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

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