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

2026-05-13

AI Summary

Computing the nonnegative rank of a nonnegative matrix is an NP-hard problem, necessitating efficient methods to determine its lower bounds. This work addresses this challenge by introducing, within a nonconvex optimization framework, the first practically implementable algorithm for the self-scaled bound (SSB) and unifying the efficient computation of four classical lower bounds—FSB, RCB, HSB, and SSB. Evaluated on multiple standard benchmark matrices, the proposed method improves upon the best-known lower bounds in the literature; in several cases, these bounds match existing upper bounds, thereby establishing the exact nonnegative rank for the first time. The study thus provides a unified and practical numerical tool for advancing research on nonnegative rank lower bounds.

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.

Problem

• nonnegative rank
• lower bounds
• NP-hard
• nonnegative matrix
• matrix factorization

Innovation

• nonnegative rank
• non-convex optimization
• lower bounds
• self-scaled bound
• rectangle covering bound