Nonnegative approximations of nonnegative tensors

TL;DR

This paper investigates the decomposition of nonnegative tensors into minimal sums of nonnegative outer products, establishing the existence of optimal solutions for approximation problems across various norms and divergences.

Contribution

It proves that nonnegative tensor approximation problems always have optimal solutions, extending to all norms and certain divergences, which was previously uncertain.

Findings

Optimal solutions exist for nonnegative tensor approximations.

The results apply to any choice of norms and Bregman divergences under mild conditions.

The work links tensor decomposition with probabilistic models like naive Bayes.

Abstract

We study the decomposition of a nonnegative tensor into a minimal sum of outer product of nonnegative vectors and the associated parsimonious naive Bayes probabilistic model. We show that the corresponding approximation problem, which is central to nonnegative PARAFAC, will always have optimal solutions. The result holds for any choice of norms and, under a mild assumption, even Bregman divergences.