An Alternating l1 approach to the compressed sensing problem

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.

💡 Research Summary

Compressed sensing (CS) exploits the fact that many natural signals are sparse in some basis, allowing accurate reconstruction from far fewer linear measurements than the ambient dimension would suggest. Mathematically, the reconstruction problem is to find the sparsest vector $x$ satisfying $Ax=b$, where $A\in\mathbb{R}^{m\times n}$ with $m\ll n$. This is an $l_0$‑minimization problem, which is combinatorial and NP‑hard. The most common convex surrogate replaces the $l_0$ norm by the $l_1$ norm, leading to the Basis Pursuit (BP) formulation \