A Projected Subgradient Method for the Computation of Adapted Metrics for Dynamical Systems

TL;DR

This paper develops an advanced projected subgradient method to compute Riemannian metrics for dynamical systems, incorporating inexact gradients and various step size strategies, with applications to the Hénon map.

Contribution

It introduces a projected subgradient algorithm with convergence guarantees for computing adapted metrics, considering inexact gradients and multiple step size rules.

Findings

Existence of minimizers for the proposed method.

Effective application to Hénon map dimension and entropy estimation.

Robustness of the method with inexact subgradients.

Abstract

In this paper, we extend a recently established subgradient method for the computation of Riemannian metrics that optimizes certain singular value functions associated with dynamical systems. This extension is threefold. First, we introduce a projected subgradient method which results in Riemannian metrics whose parameters are confined to a compact convex set and we can thus prove that a minimizer exists; second, we allow inexact subgradients and study the effect of the errors on the computed metrics; and third, we analyze the subgradient algorithm for three different choices of step sizes: constant, exogenous and Polyak. The new methods are illustrated by application to dimension and entropy estimation of the H\'enon map.