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

TL;DR

This paper introduces fast, scalable algorithms for near-separable non-negative matrix factorization based on conical hull geometry, improving robustness and efficiency over existing methods.

Contribution

The paper reformulates separable NMF as a conical hull problem and develops new algorithms that are highly scalable and noise-robust, with a parallel implementation.

Findings

Algorithms are highly scalable on shared and distributed systems.

Proposed methods demonstrate empirical robustness to noise.

New algorithms outperform existing approaches in speed and accuracy.

Abstract

The separability assumption (Donoho & Stodden, 2003; Arora et al., 2012) 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 other favorable properties in relation to existing methods. A parallel implementation of our algorithm demonstrates high scalability on shared- and distributed-memory machines.