An Alternating l1 approach to the compressed sensing problem

TL;DR

This paper introduces an alternating l1 relaxation method for compressed sensing that improves recovery rates over traditional l1 and reweighted l1 techniques by leveraging Lagrangian duality.

Contribution

It proposes a novel alternating l1 approach based on Lagrangian duality, enhancing sparse signal recovery in compressed sensing beyond existing methods.

Findings

Higher recovery rates in practice compared to plain l1 relaxation

Outperforms reweighted l1 method of Candès, Wakin, and Boyd

Effective for sparse signal reconstruction

Abstract

Compressed sensing is a new methodology for constructing sensors which allow sparse signals to be efficiently recovered using only a small number of observations. The recovery problem can often be stated as the one of finding the solution of an underdetermined system of linear equations with the smallest possible support. The most studied relaxation of this hard combinatorial problem is the $l_1$-relaxation consisting of searching for solutions with smallest $l_1$-norm. In this short note, based on the ideas of Lagrangian duality, we introduce an alternating $l_1$ relaxation for the recovery problem enjoying higher recovery rates in practice than the plain $l_1$ relaxation and the recent reweighted $l_1$ method of Cand\`es, Wakin and Boyd.