Checking the Sufficiently Scattered Condition using a Global Non-Convex Optimization Software

The sufficiently scattered condition (SSC) is a key condition in the study of identifiability of various matrix factorization problems, including nonnegative, minimum-volume, symmetric, simplex-structured, and polytopic matrix factorizations. The SSC allows one to guarantee that the computed matrix factorization is unique/identifiable, up to trivial ambiguities. However, this condition is NP-hard to check in general. In this paper, we show that it can however be checked in a reasonable amount of time in realistic scenarios, when the factorization rank is not too large. This is achieved by formulating the problem as a non-convex quadratic optimization problem over a bounded set. We use the global non-convex optimization software Gurobi, and showcase the usefulness of this code on synthetic data sets and on real-world hyperspectral images.

Robust Near-Separable Nonnegative Matrix Factorization Using Linear Optimization

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.