Sampling from multi-modal distributions on Riemannian manifolds with training-free stochastic interpolants

2026-01-31

AI Summary

Efficient sampling from multimodal, unnormalized density functions on Riemannian manifolds remains challenging, as existing methods often fail to simultaneously respect the underlying geometric structure and capture complex distributional characteristics. This work proposes a training-free sampling framework that extends the principles of diffusion models to Riemannian manifolds for the first time. By constructing geometrically compatible stochastic interpolation paths and coupling them with nonequilibrium deterministic dynamics, the method gradually transports an easily sampled noise distribution toward the target distribution. Relying solely on standard Monte Carlo techniques and incorporating iterative posterior sampling, the approach demonstrates strong empirical performance in high-dimensional, heavy-tailed, and multimodal settings, offering both theoretical rigor and broad applicability.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Sampling/Simulation-based Search
• Computer Vision: Diffusion Models for Vision

Application Category

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

Abstract

In this paper, we propose a general methodology for sampling from un-normalized densities defined on Riemannian manifolds, with a particular focus on multi-modal targets that remain challenging for existing sampling methods. Inspired by the framework of diffusion models developed for generative modeling, we introduce a sampling algorithm based on the simulation of a non-equilibrium deterministic dynamics that transports an easy-to-sample noise distribution toward the target. At the marginal level, the induced density path follows a prescribed stochastic interpolant between the noise and target distributions, specifically constructed to respect the underlying Riemannian geometry. In contrast to related generative modeling approaches that rely on machine learning, our method is entirely training-free. It instead builds on iterative posterior sampling procedures using only standard Monte Carlo techniques, thereby extending recent diffusion-based sampling methodologies beyond the Euclidean setting. We complement our approach with a rigorous theoretical analysis and demonstrate its effectiveness on a range of multi-modal sampling problems, including high-dimensional and heavy-tailed examples.

Problem

• multi-modal sampling
• Riemannian manifolds
• un-normalized densities
• stochastic interpolants
• training-free

Innovation

• Riemannian manifold sampling
• training-free interpolants
• multi-modal distributions
• diffusion-based sampling
• stochastic interpolants