Spectral Gaps of Hit-and-Run and Coordinate Hit-and-Run
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 constant, which we show can be bounded in terms of the usual Poincar\'e 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 $O(n^3C_{\mathsf{PI}}\log(M/\varepsilon))$.