Computing Lower Rank Approximations of Matrix Polynomials

TL;DR

This paper introduces an efficient iterative algorithm for finding the nearest matrix polynomial with a specified lower rank, proving existence, regularity, and convergence properties, with demonstrated numerical robustness.

Contribution

The paper develops a new algorithm for computing lower rank approximations of matrix polynomials, establishing theoretical guarantees and practical effectiveness.

Findings

Existence of minimal distance lower rank matrix polynomials proven.

Algorithm converges quadratically with a good initial guess.

Implementation shows robustness and efficiency in numerical tests.

Abstract

Given an input matrix polynomial whose coefficients are floating point numbers, we consider the problem of finding the nearest matrix polynomial which has rank at most a specified value. This generalizes the problem of finding a nearest matrix polynomial that is algebraically singular with a prescribed lower bound on the dimension given in a previous paper by the authors. In this paper we prove that such lower rank matrices at minimal distance always exist, satisfy regularity conditions, and are all isolated and surrounded by a basin of attraction of non-minimal solutions. In addition, we present an iterative algorithm which, on given input sufficiently close to a rank-at-most matrix, produces that matrix. The algorithm is efficient and is proven to converge quadratically given a sufficiently good starting point. An implementation demonstrates the effectiveness and numerical robustness of our algorithm in practice.