On Finite Alphabet Compressive Sensing

TL;DR

This paper explores compressive sensing over finite alphabets, demonstrating benefits like lower sample complexity, constructive sensing matrix design, polynomial-time exact sparse recovery, and reduced data storage, compared to traditional real-valued methods.

Contribution

It introduces a finite alphabet compressive sensing framework that leverages coding theory for improved efficiency and exact sparse recovery in polynomial time.

Findings

Lower sample complexity for sparsity below a threshold

Constructive sensing matrix design using coding theory

Polynomial-time exact $\

Abstract

This paper considers the problem of compressive sensing over a finite alphabet, where the finite alphabet may be inherent to the nature of the data or a result of quantization. There are multiple examples of finite alphabet based static as well as time-series data with inherent sparse structure; and quantizing real values is an essential step while handling real data in practice. We show that there are significant benefits to analyzing the problem while incorporating its finite alphabet nature, versus ignoring it and employing a conventional real alphabet based toolbox. Specifically, when the alphabet is finite, our techniques (a) have a lower sample complexity compared to real-valued compressive sensing for sparsity levels below a threshold; (b) facilitate constructive designs of sensing matrices based on coding-theoretic techniques; (c) enable one to solve the exact $\ell_0$-minimization problem in polynomial time rather than a approach of convex relaxation followed by sufficient conditions for when the relaxation matches the original problem; and finally, (d) allow for smaller amount of data storage (in bits).