Radon--Wasserstein Gradient Flows for Interacting-Particle Sampling in High Dimensions

2026-02-05

AI Summary

This work addresses the high computational complexity and inaccurate particle approximation commonly encountered in KL-divergence-based gradient flow methods for high-dimensional sampling. The authors propose a novel Riemannian geometric structure grounded in the Radon transform, which enables the construction of gradient flows that rely solely on one-dimensional projections. This design reduces the per-iteration computational complexity to linear in both the number of particles and the ambient dimension. By introducing the Radon–Wasserstein metric and a regularized RRW geometry, the resulting algorithm achieves scalability in high dimensions, leverages fast Fourier transform acceleration, and admits efficient implementation via one-dimensional convolutions. Theoretical analysis establishes long-term convergence guarantees, and empirical experiments confirm that the method maintains accurate particle approximations even in high-dimensional settings.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

Abstract

Gradient flows of the Kullback--Leibler (KL) divergence, such as the Fokker--Planck equation and Stein Variational Gradient Descent, evolve a distribution toward a target density known only up to a normalizing constant. We introduce new gradient flows of the KL divergence with a remarkable combination of properties: they admit accurate interacting-particle approximations in high dimensions, and the per-step cost scales linearly in both the number of particles and the dimension. These gradient flows are based on new transportation-based Riemannian geometries on the space of probability measures: the Radon--Wasserstein geometry and the related Regularized Radon--Wasserstein (RRW) geometry. We define these geometries using the Radon transform so that the gradient-flow velocities depend only on one-dimensional projections. This yields interacting-particle-based algorithms whose per-step cost follows from efficient Fast Fourier Transform-based evaluation of the required 1D convolutions. We additionally provide numerical experiments that study the performance of the proposed algorithms and compare convergence behavior and quantization. Finally, we prove some theoretical results including well-posedness of the flows and long-time convergence guarantees for the RRW flow.

Problem

• interacting-particle sampling
• high dimensions
• KL divergence gradient flows
• computational scalability
• probability measures

Innovation

• Radon–Wasserstein geometry
• interacting-particle sampling
• high-dimensional gradient flows
• KL divergence
• Fast Fourier Transform