Multiple works regarding convergence analysis of Markov chains have led to spectral gap decomposition formulas of the form \[ \mathrm{Gap}(S) \geq c_0 \left[\inf_z \mathrm{Gap}(Q_z)\right] \mathrm{Gap}(\bar{S}), \] where $c_0$ is a constant, $\mathrm{Gap}$ denotes the right spectral gap of a reversible Markov operator, $S$ is the Markov transition kernel (Mtk) of interest, $\bar{S}$ is an idealized or simplified version of $S$, and $\{Q_z\}$ is a collection of Mtks characterizing the differences between $S$ and $\bar{S}$. This type of relationship has been established in various contexts, including: 1. decomposition of Markov chains based on a finite cover of the state space, 2. hybrid Gibbs samplers, and 3. spectral independence and localization schemes. We show that multiple key decomposition results across these domains can be connected within a unified framework, rooted in a simple sandwich structure of $S$. Within the general framework, we establish new instances of spectral gap decomposition for hybrid hit-and-run samplers and hybrid data augmentation algorithms with two intractable conditional distributions. Additionally, we explore several other properties of the sandwich structure, and derive extensions of the spectral gap decomposition formula.

翻译：关于马尔可夫链收敛分析的诸多研究，已推导出形式为 \[ \mathrm{Gap}(S) \geq c_0 \left[\inf_z \mathrm{Gap}(Q_z)\right] \mathrm{Gap}(\bar{S}) \] 的谱隙分解公式，其中 $c_0$ 为常数，$\mathrm{Gap}$ 表示可逆马尔可夫算子的右谱隙，$S$ 是目标马尔可夫转移核（Mtk），$\bar{S}$ 是 $S$ 的理想化或简化版本，而 $\{Q_z\}$ 是描述 $S$ 与 $\bar{S}$ 差异的 Mtk 集合。此类关系已在多种情境中建立，包括：1. 基于状态空间有限覆盖的马尔可夫链分解，2. 混合吉布斯采样器，以及 3. 谱独立与局部化方案。本文证明，这些领域中多个关键分解结果可通过一个统一框架相互关联，该框架植根于 $S$ 的简单夹层结构。在此通用框架内，我们为混合命中-游走采样器以及具有两个难处理条件分布的混合数据增强算法建立了新的谱隙分解实例。此外，我们探讨了夹层结构的若干其他性质，并推导了谱隙分解公式的扩展形式。