Lower bounds on nonnegative rank via nonnegative nuclear norms

Abstract

The nonnegative rank of an entrywise nonnegative matrix \(A \in \mathbb {R}^{m \times n}_+\) is the smallest integer \(r\) such that \(A\) can be written as \(A=UV\) where \(U \in \mathbb {R}^{m \times r}_+\) and \(V \in \mathbb {R}^{r \times n}_+\) 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 solely 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.