Wasserstein-Fisher-Rao gradient flows preserve log-concavity and speed convergence
Preservation of Log-Concavity and Convergence of Wasserstein-Fisher-Rao Gradient Flows
Machine Learning
Summary
Sampling from complex probability distributions is important for many applications but can be slow. The authors studied a method called Wasserstein-Fisher-Rao (WFR) gradient flows that mixes movement and birth-death dynamics to explore and select samples more effectively. They found that for certain well-behaved distributions, the WFR method keeps useful mathematical properties that other methods lose, which helps them show faster and more reliable convergence. This understanding helps improve algorithms for Bayesian inference and other sampling tasks.
What this means in practice
- •For machine learning engineers: Develop faster algorithms for sampling from complex probability models in Bayesian inference without needing carefully chosen starting points.
- •For statistical data analysts: Improve convergence guarantees when applying sampling methods to estimate probabilistic models more reliably in high-dimensional settings.
Authors
Francesca Romana Crucinio, Sahani Pathiraja
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.