凸多胞体上的Gibbs/Metropolis算法
Gibbs/Metropolis algorithms on a convex polytope
摘要：
本文给出了凸多胞体上均匀分布采样的Metropolis算法的自然形式的收敛速度。奇异提案分布基于在固定的有限集方向中的一个方向上局部移动的行走，需要一些新的工具。利用Nash不等式和Weyl不等式得到了谱和特征函数的上界。马尔可夫链的上特征值与一个新拉普拉斯算子的多面体的Neuman特征值密切相关。
英文摘要：
This paper gives sharp rates of convergence for natural versions of the Metropolis algorithm for sampling from the uniform distribution on a convex polytope. The singular proposal distribution, based on a walk moving locally in one of a fixed, finite set of directions, needs some new tools. We get useful bounds on the spectrum and eigenfunctions using Nash and Weyl-type inequalities. The top eigenvalues of the Markov chain are closely related to the Neuman eigenvalues of the polytope for a novel Laplacian.
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