A Quadratically Convergent Algorithm for Structured Low-Rank Approximation

TL;DR

This paper introduces a Newton-like iterative algorithm with quadratic convergence for structured low-rank approximation problems, applicable across various applications, and demonstrates its effectiveness through implementation and experiments.

Contribution

The paper presents a novel quadratically convergent Newton-like method for structured low-rank approximation, with theoretical guarantees and practical validation across multiple applications.

Findings

Quadratic convergence of the proposed algorithm under mild assumptions.

The algorithm is competitive with state-of-the-art methods.

Successful application to problems like GCD, matrix completion, and denoising.

Abstract

Structured Low-Rank Approximation is a problem arising in a wide range of applications in Numerical Analysis and Engineering Sciences. Given an input matrix $M$, the goal is to compute a matrix $M'$ of given rank $r$ in a linear or affine subspace $E$ of matrices (usually encoding a specific structure) such that the Frobenius distance $\lVert M-M'\rVert$ is small. We propose a Newton-like iteration for solving this problem, whose main feature is that it converges locally quadratically to such a matrix under mild transversality assumptions between the manifold of matrices of rank $r$ and the linear/affine subspace $E$. We also show that the distance between the limit of the iteration and the optimal solution of the problem is quadratic in the distance between the input matrix and the manifold of rank $r$ matrices in $E$. To illustrate the applicability of this algorithm, we propose a Maple implementation and give experimental results for several applicative problems that can be modeled by Structured Low-Rank Approximation: univariate approximate GCDs (Sylvester matrices), low-rank Matrix completion (coordinate spaces) and denoising procedures (Hankel matrices). Experimental results give evidence that this all-purpose algorithm is competitive with state-of-the-art numerical methods dedicated to these problems.