Curvature-based rejection sampling

TL;DR

The paper introduces CURS, a curvature-based rejection sampling method for densities on Riemannian manifolds that depend only on distance to a center, uniting rejection sampling with Bishop’s volume comparison to yield exact samples at moderate cost in low-to-moderate dimensions. It develops geodesic spherical coordinates, derives a tractable joint density $p(r,s)$, and constructs a curvature-informed proposal $g(r,s)$ to enable rejection-based sampling. The framework is specialized to covariance-matrix spaces with affine-invariant metrics, and the authors present General CURS and Sharp CURS variants, plus accommodations for the cut locus via truncation to the injectivity domain. Together, these results offer a practical, geometry-aware tool for exact sampling on negatively curved symmetric spaces and related manifolds, with analytic and empirical insights into acceptance probabilities and efficiency.

Abstract

The present work introduces curvature-based rejection sampling (CURS). This is a method for sampling from a general class of probability densities defined on Riemannian manifolds. It can be used to sample from any probability density which ``depends only on distance". The idea is to combine the statistical principle of rejection sampling with the geometric principle of volume comparison. CURS is an exact sampling method and (assuming the underlying Riemannian manifold satisfies certain technical conditions) it has a particularly moderate computational cost. The aim of the present work is to show that there are many applications where CURS should be the user's method of choice for dealing with relatively low-dimensional scenarios.

Figures (3)

• Figure 1: Theoretical (solid line) and empirical (dot points) acceptance probabilities, plotted against $\sigma$
• Figure 2: Theoretical (solid line) and empirical (dot points) acceptance probabilities, plotted against $\delta$
• Figure 3: Proposal and acceptance regions for CURS ($M = U(2)$)