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与统计计算中至关重要的马尔可夫链蒙特卡洛算法。然而，其最基本的性质，例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隙，则所有版本均具有谱隙。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-哈尔珀林循环交替投影法的收敛速率。此外，我们给出了定量结果：若随机扫描吉布斯采样器的谱隙随维度呈多项式缩放，则任意确定性扫描的谱隙也具有相同缩放性质。