Near-separable Non-negative Matrix Factorization with $\ell_1$- and Bregman Loss Functions
This addresses computer vision tasks like foreground-background separation with incremental improvements in speed and robustness.
The paper tackles the problem of near-separable non-negative matrix factorization (NMF) by extending conical hull procedures to robust approximations and Bregman divergences, achieving performance matching Robust PCA on foreground-background separation with an order of magnitude faster training time.
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 several extensions of the conical hull procedures of Kumar et al. (2013) for robust ($\ell_1$) approximations and Bregman divergences. 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.