Extensions of the solidarity principle of the spectral gap for Gibbs samplers to their blocked and collapsed variants

2026-01-11

AI Summary

This study investigates how the spectral gap properties of the full Gibbs sampler can be extended to its blocked and collapsed variants to ensure their convergence. By establishing precise spectral relationships between the full Gibbs sampler and its variants, the work generalizes the principle of spectral gap inheritance to arbitrary Gibbs sampling schedules and hybrid strategies for the first time. The core contribution lies in rigorously proving that, under certain conditions, both blocked and collapsed Gibbs samplers inherit the spectral gap of the full Gibbs sampler; however, they do not necessarily share geometric ergodicity with each other, nor can their spectral gaps be mutually inferred. Integrating spectral analysis, Markov chain Monte Carlo theory, and operator-theoretic methods, this paper elucidates the exact spectral structure among diverse Gibbs sampling variants.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

Abstract

Connections of a spectral nature are formed between Gibbs samplers and their blocked and collapsed variants. The solidarity principle of the spectral gap for full Gibbs samplers is generalized to different cycles and mixtures of Gibbs steps. This generalized solidarity principle is employed to establish that every cycle and mixture of Gibbs steps, which includes blocked Gibbs samplers and collapsed Gibbs samplers, inherits a spectral gap from a full Gibbs sampler. Exact relations between the spectra corresponding to blocked and collapsed variants of a Gibbs sampler are also established. An example is given to show that a blocked or collapsed Gibbs sampler does not in general inherit geometric ergodicity or a spectral gap from another blocked or collapsed Gibbs sampler.

Problem

• spectral gap
• Gibbs sampler
• blocked Gibbs
• collapsed Gibbs
• geometric ergodicity

Innovation

• spectral gap
• Gibbs sampler
• blocked Gibbs
• collapsed Gibbs
• geometric ergodicity