Computing Lower Bounds on the Nonnegative Rank via Non-Convex Optimization Solvers

TL;DR

This paper introduces non-convex optimization methods to compute lower bounds on the nonnegative rank of matrices, including the first algorithm for the self-scaled bound, improving bounds and establishing exact ranks in some cases.

Contribution

It proposes novel non-convex formulations for four key lower bounds on nonnegative rank, including the first algorithm for the self-scaled bound, enhancing bound accuracy and computational efficiency.

Findings

Improved lower bounds on nonnegative rank for certain matrices.

Established exact nonnegative rank in some cases by matching bounds.

Provided competitive alternative methods on benchmark matrices.

Abstract

The nonnegative rank of a nonnegative matrix $X$ is the smallest number of nonnegative rank-one factors that sum to $X$. Since computing the nonnegative rank is NP-hard, it is common to circumvent this issue by computing lower and upper bounds. In this paper, we propose non-convex formulations and practical implementations for four important lower bounds for the nonnegative rank, namely the fooling set bound (FSB), the rectangle covering bound (RCB), the hyperplane separation bound (HSB), and the self-scaled bound (SSB). In particular, our algorithm for computing the SSB is the first available in the literature, to the best of our knowledge. It allows us to improve the best known lower bound on the nonnegative rank for some matrices. In some cases, they coincide with the best known upper bound, thereby establishing their exact nonnegative rank for the first time. Moreover, on canonical benchmarks, we show that our non-convex approaches provide a meaningful and often competitive alternative to standard methods. The paper also provides a consolidated reference for the current state of several classical lower bounds on a large number of benchmark matrices.