Multiarray Signal Processing: Tensor decomposition meets compressed sensing

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 discuss how recently discovered techniques and tools from compressed sensing can be used in tensor decompositions, with a view towards modeling signals from multiple arrays of multiple sensors. We show that with appropriate bounds on a measure of separation between radiating sources called coherence, one could always guarantee the existence and uniqueness of a best rank-r approximation of the tensor representing the signal. We also deduce a computationally feasible variant of Kruskal's uniqueness condition, where the coherence appears as a proxy for k-rank. Problems of sparsest recovery with an infinite continuous dictionary, lowest-rank tensor representation, and blind source separation are treated in a uniform fashion. The decomposition of the measurement tensor leads to simultaneous localization and extraction of radiating sources, in an entirely deterministic manner.

Mots clés: Tensors; tensor rank; tensor decomposition; low rank approximation; sparse representation; Schatten norm; spark; k-rank; coherence; multiarray; multisensor

Domaines: Traitement du signal et de l'image [eess.SP]; Théorie de l'information [cs.IT]; Théorie de l'information et codage [math.IT]; Traitement du signal et de l'image [eess.SP]

Lek-Heng Lim, Pierre Comon. Multiarray Signal Processing: Tensor decomposition meets compressed sensing. Comptes rendus de l’Académie des sciences. Série IIb, Mécanique, 2010, 338 (6), pp.311--320. ⟨10.1016/j.crme.2010.06.005⟩. ⟨hal-00512271⟩