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

2026-08-17

AI Summary

This study addresses the slow convergence of Hit-and-Run sampling, which is constrained by Poincaré constants and excessive dimension dependence. By employing functional isoperimetric inequalities and dual variational methods, we establish a connection between spectral gaps and KLS constants to overcome the limitations of traditional conductance analysis. We prove that the mixing time complexity for near-isotropic bodies improves from cubic to O(n² log n), while the coordinate version achieves O(n³ C_PI log(M/ε)). This work effectively resolves critical bottlenecks in high-dimensional sampling, significantly enhancing algorithmic efficiency and providing new theoretical foundations for related open problems in the field.

Abstract

For any convex body $\mathcal{K}\subset\mathbb{R}^{n}$ containing a unit ball, the spectral gap of Hit-and-Run is $Ω(1/(n^2 C_{\mathsf{PI}}))$, 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 $O(n^3C_{\mathsf{PI}}\log(M/\varepsilon))$.

Problem

• Hit-and-Run
• Spectral Gap
• Mixing Time
• Poincaré Constant
• KLS Conjecture

Innovation

• Spectral Gap
• Hit-and-Run
• Poincaré Constant
• Functional Isoperimetry
• Mixing Time