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Gradient systems and asymmetric relaxations in view of Riemannian geometry

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

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.

Gradient systems and asymmetric relaxations in view of Riemannian geometry

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.

Paper Structure

This paper contains 6 sections, 4 theorems, 36 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Gradient flows on dually flat manifolds
  3. Gradient flows follow pregeodesics on Riemannian manifolds
  4. Asymmetric relaxations on Riemannian manifolds
  5. Universal asymmetry in relaxations of Gaussian chains
  6. Concluding remarks

Key Result

Theorem 3.1

For any function $f$, there exists a symmetric connection $\tilde{\nabla}^f$ defined on $M$ minus the critical points of $f$ by for some real function $\lambda$ and satisfying Eq. eq:gradientpregeodesic. Hence, every gradient curve of $f$ is a pregeodesic of $\tilde{\nabla}^f$. $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: Geodesic projection property. The $\nabla^f$-geodesic $\gamma$ from $q$ to $\hat{p}$ (left) meets the submanifold $S$ orthogonally at the minimizer $\hat{p}$ of $f$ on $S$ (right).
  • Figure 2: An illustration of our asymmetry criterion. Upper left:$\Delta f(t)=f(\gamma_1(t))-f(\gamma_2(t))$ attains its maximum at $t=t_*$. Upper right: Two gradient descent curves $\gamma_1$ and $\gamma_2$ with $f$-equidistant initial conditions. At $t_*$, their velocities (red) have equal length. Lower left: Heat map of $-C^f(\operatorname{grad }f,\operatorname{grad} f,\operatorname{grad }f)$ (lighter = larger). At $t_*$, $C^f(\gamma_1,\gamma_1,\gamma_1)>C^f(\gamma_2,\gamma_2,\gamma_2)$, so $\gamma_2$ relaxes faster. Lower right: Covariant acceleration (blue) at $t_*$. Its tangential component (red) is negative for both curves, larger in magnitude for $\gamma_2$, explaining its faster relaxation.

Theorems & Definitions (9)

  • Theorem 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Definition 4.1
  • Theorem 4.2
  • proof
  • Corollary 4.3
  • proof