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On spectral gap decomposition for Markov chains

Qian Qin

TL;DR

The paper develops a unified spectral-gap decomposition framework for reversible Markov chains by expressing the target kernel as a sandwich: $S = P^*QP$, with an idealized benchmark $\bar S = P^*E^*E P$ and a family of $Q_z$ kernels that capture deviations. A main result bounds $\mathrm{Gap}(S)$ below by $c_0\big(\inf_z \mathrm{Gap}(Q_z)\big)\mathrm{Gap}(\bar S)$, tying together spectral-gap formulas from partitioning, hybrid data augmentation, and localization into a single structure. It then extends to non-reversible chains and weak Poincaré inequalities, and shows how seven key decomposition results are special cases of the framework, including two new instances: hybrid hit-and-run and hybrid data augmentation with two intractable conditionals. The work provides a practical toolkit for analyzing and designing efficient hybrid MCMC algorithms with explicit gap or variance bounds, and demonstrates these ideas with quantitative bounds and simulations in well-conditioned settings.

Abstract

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.

On spectral gap decomposition for Markov chains

TL;DR

The paper develops a unified spectral-gap decomposition framework for reversible Markov chains by expressing the target kernel as a sandwich: , with an idealized benchmark and a family of kernels that capture deviations. A main result bounds below by , tying together spectral-gap formulas from partitioning, hybrid data augmentation, and localization into a single structure. It then extends to non-reversible chains and weak Poincaré inequalities, and shows how seven key decomposition results are special cases of the framework, including two new instances: hybrid hit-and-run and hybrid data augmentation with two intractable conditionals. The work provides a practical toolkit for analyzing and designing efficient hybrid MCMC algorithms with explicit gap or variance bounds, and demonstrates these ideas with quantitative bounds and simulations in well-conditioned settings.

Abstract

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 is a constant, denotes the right spectral gap of a reversible Markov operator, is the Markov transition kernel (Mtk) of interest, is an idealized or simplified version of , and is a collection of Mtks characterizing the differences between and . 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 . 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.

Paper Structure

This paper contains 40 sections, 22 theorems, 117 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $L^2$ space with respect to a probability measure
  4. Linear operator associated with a Markov transition kernel
  5. Examples of Spectral Gap Decomposition, Old and New
  6. Markov chain decomposition based on state space partitioning
  7. Markov chain decomposition, a second form
  8. Hybrid data augmentation algorithms with one intractable conditional
  9. Random-scan hybrid Gibbs samplers
  10. Approximate variance conservation in localization schemes
  11. Hybrid hit-and-run samplers
  12. Hybrid data augmentation algorithms with two intractable conditionals
  13. A Unified Framework
  14. A sandwich structure
  15. Decomposition of spectral gaps and norms
  16. ...and 25 more sections

Key Result

Proposition 1

caracciolo1992two In the context of this subsection,

Figures (2)

  • Figure 1: Left: $\mathrm{var}_S(f_i)/\mathrm{var}_{\bar{S}(f_i)}$ plotted against $\xi$. Right: $\mathrm{var}_{S_{\lambda}}(f_i)/\mathrm{var}_{\bar{S}(f_i)}$ plotted against $\xi$. The dashed horizontal lines have height 1.
  • Figure 2: Left: Distribution of the number of Metropolis-Hastings steps per iteration when $S_{\lambda}$ is implemented on a data set with $\xi \approx 226$. Right: $\tau(S_{\lambda}) \, \mathrm{var}_{S_{\lambda}}(f_i)/\|f_i - \pi f_i\|_{\pi}^2$ plotted against $\tau(S) \, \mathrm{var}_S(f_i)/\|f_i - \pi f_i\|_{\pi}^2$. The dashed line goes through the origin and has slope 1.

Theorems & Definitions (36)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Remark 8
  • Lemma 9
  • Remark 10
  • ...and 26 more