Tensor Decompositions, Alternating Least Squares and other Tales

Pierre Comon (1) , Xavier Luciani (1, 2) , André L. F. de Almeida (1)

1 Laboratoire d'Informatique, Signaux, et Systèmes de Sophia-Antipolis (I3S) / Equipe SIGNAL
2 PROTEE - Processus de Transfert et d'Echanges dans l'Environnement - EA 3819

This work was originally motivated by a classification of tensors proposed by Richard Harshman. In particular, we focus on simple and multiple ''bottlenecks'', and on ''swamps''. Existing theoretical results are surveyed, some numerical algorithms are described in details, and their numerical complexity is calculated. In particular, the interest in using the ELS enhancement in these algorithms is discussed. Computer simulations feed this discussion.

Mots clés: Tensor rank; Canonical Decomposition; Candecomp; Parafac; Three-way Array; compression; dimension reduction; Levenberg-Marquardt; computational complexity; Performance measure; Indeterminacy; greedy algorithm

Domaines: Traitement du signal et de l'image [eess.SP]; Chemo-informatique; Sciences de l'environnement; Traitement du signal et de l'image [eess.SP]

Pierre Comon, Xavier Luciani, André L. F. de Almeida. Tensor Decompositions, Alternating Least Squares and other Tales. Journal of Chemometrics, 2009, 23, pp.393-405. ⟨10.1002/cem.1236⟩. ⟨hal-00410057⟩