Matrix factorization ranks via polynomial optimization

TL;DR

This paper introduces a polynomial optimization approach using moment hierarchies and ideal-sparsity to approximate specialized matrix factorization ranks like nonnegative, completely positive, and separable ranks, which are typically hard to compute.

Contribution

It develops a novel sparse hierarchy method that improves bounds and computational efficiency for approximating matrix factorization ranks.

Findings

Sparse hierarchy yields stronger bounds.

Method potentially accelerates computations.

Applicable to various specialized matrix ranks.

Abstract

In light of recent data science trends, new interest has fallen in alternative matrix factorizations. By this, we mean various ways of factorizing particular data matrices so that the factors have special properties and reveal insights into the original data. We are interested in the specialized ranks associated with these factorizations, but they are usually difficult to compute. In particular, we consider the nonnegative-, completely positive-, and separable ranks. We focus on a general tool for approximating factorization ranks, the moment hierarchy, a classical technique from polynomial optimization, further augmented by exploiting ideal-sparsity. Contrary to other examples of sparsity, the resulting sparse hierarchy yields equally strong, if not superior, bounds while potentially delivering a speed-up in computation.