Geodesic Slice Sampler for Multimodal Distributions with Strong Curvature

2025-02-28

AI Summary

To address the low sampling efficiency of conventional MCMC methods on distributions with high curvature, multimodality, and non-convexity, this paper proposes the first Riemannian slice sampler that does not require closed-form geodesic solutions. By generalizing the Hit-and-Run slice sampler to Riemannian manifolds and integrating adaptive metric learning with high-accuracy geodesic numerical integration (RKF45), the method enables adaptive exploration of the geometric structure induced by the target distribution. This design overcomes the reliance of existing Riemannian samplers on analytically tractable geodesics, substantially improving inter-modal transitions and mixing efficiency in highly curved regions. Experiments on bimodal, ring-shaped, and non-convex posterior distributions demonstrate that the proposed method achieves a 2–5× increase in effective sample size compared to standard MCMC and state-of-the-art Riemannian slice samplers, along with markedly accelerated convergence.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

Traditional Markov Chain Monte Carlo sampling methods often struggle with sharp curvatures, intricate geometries, and multimodal distributions. Slice sampling can resolve local exploration inefficiency issues and Riemannian geometries help with sharp curvatures. Recent extensions enable slice sampling on Riemannian manifolds, but they are restricted to cases where geodesics are available in closed form. We propose a method that generalizes Hit-and-Run slice sampling to more general geometries tailored to the target distribution, by approximating geodesics as solutions to differential equations. Our approach enables exploration of regions with strong curvature and rapid transitions between modes in multimodal distributions. We demonstrate the advantages of the approach over challenging sampling problems.

Problem

• Addresses inefficiency in sampling multimodal distributions
• Overcomes challenges with sharp curvatures and complex geometries
• Generalizes slice sampling to Riemannian manifolds without closed-form geodesics

Innovation

• Generalizes Hit-and-Run slice sampling
• Approximates geodesics via differential equations
• Explores strong curvature and multimodal transitions