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 $S(A_1),...,S(A_N)$ in the sense of finding the Euclidean least-squares distance $$\min \{\Vert X_1+...+X_N-A_0\Vert :X_j\in S(A_j),j=1,>...,N\}.$$ Connections of the results to different pure and applied areas are discussed.

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