Continuous Compressed Sensing With a Single or Multiple Measurement Vectors

TL;DR

This paper introduces a novel approach to continuous compressed sensing (CCS) for frequency-sparse signals, establishing a link with low rank matrix completion, providing theoretical guarantees, and proposing efficient algorithms validated through simulations.

Contribution

It establishes a theoretical connection between CCS and LRMC, and develops practical algorithms with guarantees for recovering frequency-sparse signals from partial samples.

Findings

Theoretical guarantees for exact recovery in CCS.

Efficient algorithms based on convex and nonconvex relaxations.

Validated performance through numerical simulations.

Abstract

We consider the problem of recovering a single or multiple frequency-sparse signals, which share the same frequency components, from a subset of regularly spaced samples. The problem is referred to as continuous compressed sensing (CCS) in which the frequencies can take any values in the normalized domain [0,1). In this paper, a link between CCS and low rank matrix completion (LRMC) is established based on an $\ell_0$-pseudo-norm-like formulation, and theoretical guarantees for exact recovery are analyzed. Practically efficient algorithms are proposed based on the link and convex and nonconvex relaxations, and validated via numerical simulations.