Artificial Intelligence · 18.08.2026, 16:40 UTC
Spectral Gaps of Hit-and-Run and Coordinate Hit-and-Run
| Schweregrad | info |
|---|---|
| Kategorie | Artificial Intelligence |
| Quelle | arXiv cs.LG ↗ |
| Veröffentlicht | 18.08.2026 UTC |
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arXiv:2608.16878v1 Announce Type: cross Abstract: For any convex body $\mathcal{K}\subset\mathbb{R}^{n}$ containing a unit ball, the spectral gap of Hit-and-Run is $\Omega(1/(n^2 C_{\mathsf{PI}}))$, where $C_{\mathsf{PI}}$ is the Poincar\'e constant of the uniform distribution $\pi$ over $\mathcal{K}$. This implies that Hit-and-Run converges to a distribution within $\chi^2$-divergence $\varepsilon$ of the uniform distribution $\pi$ in $O(n^2 C_{\mathsf{PI}}\log(M/\varepsilon))$ steps from any starting distribution $\pi_0$ with $M=\chi^2(\pi_{0}\,\|\,\pi)$, thus refining the known bound of $O(n^2 R^2 \log(M/\varepsilon))$ by Lov\'asz 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 Poincar\'e/KLS constants as was done for the Ball walk by Kannan, Lov\'asz 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 Babu\v{s}ka--Aziz constant studied in the analysis of PDEs; it is asymptotically bounded by the improved Poincar\'e …