Nonnegative approximations of nonnegative tensors

Lek-Heng Lim (1) , Pierre Comon (2)

1 Department of Mathematics [Berkeley]
2 Laboratoire d'Informatique, Signaux, et Systèmes de Sophia-Antipolis (I3S) / Equipe SIGNAL

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.

Mots clés: tensor; Nonnegative tensors; nonnegative hypermatrices; nonnegative tensor decompositions; nonnegative tensor rank; probabilistic latent semantic indexing; Candecomp; Parafac; tensor norm; Bregman divergence

Domaines: Traitement du signal et de l'image [eess.SP]; Traitement du signal et de l'image [eess.SP]

Lek-Heng Lim, Pierre Comon. Nonnegative approximations of nonnegative tensors. Journal of Chemometrics, 2009, 23, pp.432-441. ⟨10.1002/cem.1244⟩. ⟨hal-00410056⟩