Work

Gradient Systems and Asymmetric Relaxations in View of Riemannian Geometry

Year: 2026

Type: article

Abstract: In dually flat manifolds, there is a deep connection between gradient flows and pregeodesics. This was one of the many important contributions of Amari to information geometry. In this paper, we extend the study of this relationship to general Riemannian manifolds. Our result does not impose conditions of flatness on the connection or symmetry on its non-metricity tensor, thus broadening the geometric setting beyond Hessian manifolds. Within this framework, we provide a criterion for comparing relaxation along two different gradient descent curves of a function, formulated in terms of the non-metricity tensor of a connection for which the gradient curves are pregeodesics. We use it to study Gaussian chains, whose relaxation trajectories coincide with gradient descent curves in the space of Gaussian distributions. Thus, we recover a recent result that establishes a universal asymmetry: warming up is faster than cooling down. Our work illustrates how geometric insights rooted in Amari's legacy offer new perspectives for optimization problems and stochastic processes.

Source: Entropy

DOI: 10.3390/e28050516

Authors: Alessandro Bravetti, Miguel Ángel García Ariza, José Roberto Romero-Arias

Institutions: Università di Camerino, Universidad Nacional Autónoma de México

Language: English

Topic: Geometric Analysis and Curvature Flows

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $3,137

Funders: Universidad Nacional Autónoma de México, Secretaría de Ciencia, Tecnología e Innovación del Distrito Federal, Stony Brook University, Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México