Gibbs samplers are preeminent Markov chain Monte Carlo algorithms used in computational physics and statistical computing. Yet, their most fundamental properties, such as relations between convergence characteristics of their various versions, are not well understood. In this paper we prove the solidarity of their spectral gaps: if any of the random scan or $d!$ deterministic scans has a~spectral gap then all of them have. Our methods rely on geometric interpretation of the Gibbs samplers as alternating projection algorithms and analysis of the rate of convergence in the von Neumann--Halperin method of cyclic alternating projections. In addition, we provide a quantitative result: if the spectral gap of the random scan Gibbs sampler scales polynomially with dimension, so does the spectral gap of any of the deterministic scans.

翻译：吉布斯采样器是计算物理学和统计计算中常用的马尔可夫链蒙特卡洛算法。然而，其最基本性质（如不同版本收敛特性之间的关系）尚未得到充分理解。本文证明了其谱间隙的团结性：若随机扫描或$d!$种确定性扫描中任意一种具有谱间隙，则所有扫描均具有该性质。我们的方法基于将吉布斯采样器几何解释为交替投影算法，并分析了冯·诺依曼-哈尔佩林循环交替投影方法的收敛速率。此外，我们提供定量结果：若随机扫描吉布斯采样器的谱间隙随维度呈多项式缩放，则任何确定性扫描的谱间隙也呈多项式缩放。