Sparse Representations and Low-Rank Tensor Approximation

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

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

We discuss a technique that allows blind recovery of signals or blind identification of mixtures in instances where such recovery or identification were previously thought to be impossible: (i) closely located or highly correlated sources in antenna array processing, (ii) highly correlated spreading codes in CDMA radio communication, (iii) nearly dependent spectra in fluorescent spectroscopy. This has important implications -- in the case of antenna array processing, it allows for joint localization and extraction of multiple sources from the measurement of a noisy mixture recorded on multiple sensors in an entirely deterministic manner. In the case of CDMA, it allows the possibility of having a number of users larger than the spreading gain. In the case of fluorescent spectroscopy, it allows for detection of nearly identical chemical constituents. The proposed technique involves the solution of a bounded coherence low-rank multilinear approximation problem. We show that bounded coherence allows us to establish existence and uniqueness of the recovered solution.

Mots clés: Identification aveugle; mélanges linéaires sous-déterminés; séparation aveugle de sources; décompositions polyadiques tensorielles; tenseurs; rang tensoriel; localisation; meilleure approximation de rang faible; représentation parcimonieuse; spark; échantillonnage compressé; rang de Kruskal; cohérence; antennes multiples; capteurs multiples

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

Pierre Comon, Lek-Heng Lim. Sparse Representations and Low-Rank Tensor Approximation. 2011. ⟨hal-00703494⟩