Counting real critical points of the distance to orthogonally invariant matrix sets

TL;DR

This paper introduces a framework for counting real critical points of the Euclidean distance to orthogonally invariant matrix sets, simplifying calculations via transfer principles in singular value space.

Contribution

It provides a novel method to compute and count critical points for matrix sets, connecting to Euclidean distance degree concepts and simplifying complex algebraic calculations.

Findings

Framework for counting critical points using transfer principles

Simplified formulas for specific matrix sets

Comparison with Euclidean distance degree concepts

Abstract

Minimizing the Euclidean distance to a set arises frequently in applications. When the set is algebraic, a measure of complexity of this optimization problem is its number of critical points. In this paper we provide a general framework to compute and count the real smooth critical points of a data matrix on an orthogonally invariant set of matrices. The technique relies on "transfer principles" that allow calculations to be done in the space of singular values of the matrices in the orthogonally invariant set. The calculations often simplify greatly and yield transparent formulas. We illustrate the method on several examples, and compare our results to the recently introduced notion of Euclidean distance degree of an algebraic variety.