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

• URL: http://arxiv.org/abs/2504.20400v1
• Date: Tue, 29 Apr 2025 03:54:56 GMT
• Title: Evolution of Gaussians in the Hellinger-Kantorovich-Boltzmann gradient flow
• Authors: Matthias Liero, Alexander Mielke, Oliver Tse, Jia-Jie Zhu,
• Abstract summary: We derive explicit ordinary equations govern the evolution of mean, covariance, and mass under HK-Boltzmann flow.<n>We demonstrate exponential convergence to equilibrium through constructing a gradient structure associated with a Polyak-Lojasiewicz-type inequalities.<n>We extend our analysis to non-Gaussian targets exhibiting strong log-lambda-concavity.
• Score: 44.500314182711236
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

Related papers

• Hellinger-Kantorovich Gradient Flows: Global Exponential Decay of Entropy Functionals [52.154685604660465]
  We investigate a family of gradient flows of positive and probability measures, focusing on the Hellinger-Kantorovich (HK) geometry.<n>A central contribution is a complete characterization of global exponential decay behaviors of entropy functionals under Otto-Wasserstein and Hellinger-type gradient flows.
  arXiv  Detail & Related papers  (2025-01-28T16:17:09Z)
• Reflection coupling for unadjusted generalized Hamiltonian Monte Carlo in the nonconvex stochastic gradient case [0.0]
  Contraction in Wasserstein 1-distance with explicit rates is established for generalized Hamiltonian Monte Carlo with gradients under possibly non diffusion conditions. The algorithms considered include kinetic splitting schemes of Langevin, commonly used in molecular dynamics simulations.
  arXiv  Detail & Related papers  (2023-10-28T18:25:59Z)
• Curvature-Independent Last-Iterate Convergence for Games on Riemannian Manifolds [77.4346324549323]
  We show that a step size agnostic to the curvature of the manifold achieves a curvature-independent and linear last-iterate convergence rate. To the best of our knowledge, the possibility of curvature-independent rates and/or last-iterate convergence has not been considered before.
  arXiv  Detail & Related papers  (2023-06-29T01:20:44Z)
• Convergence and concentration properties of constant step-size SGD through Markov chains [0.0]
  We consider the optimization of a smooth and strongly convex objective using constant step-size gradient descent (SGD) We show that, for unbiased gradient estimates with mildly controlled variance, the iteration converges to an invariant distribution in total variation distance. All our results are non-asymptotic and their consequences are discussed through a few applications.
  arXiv  Detail & Related papers  (2023-06-20T12:36:28Z)
• Universal coding, intrinsic volumes, and metric complexity [3.4265828682659705]
  In this assignment, the goal is to predict, or equivalently, a sequence of real-valued observations almost as well as the best distribution with mean-valued subsets of $mathbbRn. As part of our analysis, we also characterize the minimax redundancy for a general constraint set. We contrast our findings with classical results in information theory.
  arXiv  Detail & Related papers  (2023-03-13T16:54:04Z)
• Page curves and typical entanglement in linear optics [0.0]
  We study entanglement within a set of squeezed modes that have been evolved by a random linear optical unitary. We prove various results on the typicality of entanglement as measured by the R'enyi-2 entropy. Our main make use of a symmetry property obeyed by the average and the variance of the entropy that dramatically simplifies the averaging over unitaries.
  arXiv  Detail & Related papers  (2022-09-14T18:00:03Z)
• Nonconvex Stochastic Scaled-Gradient Descent and Generalized Eigenvector Problems [98.34292831923335]
  Motivated by the problem of online correlation analysis, we propose the emphStochastic Scaled-Gradient Descent (SSD) algorithm. We bring these ideas together in an application to online correlation analysis, deriving for the first time an optimal one-time-scale algorithm with an explicit rate of local convergence to normality.
  arXiv  Detail & Related papers  (2021-12-29T18:46:52Z)
• On Convergence of Training Loss Without Reaching Stationary Points [62.41370821014218]
  We show that Neural Network weight variables do not converge to stationary points where the gradient the loss function vanishes. We propose a new perspective based on ergodic theory dynamical systems.
  arXiv  Detail & Related papers  (2021-10-12T18:12:23Z)
• Spectral clustering under degree heterogeneity: a case for the random walk Laplacian [83.79286663107845]
  This paper shows that graph spectral embedding using the random walk Laplacian produces vector representations which are completely corrected for node degree. In the special case of a degree-corrected block model, the embedding concentrates about K distinct points, representing communities.
  arXiv  Detail & Related papers  (2021-05-03T16:36:27Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.