Nonnegative approximations of nonnegative tensors
Provides a fundamental existence result for nonnegative tensor factorization, which is important for practitioners and theorists working on tensor decompositions and probabilistic models.
The paper proves that the nonnegative tensor approximation problem underlying nonnegative PARAFAC always has optimal solutions for any norm and, under mild assumptions, for Bregman divergences, establishing a theoretical foundation for this decomposition.
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.