Entropy 2026, 28(5), 516; https://doi.org/10.3390/e28050516 (registering DOI) - 2 May 2026
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
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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.
Full article
(This article belongs to the Special Issue SUURI of Information Geometry: Dedicated to SUURI Engineer Professor Shun’ichi Amari on the Occasion of His 90th Birthday)
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