Identifiability and Error Bound: Metric and Geometric Perspectives

TL;DR

This paper explores the Error Bound property in optimization, demonstrating its equivalence on ambient space and identifiable manifolds through metric and geometric analyses, with applications to -regularized problems.

Contribution

It establishes the equivalence of local Error Bound conditions on ambient space and identifiable manifolds using metric and geometric perspectives.

Findings

Proves local EB equivalence between ambient space and manifolds under mild conditions.

Provides metric and geometric analyses for understanding EB property.

Recovers known EB results for -regularized optimization.

Abstract

Identifiability means that iterates generated by optimization algorithms are eventually confined to an identifiable set. This property is computationally useful because minimizing a nonsmooth function near a critical point reduces to minimizing its smooth restriction on the corresponding identifiable manifold. Motivated by this reduction, we study the Error Bound (EB) property from both ambient and manifold viewpoints. Under mild assumptions in Euclidean space, we prove that local EB on $(\mathbb{R}^n,d)$ is equivalent to local EB on an identifiable manifold $(\mathcal{M},d)$. We establish this result from two complementary perspectives: a metric analysis based on slope and linear growth away from $\mathcal{M}$, and a geometric analysis based on subdifferentials, partial smoothness, and $\mathcal{VU}$-theory. As an application, we recover the EB equivalence for $\ell_1$-regularized optimization in the literature.