Segmented convex-hull algorithms for near-separable NMF and NTF

Abstract

Many computational problems in machine learning can be represented with near-separable matrix factorization models. In a geometric approach, linear separability means that the entire set of data points can be modeled as a conical combination of a few data points, referred to as the extreme rays that express meaningful features. In this study, we propose segmented convex-hull algorithms for estimating the extreme rays of the simplicial cone generated by observations in the near-separable and inconsistent non-negative matrix factorization (NMF) and non-negative tensor factorization (NTF) models. The segmentation is based on the concept of hierarchical convex-hull NMF. The proposed algorithms are used to solve near-separable noisy blind source separation problems and classification problems. For the former, the experimental results demonstrate that they significantly outperform the state-of-the-art geometry-based NMF algorithms and the basic hierarchical alternating least squares NTF, if observations are noisy with signal-to-noise ratio lower than 100 dB. For the latter, the classification results also provide strong evidence for the effectiveness of the proposed approach with respect to existing geometry-based NMF algorithms.