Multiarray Signal Processing: Tensor decomposition meets compressed sensing

TL;DR

This paper explores how compressed sensing techniques can enhance tensor decomposition methods for multiarray signal processing, enabling deterministic source localization and extraction with guarantees on uniqueness under certain coherence bounds.

Contribution

It introduces a novel framework combining compressed sensing and tensor decomposition, providing conditions for unique signal representation and a practical approach for source separation.

Findings

Guarantees on the existence and uniqueness of tensor approximations

A feasible variant of Kruskal's condition using coherence

Deterministic localization and extraction of sources

Abstract

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.