Sparse Representations and Low-Rank Tensor Approximation

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.