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Gradient Systems and Asymmetric Relaxations in View of Riemannian Geometry.

Alessandro Bravetti1, Miguel Ángel García Ariza2, José Roberto Romero-Arias2

  • 1School of Science and Technology, University of Camerino, 62032 Camerino, Italy.

Entropy (Basel, Switzerland)
|May 26, 2026
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Summary

This study extends the connection between gradient flows and pregeodesics to general Riemannian manifolds, offering new insights into optimization and stochastic processes. It reveals a universal asymmetry in relaxation, where warming up is faster than cooling down.

Keywords:
asymmetric relaxationsgradient flowsinformation geometrynon-metricity tensor

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Area of Science:

  • Differential Geometry
  • Information Geometry
  • Optimization Theory

Background:

  • Amari's work established a link between gradient flows and pregeodesics in dually flat manifolds.
  • This relationship is crucial for understanding information geometry and optimization.

Purpose of the Study:

  • To generalize the study of gradient flows and pregeodesics to arbitrary Riemannian manifolds.
  • To develop a framework beyond Hessian manifolds by relaxing flatness and symmetry conditions.
  • To provide a geometric criterion for comparing relaxation dynamics.

Main Methods:

  • Extending Amari's framework to general Riemannian manifolds.
  • Formulating a criterion based on the non-metricity tensor for gradient descent curves.
  • Analyzing Gaussian chains as a case study.

Main Results:

  • A generalized relationship between gradient flows and pregeodesics is established for non-flat, non-symmetric connections.
  • A novel criterion for comparing relaxation along gradient descent curves is derived.
  • The universal asymmetry in relaxation (warming up faster than cooling down) for Gaussian chains is recovered.

Conclusions:

  • Geometric insights from Amari's legacy provide new perspectives on optimization and stochastic processes.
  • The generalized framework offers broader applicability beyond traditional information geometry.
  • The study highlights the utility of non-metricity tensor in analyzing dynamic processes.