Near-separable Non-negative Matrix Factorization with $\ell_1$- and Bregman Loss Functions

TL;DR

This paper extends near-separable non-negative matrix factorization algorithms to include robust $ ext{l}_1$ and Bregman divergence methods, improving scalability and noise robustness for applications like computer vision and exemplar selection.

Contribution

It introduces new conical hull algorithms for robust $ ext{l}_1$ and Bregman divergence NMF, enhancing noise tolerance and computational efficiency.

Findings

Robust near-separable NMF matches Robust PCA performance in vision tasks.

The methods are significantly faster than existing algorithms.

Applications include foreground-background separation and exemplar selection.

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 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.