# Fast Conical Hull Algorithms for Near-separable Non-negative Matrix Factorization

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Abhishek Kumar,
Vikas Sindhwani,
Prabhanjan Kambadur
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Proceedings of the 30th International Conference on Machine Learning, PMLR 28(1):231-239, 2013.

#### Abstract

The separability assumption (Arora et al., 2012; Donoho & Stodden, 2003) turns non-negative matrix factorization (NMF) into a tractable problem. Recently, a new class of provably-correct NMF algorithms have emerged under this assumption. In this paper, we reformulate the separable NMF problem as that of finding the extreme rays of the conical hull of a finite set of vectors. From this geometric perspective, we derive new separable NMF algorithms that are highly scalable and empirically noise robust, and have several favorable properties in relation to existing methods. A parallel implementation of our algorithm scales excellently on shared and distributed-memory machines.

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