Robust Near-Separable Nonnegative Matrix Factorization Using Linear Optimization

TL;DR

This paper introduces an improved linear programming approach for near-separable nonnegative matrix factorization that is more robust, does not require prior normalization or known rank, and effectively handles noise and outliers.

Contribution

The authors propose a generalized LP model for NMF that overcomes key limitations of previous methods, enhancing robustness and applicability.

Findings

Outperforms Hottopixx in synthetic tests

More tolerant to noise and outliers

Automatically detects factorization rank

Abstract

Nonnegative matrix factorization (NMF) has been shown recently to be tractable under the separability assumption, under which all the columns of the input data matrix belong to the convex cone generated by only a few of these columns. Bittorf, Recht, R\'e and Tropp (`Factoring nonnegative matrices with linear programs', NIPS 2012) proposed a linear programming (LP) model, referred to as Hottopixx, which is robust under any small perturbation of the input matrix. However, Hottopixx has two important drawbacks: (i) the input matrix has to be normalized, and (ii) the factorization rank has to be known in advance. In this paper, we generalize Hottopixx in order to resolve these two drawbacks, that is, we propose a new LP model which does not require normalization and detects the factorization rank automatically. Moreover, the new LP model is more flexible, significantly more tolerant to noise, and can easily be adapted to handle outliers and other noise models. Finally, we show on several synthetic datasets that it outperforms Hottopixx while competing favorably with two state-of-the-art methods.