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Uniform ergodicity of geodesic slice sampling

Mareike Hasenpflug

TL;DR

This work addresses sampling from target distributions on Riemannian manifolds using geodesic slice sampling and proves a uniform ergodicity bound with explicit constants. The contraction factor $\rho$ depends on hyperparameters $(m,w)$, geometric quantities (via $\kappa$ and $\omega_{d-1}$), and the target through its level-set function, ensuring a practically interpretable convergence rate. The proof combines a stepping-out and shrinkage construction with Ricci curvature bounds and Bishop-Gromov volume comparison to establish a small-set condition and derive the bound. The results offer guidance on parameter tuning and relate to known methods in special cases such as sampling on spheres and the hit-and-run algorithm as a limiting scenario.

Abstract

Geodesic slice sampling, introduced in Durmus et al., 2024, is a slice sampling based Markov chain Monte Carlo method for approximate sampling from distributions on Riemannian manifolds. We prove that it is uniformly ergodic for distributions with compact support that have a bounded density with respect to the Riemannian measure. The constants in our convergence bound are available explicitly, and we investigate their dependence on the hyperparameters of the geodesic slice sampler, the target distribution and the underlying domain.

Uniform ergodicity of geodesic slice sampling

TL;DR

This work addresses sampling from target distributions on Riemannian manifolds using geodesic slice sampling and proves a uniform ergodicity bound with explicit constants. The contraction factor depends on hyperparameters , geometric quantities (via and ), and the target through its level-set function, ensuring a practically interpretable convergence rate. The proof combines a stepping-out and shrinkage construction with Ricci curvature bounds and Bishop-Gromov volume comparison to establish a small-set condition and derive the bound. The results offer guidance on parameter tuning and relate to known methods in special cases such as sampling on spheres and the hit-and-run algorithm as a limiting scenario.

Abstract

Geodesic slice sampling, introduced in Durmus et al., 2024, is a slice sampling based Markov chain Monte Carlo method for approximate sampling from distributions on Riemannian manifolds. We prove that it is uniformly ergodic for distributions with compact support that have a bounded density with respect to the Riemannian measure. The constants in our convergence bound are available explicitly, and we investigate their dependence on the hyperparameters of the geodesic slice sampler, the target distribution and the underlying domain.

Paper Structure

This paper contains 8 sections, 6 theorems, 68 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and manifold setting
  3. Basic notation.
  4. Manifold setting.
  5. Main result
  6. The stepping-out distribution
  7. Lower bound for the reeled shrinkage kernel
  8. Acknowledgements

Key Result

Theorem 1

Let $(\mathsf{M}, \mathfrak{g})$ be a geodesically complete, connected Riemannian manifold. Assume that $\pi$ is defined as in Eq: target density with unnormalised target density $p: \mathsf{M}\to [0, \infty)$ satisfying Fix hyperparameters $w \in (0, \infty)$ and $m \in \mathbb{N} \cup \{\infty\}$ such that Assumption A: hyperparameters holds and such that there exists an $\upvarepsilon > 0$ sat

Figures (1)

  • Figure 1: Illustration of the different regimes for $q$ defined in \ref{['Eq: Ergodicity constant term depending on hyperparameters']}. The horizontal axis shows the hyperparameter $w$, the vertical axis shows $q(m, \cdot)$, where different values of the hyperparameter $m$ correspond to different coloured graphs. All plots are drawn for $\Updelta > 0$. Note that $y$-axis scaling is not comparable between plots.

Theorems & Definitions (17)

  • Theorem 1
  • proof
  • Corollary 2
  • proof
  • Remark 3: Hyperparameters
  • Remark 4: Target distribution
  • Remark 5: State space and ambient Riemannian manifold
  • Example 6: Geodesic slice sampling on the sphere
  • Example 7: Hit-and-run algorithm
  • Lemma 8
  • ...and 7 more