A Fast Gradient Method for Nonnegative Sparse Regression with Self Dictionary

A nonnegative matrix factorization (NMF) can be computed efficiently under the separability assumption, which asserts that all the columns of the given input data matrix belong to the cone generated by a (small) subset of them. The provably most robust methods to identify these conic basis columns are based on nonnegative sparse regression and self dictionaries, and require the solution of large-scale convex optimization problems. In this paper we study a particular nonnegative sparse regression model with self dictionary. As opposed to previously proposed models, this model yields a smooth optimization problem where the sparsity is enforced through linear constraints. We show that the Euclidean projection on the polyhedron defined by these constraints can be computed efficiently, and propose a fast gradient method to solve our model. We compare our algorithm with several state-of-the-art methods on synthetic data sets and 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.Comment: 27 page; 4 figures. New Example, new experiment on the Swimmer data se