论文标题
光谱望远镜：基于层次结构的随机扫描吉布斯采样器的收敛速率边界
Spectral Telescope: Convergence Rate Bounds for Random-Scan Gibbs Samplers Based on a Hierarchical Structure
论文作者
论文摘要
随机扫描吉布斯采样器具有自然的分层结构。该结构将靶向较高维分布的吉布斯采样器连接到针对较低维数的gibbs采样器。这导致了其光谱差距的准检测属性。基于此属性，我们在通用域上的吉布斯采样器的光谱差距和收敛速率上得出了三个新界限。这三个边界将链的光谱差距分别与目标分布的相关结构，一类随机行走链和影响矩阵的集合有关。值得注意的是，我们的结果之一概括了光谱独立性的技术，该技术因其在有限域上的成功而受到了广泛的状态空间的关注。我们通过针对$ n $ -cube的角落的均匀分布的采样器来说明我们的方法。
Random-scan Gibbs samplers possess a natural hierarchical structure. The structure connects Gibbs samplers targeting higher dimensional distributions to those targeting lower dimensional ones. This leads to a quasi-telescoping property of their spectral gaps. Based on this property, we derive three new bounds on the spectral gaps and convergence rates of Gibbs samplers on general domains. The three bounds relate a chain's spectral gap to, respectively, the correlation structure of the target distribution, a class of random walk chains, and a collection of influence matrices. Notably, one of our results generalizes the technique of spectral independence, which has received considerable attention for its success on finite domains, to general state spaces. We illustrate our methods through a sampler targeting the uniform distribution on a corner of an $n$-cube.
