Orthogonal Invariance and Identifiability

TL;DR

This paper explores how orthogonally invariant functions of symmetric matrices inherit properties from their diagonal restrictions, focusing on identifiability and its role in optimization methods.

Contribution

It provides a theoretical analysis of identifiability in orthogonally invariant functions, linking it to properties like partial smoothness and active set methods.

Findings

Polyhedral functions are always partly smooth.

Many eigenvalue optimization functions exhibit partial smoothness.

Identifiability underpins active set methods in nonsmooth optimization.

Abstract

Orthogonally invariant functions of symmetric matrices often inherit properties from their diagonal restrictions: von Neumann's theorem on matrix norms is an early example. We discuss the example of "identifiability", a common property of nonsmooth functions associated with the existence of a smooth manifold of approximate critical points. Identifiability (or its synonym, "partial smoothness") is the key idea underlying active set methods in optimization. Polyhedral functions, in particular, are always partly smooth, and hence so are many standard examples from eigenvalue optimization.