A Deterministic Optimal Solution to Nonnegative Matrix Factorization

Karan Vikyath Veeranna Rupashree, Daniel L. Pimentel-Alarcón

Keywords: polyhedral, geometric, extreme rays

Abstract: This paper presents a new approach to nonnegative matrix factorization (NMF) that directly focuses on the subspace structure of the data instead of the specific samples. The main idea is to find the borders of the data's principal subspace with the nonnegative orthant, which we call nonnegative subspace edges (NoSEs), and construct the factorization according to these NoSEs. We introduce a deterministic algorithm to find NoSEs in linear time, and straight-forward techniques to obtain the desired NMF from these NoSEs. We show that this approach defines a deterministic, optimal, unique, and well-posed solution to NMF. To understand the importance of this result, consider the Moore-Penrose pseudo-inverse, which determines an optimal (minimum-norm) solution to ill-posed linear systems. Analogously, NoSEs provide an optimal (widest-cone) solution to the ill-posed problem of NMF.

Primary Area: unsupervised, self-supervised, semi-supervised, and supervised representation learning

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Submission Number: 12097