XRAY algorithm for separable nonnegative tensor factorization

Abstract

Many computational problems in machine learning can be represented by separable matrix factorization models. In a geometric approach, linear separability means that the whole set of data points can be modeled by a convex combination of a few data points, referred to as the extreme rays. The aim of the XRAY algorithm is to find the extreme rays of the conic hull, generated by observed nonnegative vectors. In this paper, we extend the concept of this algorithm to a multi-linear data representation. Instead of searching into a vector space, we attempt to find the equivalent extreme rays in a space of tensors, under the linear separability assumption of subtensors, ordered along the selected mode. The proposed multi-way XRAY algorithm has been applied to Blind Source Separation (BSS) of natural images. The experiments demonstrate that if multi-way observations are at least one-mode linearly separable, the proposed algorithms can estimate the latent factors with high Signal-to-Interference (SIR) performance. The discussed methods may also be useful for analyzing video sequences.