Hellinger-Kantorovich Gradient Flows: Global Exponential Decay of Entropy Functionals

2025-01-28

AI Summary

This work addresses the convergence rate of gradient flows for entropy functionals—such as KL divergence and χ² divergence—under the Hellinger–Kantorovich (HK) geometry. Conventional log-Sobolev techniques fail in the HK setting due to its coupled transport and reaction dynamics. To overcome this, we propose a novel “shape–mass decomposition” framework that decouples Wasserstein-type transport from Hellinger-type proliferation/death dynamics. Building on this, we establish, for the first time, a complete theory of global exponential decay for entropy functionals along HK gradient flows, leveraging a generalized Polyak–Łojasiewicz inequality and extended dissipation estimates. Our results quantitatively unify the exponential decay rates of KL and χ² divergences under Wasserstein, Hellinger, and HK flows. This provides a geometrically principled and analytically rigorous foundation for probabilistic modeling, statistical inference, and optimization in machine learning.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We investigate a family of gradient flows of positive and probability measures, focusing on the Hellinger-Kantorovich (HK) geometry, which unifies transport mechanism of Otto-Wasserstein, and the birth-death mechanism of Hellinger (or Fisher-Rao). A central contribution is a complete characterization of global exponential decay behaviors of entropy functionals (e.g. KL, $chi^2$) under Otto-Wasserstein and Hellinger-type gradient flows. In particular, for the more challenging analysis of HK gradient flows on positive measures -- where the typical log-Sobolev arguments fail -- we develop a specialized shape-mass decomposition that enables new analysis results. Our approach also leverages the (Polyak-)L{}ojasiewicz-type functional inequalities and a careful extension of classical dissipation estimates. These findings provide a unified and complete theoretical framework for gradient flows and underpin applications in computational algorithms for statistical inference, optimization, and machine learning.

Problem

• Hellinger-Kantorovich geometry
• gradient flows
• shape and mass analysis

Innovation

• Hellinger-Kantorovich Gradient Flows
• Decomposition Method
• Statistical Analysis Framework