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

Understand

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.