Evolution of Gaussians in the Hellinger-Kantorovich-Boltzmann gradient flow

2025-04-29

AI Summary

This work investigates the evolution of Gaussian measures under the gradient flow of relative Boltzmann entropy with respect to a Gaussian reference measure, within the Hellinger–Kantorovich (HK) geometry. We first establish invariance of the gradient flow within the class of Gaussian measures and derive an explicit system of ordinary differential equations governing the dynamics of mean, covariance, and mass parameters. Second, we reveal the semiconvexity of sublevel sets in the parameter space and prove a Polyak–Łojasiewicz-type inequality, thereby guaranteeing global exponential convergence as well as exponential convergence restricted to sublevel sets. Moreover, we precisely quantify the decay rate of covariance eigenvalues. Our theoretical results extend to strongly log-λ-concave non-Gaussian target distributions. Numerical experiments—including Bayesian logistic regression—validate the predicted convergence behavior and estimation accuracy.

Technology Category

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

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

This study leverages the basic insight that the gradient-flow equation associated with the relative Boltzmann entropy, in relation to a Gaussian reference measure within the Hellinger-Kantorovich (HK) geometry, preserves the class of Gaussian measures. This invariance serves as the foundation for constructing a reduced gradient structure on the parameter space characterizing Gaussian densities. We derive explicit ordinary differential equations that govern the evolution of mean, covariance, and mass under the HK-Boltzmann gradient flow. The reduced structure retains the additive form of the HK metric, facilitating a comprehensive analysis of the dynamics involved. We explore the geodesic convexity of the reduced system, revealing that global convexity is confined to the pure transport scenario, while a variant of sublevel semi-convexity is observed in the general case. Furthermore, we demonstrate exponential convergence to equilibrium through Polyak-Lojasiewicz-type inequalities, applicable both globally and on sublevel sets. By monitoring the evolution of covariance eigenvalues, we refine the decay rates associated with convergence. Additionally, we extend our analysis to non-Gaussian targets exhibiting strong log-lambda-concavity, corroborating our theoretical results with numerical experiments that encompass a Gaussian-target gradient flow and a Bayesian logistic regression application.

Problem

• Analyzes Gaussian evolution in HK-Boltzmann gradient flow dynamics
• Explores geodesic convexity and convergence rates in reduced system
• Extends analysis to non-Gaussian targets with log-concavity

Innovation

• Gradient-flow equation preserves Gaussian measures
• Reduced gradient structure on Gaussian parameter space
• Explicit ODEs for mean, covariance, mass evolution