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

Robust Near-Separable Nonnegative Matrix Factorization Using Linear Optimization