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

Adrian Lewis, Tonghua Tian

TL;DR

The paper develops a unified, metric-space framework for identifiability and its interaction with the Kurdyka-Łojasiewicz property, connecting discrete-time iterates and continuous-time subgradient dynamics. It introduces a slope-based notion of identifiability, establishes a linear-growth principle via the Ekeland variational principle, and shows KL properties can be inherited from restrictions to identifiable sets. It then specializes to Euclidean spaces, detailing four nonsmooth function classes, max-function analyses, and the relationship between identifiability and partial smoothness. Finally, it proves that subgradient curves identify onto identifiable manifolds in continuous time and describes the resulting reduced dynamics, providing a cohesive theory for structure discovery and convergence in nonsmooth optimization.

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.

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

TL;DR

The paper develops a unified, metric-space framework for identifiability and its interaction with the Kurdyka-Łojasiewicz property, connecting discrete-time iterates and continuous-time subgradient dynamics. It introduces a slope-based notion of identifiability, establishes a linear-growth principle via the Ekeland variational principle, and shows KL properties can be inherited from restrictions to identifiable sets. It then specializes to Euclidean spaces, detailing four nonsmooth function classes, max-function analyses, and the relationship between identifiability and partial smoothness. Finally, it proves that subgradient curves identify onto identifiable manifolds in continuous time and describes the resulting reduced dynamics, providing a cohesive theory for structure discovery and convergence in nonsmooth optimization.

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.
Paper Structure (12 sections, 38 theorems, 88 equations, 1 figure)

This paper contains 12 sections, 38 theorems, 88 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Identifiability in metric spaces
  3. Critical sequences in metric spaces
  4. Identifiable sets in metric spaces
  5. Linear growth around identifiable sets
  6. The Kurdyka-Łojasiewicz property
  7. Identifiability in Euclidean spaces
  8. Four classes of nonsmooth functions
  9. Max functions
  10. Identifiability and partial smoothness
  11. Subgradient curves and identification
  12. Trajectories for primal lower nice functions

Key Result

Proposition 2.2

Functions $f$ and $g$ on a metric space always satisfy for all points $x$ and scalars $\delta > 0$. Furthermore, if $f$ is $L$-lipschitz, then $|\nabla f|(x) \le L$.

Figures (1)

  • Figure 1: Subgradient curves for the objective $5 |x_2-x_1^2| + x_1^2$.

Theorems & Definitions (58)

  • Definition 2.1
  • Proposition 2.2
  • Theorem 2.3: Slope characterization of length spaces aze-corvellec
  • Proposition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7: Distance formula
  • Lemma 2.9
  • Lemma 2.10
  • Proposition 2.11: Proximal images of minimizing sequences
  • ...and 48 more