Exact solutions in low-rank approximation with zeros

TL;DR

This paper explores the algebraic structure of critical points in low-rank matrix approximation problems with fixed zero patterns, revealing relations and complexity aspects relevant to nonnegative matrix factorization.

Contribution

It provides a detailed algebraic analysis of critical points, including relations they satisfy and their count, linking to the complexity of nonnegative matrix factorization.

Findings

Characterization of algebraic relations among critical points

Analysis of the number of critical points in the problem

Insights into the complexity of nonnegative matrix factorization

Abstract

Low-rank approximation with zeros aims to find a matrix of fixed rank and with a fixed zero pattern that minimizes the Euclidean distance to a given data matrix. We study the critical points of this optimization problem using algebraic tools. In particular, we describe special linear, affine, and determinantal relations satisfied by the critical points. We also investigate the number of critical points and how this number is related to the complexity of nonnegative matrix factorization problem.