in and connect a model key to generate a briefing for this paper.

Socratic margin

Ask the abstract

Questions are grounded in the title and abstract. If the PDF is required, arXivDesk will say so.

Sign in to ask questions about this paper.

Nearby in the stack

arXiv did not return neighbours for this paper. Try again in a bit.

, where

C_{\mathsf{PI}}

is the Poincaré constant of the uniform distribution

π

over

\mathcal{K}

. This implies that Hit-and-Run converges to a distribution within

χ^2

-divergence

\varepsilon

of the uniform distribution

π

in

O(n^2 C_{\mathsf{PI}}\log(M/\varepsilon))

steps from any starting distribution

π_0

with

M=χ^2(π_{0}\,\|\,π)

, thus refining the known bound of

O(n^2 R^2 \log(M/\varepsilon))

by Lovász and Vempala (2004) in terms of the outer radius

R

; for nearly isotropic bodies, together with progress on the KLS conjecture, the complexity is

O(n^2\log n\log(M/\varepsilon))

, improving the dimension dependence from cubic to nearly quadratic while maintaining logarithmic dependence on the initial distance. It was an open problem to connect the convergence of Hit-and-Run to Poincaré/KLS constants as was done for the Ball walk by Kannan, Lovász and Simonovits (1997). Unlike Hit-and-Run, the Ball walk has an unavoidable linear dependence on (a stronger notion) of the initial warmness. We directly bound the spectral gap of the Hit-and-Run Markov chain by connecting it to functional isoperimetric constants, inspired by the recent analysis of In-and-Out. Rewriting the spectral gap in terms of dual certificates leads to the Babuška--Aziz constant studied in the analysis of PDEs; it is asymptotically bounded by the improved Poincaré constant, which we show can be bounded in terms of the usual Poincaré constant. The proof is based on duality and calculus, unlike known proofs of convergence for Hit-and-Run which are based on bounding the conductance. The same technique can be applied to Coordinate Hit-and-Run, resulting in a much improved mixing time of

O(n^3C_{\mathsf{PI}}\log(M/\varepsilon))

.

arXiv did not return neighbours for this paper. Try again in a bit.

Nearby in the stack

Spectral Gaps of Hit-and-Run and Coordinate Hit-and-Run

Claims

Methods

Limitations

Nearby in the stack