Lower bounds on nonnegative rank via nonnegative nuclear norms

TL;DR

This paper introduces a new lower bound on the nonnegative rank of matrices using a nonnegative nuclear norm, applicable to arbitrary matrices and computable via semidefinite programming, improving upon existing bounds.

Contribution

It proposes a novel nonnegative nuclear norm-based lower bound for nonnegative rank that does not depend on matrix sparsity patterns and can be efficiently computed.

Findings

The new bound outperforms existing bounds on certain matrices.

The bound is expressed as a copositive programming problem.

Relaxations allow polynomial-time computation.

Abstract

The nonnegative rank of an entrywise nonnegative matrix A of size mxn is the smallest integer r such that A can be written as A=UV where U is mxr and V is rxn and U and V are both nonnegative. The nonnegative rank arises in different areas such as combinatorial optimization and communication complexity. Computing this quantity is NP-hard in general and it is thus important to find efficient bounding techniques especially in the context of the aforementioned applications. In this paper we propose a new lower bound on the nonnegative rank which, unlike most existing lower bounds, does not explicitly rely on the matrix sparsity pattern and applies to nonnegative matrices with arbitrary support. The idea involves computing a certain nuclear norm with nonnegativity constraints which allows to lower bound the nonnegative rank, in the same way the standard nuclear norm gives lower bounds on the standard rank. Our lower bound is expressed as the solution of a copositive programming problem and can be relaxed to obtain polynomial-time computable lower bounds using semidefinite programming. We compare our lower bound with existing ones, and we show examples of matrices where our lower bound performs better than currently known ones.