Linear under-determined systems with sparse solutions: Redirecting a challenge?

TL;DR

This paper reviews a decade of research on linear under-determined systems with sparse solutions, discusses past challenges, and questions whether the research focus should be redirected to new directions.

Contribution

It provides a concise overview of recent progress and challenges in sparse solutions for under-determined systems, and offers perspectives on future research directions.

Findings

Summarizes key achievements in the last 10 years

Identifies main challenges faced in the field

Raises questions about future research focus

Abstract

Seminal works \cite{CRT,DonohoUnsigned,DonohoPol} generated a massive interest in studying linear under-determined systems with sparse solutions. In this paper we give a short mathematical overview of what was accomplished in last 10 years in a particular direction of such a studying. We then discuss what we consider were the main challenges in last 10 years and give our own view as to what are the main challenges that lie ahead. Through the presentation we arrive to a point where the following natural rhetoric question arises: is it a time to redirect the main challenges? While we can not provide the answer to such a question we hope that our small discussion will stimulate further considerations in this direction.