# Spectral Telescope: Convergence Rate Bounds for Random-Scan Gibbs Samplers Based on a Hierarchical Structure

arXiv: https://arxiv.org/abs/2208.11299 · HTML: https://mlsanity.com/paper/2208.11299/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Aug 2022.

## Built on by (most cited first, 2 of 2)

- [Scalability of Metropolis-within-Gibbs schemes for high-dimensional Bayesian models](https://mlsanity.com/paper/2403.09416/) (2024)
- [Spectral gap of Metropolis-within-Gibbs under log-concavity](https://mlsanity.com/paper/2509.26175/) (2025)

## Abstract (arXiv)

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.