Least-Squares Approximation by Elements from Matrix Orbits Achieved by Gradient Flows on Compact Lie Groups

TL;DR

This paper develops gradient-flow algorithms on compact Lie groups to efficiently approximate a matrix by sums of matrices from specified matrix orbits, with applications across various fields.

Contribution

It introduces novel gradient-flow methods on Lie groups for least-squares approximation using matrix orbits, expanding computational tools in matrix analysis.

Findings

Algorithms successfully compute best approximations within matrix orbits.

Connections established to applications in pure and applied mathematics.

Efficient convergence demonstrated for the proposed gradient flows.

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 $$\min \{\|X_1+ ... + X_N - A_0\|: X_j \in S(A_j), j = 1, >..., N\}.$$ Connections of the results to different pure and applied areas are discussed.