Preservation of Log-Concavity and Convergence of Wasserstein-Fisher-Rao Gradient Flows

2026-09-16

AI Summary

研究利用Wasserstein-Fisher-Rao梯度流解决从仅知归一化常数的概率分布中采样的收敛问题，通过保持强对数凹性并结合两种动力学方法加速收敛。

Abstract

We study the convergence of Wasserstein-Fisher-Rao (WFR) gradient flows for sampling from probability distributions known up to a normalisation constant. By combining Wasserstein transport with Fisher-Rao birth-death dynamics, WFR flows balance exploration and selection. These flows have been recognised as a promising mechanism to accelerate convergence beyond Langevin dynamics. We show that for a class of strongly log-concave target distributions satisfying additional curvature conditions, WFR flows preserve strong log-concavity, in contrast to Wasserstein flows which enjoy this property only in the Gaussian setting. Exploiting this result, we derive explicit non-asymptotic convergence rates for the symmetrised Kullback-Leibler divergence, without requiring a warm-start as required in current estimates. In particular, we show that the convergence rate decomposes additively into Wasserstein and Fisher-Rao contributions, thereby confirming a recent conjecture within this setting. These results provide refined convergence guarantees and further develop the theoretical foundations of WFR gradient flows for sampling and Bayesian inference.

Problem

• Wasserstein-Fisher-Rao
• log-concavity
• convergence
• gradient flows
• Kullback-Leibler divergence

Innovation

• Wasserstein-Fisher-Rao
• log-concavity
• convergence rate
• Kullback-Leibler divergence
• Bayesian inference