On rank-2 Nonnegative Matrix Factorizations and their variants

TL;DR

This paper explores the theory and practical algorithms for finding the best nonnegative rank-2 approximation of matrices, introducing explicit parametrizations and improved initialization methods to enhance computational efficiency and solution quality.

Contribution

It provides an explicit parametrization of all nonnegative rank-2 factorizations and proposes a new initialization method to improve the efficiency and accuracy of existing algorithms.

Findings

The explicit parametrization enables better understanding of nonnegative rank-2 factorizations.

The proposed initialization improves the convergence and quality of the Alternating Nonnegative Least Squares method.

Numerical experiments demonstrate the effectiveness of the new approach across various scenarios.

Abstract

We consider the problem of finding the best nonnegative rank-2 approximation of an arbitrary nonnegative matrix. We first revisit the theory, including an explicit parametrization of all possible nonnegative factorizations of a nonnegative matrix of rank 2. Based on this result, we construct a cheaply computable (albeit suboptimal) nonnegative rank-2 approximation for an arbitrary nonnegative matrix input. This can then be used as a starting point for the Alternating Nonnegative Least Squares method to find a nearest approximate nonnegative rank-2 factorization of the input; heuristically, our newly proposed initial value results in both improved computational complexity and enhanced output quality. We provide extensive numerical experiments to support these claims. Motivated by graph-theoretical applications, we also study some variants of the problem, including matrices with symmetry constraints.