Evolution of Gaussians in the Hellinger--Kantorovich--Boltzmann gradient flow
Authors
- Liero, Matthias
ORCID: 0000-0002-0963-2915 - Mielke, Alexander
ORCID: 0000-0002-4583-3888 - Tse, Oliver
ORCID: 0000-0001-7577-3110 - Zhu, Jia-Jie
2020 Mathematics Subject Classification
- 49Q22 35Q49
Keywords
- Gaussian measures, Hellinger--Kantorovich distance, gradient-flow equation, relative Boltzmann entropy, Kullback--Leibler divergence
DOI
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--Łojasiewicz-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.
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