Factorization approach to structured low-rank approximation with applications

TL;DR

This paper introduces a factorization-based method for structured low-rank matrix approximation, effectively handling missing data and small target ranks, with applications in system identification and tensor decomposition.

Contribution

It proposes a novel local optimization algorithm that enforces structure via penalty terms, addressing limitations of kernel-based methods for small ranks.

Findings

Consistently good performance on numerical examples

Effective handling of missing or fixed elements

Outperforms existing approaches in key applications

Abstract

We consider the problem of approximating an affinely structured matrix, for example a Hankel matrix, by a low-rank matrix with the same structure. This problem occurs in system identification, signal processing and computer algebra, among others. We impose the low-rank by modeling the approximation as a product of two factors with reduced dimension. The structure of the low-rank model is enforced by introducing a penalty term in the objective function. The proposed local optimization algorithm is able to solve the weighted structured low-rank approximation problem, as well as to deal with the cases of missing or fixed elements. In contrast to approaches based on kernel representations (in linear algebraic sense), the proposed algorithm is designed to address the case of small targeted rank. We compare it to existing approaches on numerical examples of system identification, approximate greatest common divisor problem, and symmetric tensor decomposition and demonstrate its consistently good performance.