随机游走（Hit-and-Run）与坐标随机游走（Coordinate Hit-and-Run）的谱间隙
Spectral Gaps of Hit-and-Run and Coordinate Hit-and-Run
- University of Michigan（密歇根大学）
- Georgia Tech（佐治亚理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究通过将Hit-and-Run马尔可夫链的谱间隙与函数等周常数关联，改进了其收敛界，优化了维度依赖关系，并将相同技术应用于坐标Hit-and-Run以提升混合时间。
AI中文摘要：
对于任意包含单位球的凸体$\mathcal{K}\subset\mathbb{R}^{n}$，Hit-and-Run（随机游走）的谱间隙为$Ω(1/(n^2 C_{\mathsf{PI}}))$，其中$C_{\mathsf{PI}}$是$\mathcal{K}$上均匀分布$π$的庞加莱常数。这意味着，从任意满足$M=χ^2(π_{0}\,\|\,π)$的初始分布$π_0$出发，经过$O(n^2 C_{\mathsf{PI}}\log(M/\varepsilon))$步后，Hit-and-Run收敛到与均匀分布$π$的$χ^2$散度为$\varepsilon$的分布，从而改进了Lovász和Vempala（2004）基于外半径$R$得到的已知界$O(n^2 R^2 \log(M/\varepsilon))$；对于近迷向体，结合KLS猜想的研究进展，其复杂度为$O(n^2\log n\log(M/\varepsilon))$，将维度依赖从三次提升至近二次，同时保持对初始距离的对数依赖。此前，能否像Kannan、Lovász和Simonovits（1997）对Ball walk（球游走）所做的那样，将Hit-and-Run的收敛性与庞加莱/KLS常数关联起来，一直是一个公开问题。与Hit-and-Run不同，Ball walk对（更强意义下的）初始暖度存在不可避免的线性依赖。受近期对In-and-Out（进-出走）分析的启发，我们通过将Hit-and-Run马尔可夫链的谱间隙与函数等周常数相关联，直接对其谱间隙进行了界定。将谱间隙用对偶证书重写后，得到了偏微分方程分析中研究的Babuška-Aziz常数；该常数可由改进的庞加莱常数渐近界定，而我们证明这一改进的庞加莱常数可通过常规庞加莱常数进行界定。该证明基于对偶性和微积分，不同于已知的基于电导率界定的Hit-and-Run收敛性证明。相同的技术可应用于Coordinate Hit-and-Run（坐标随机游走），使其混合时间大幅提升至$O(n^3C_{\mathsf{PI}}\log(M/\varepsilon))$。
英文摘要：
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 $O(n^3C_{\mathsf{PI}}\log(M/\varepsilon))$.
