Least-squares approximation by elements from matrix orbits achieved by gradient flows on compact lie groups

Abstract

Let $S(A)$ denote the orbit of a complex or real matrix $A$ under a certain equivalence relation such as unitary similarity, unitary equivalence, unitary congruences etc. Efficient gradient-flow algorithms are constructed to determine the best approximation of a given matrix $A_0$ by the sum of matrices in $S(A_1), \dots , S(A_N)$ in the sense of finding the Euclidean least-squares distance \[ \min \Big \{\big \|X_1+ \cdots + X_N - A_0\big \|: X_j \in S(A_j), \ j = 1, \dots , N\Big \}.\] Connections of the results to different pure and applied areas are discussed.