Identifiability, the KL property in metric spaces, and subgradient curves

TL;DR

This paper explores the concept of identifiability in optimization, linking it to the KL property and subgradient curves, providing new metric-based perspectives applicable beyond Euclidean spaces.

Contribution

It introduces a simple metric property for identifiability and analyzes its continuous-time analog for subgradient descent, extending classical ideas to broader settings.

Findings

Identifiability can be characterized by a simple metric property.

The KL property influences convergence in both discrete and continuous optimization.

Continuous-time subgradient curves exhibit behaviors analogous to discrete identifiability.

Abstract

Identifiability, and the closely related idea of partial smoothness, unify classical active set methods and more general notions of solution structure. Diverse optimization algorithms generate iterates in discrete time that are eventually confined to identifiable sets. We present two fresh perspectives on identifiability. The first distills the notion to a simple metric property, applicable not just in Euclidean settings but to optimization over manifolds and beyond; the second reveals analogous continuous-time behavior for subgradient descent curves. The Kurdya-Lojasiewicz property typically governs convergence in both discrete and continuous time: we explore its interplay with identifiability.