Near-separable non-negative matrix factorization with ℓ₁ and Bregman loss functions

Abstract

Recently, a family of tractable NMF algorithms have been proposed under the assumption that the data matrix satisfies a separability condition (Donoho & Stodden, 2003; Arora et al., 2012). Geometrically, this condition reformulates the NMF problem as that of finding the extreme rays of the conical hull of a finite set of vectors. In this paper, we develop separable NMF algorithms with li loss and Bregman divergences, by extending the conical hull procedures proposed in our earlier work (Kumar et al., 2013). Our methods inherit all the advantages of (Kumar et al., 2013) including scalability and noise-tolerance. We show that on foreground-background separation problems in computer vision, robust near-separable NMFs match the performance of Robust PCA, considered state of the art on these problems, with an order of magnitude faster training time. We also demonstrate applications in exemplar selection settings.