An Alternating $l_1$ Approach to the Compressed Sensing Problem

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 l1-relaxation consisting of searching for solutions with smallest l1-norm. In this short note, based on the ideas of Lagrangian duality, we introduce an alternating l1 relaxation for the recovery problem enjoying higher recovery rates in practice than the plain l1 relaxation and the recent reweighted l1 method of Candès, Wakin and Boyd.

On the Lagrangian biduality of sparsity minimization problems

“…Our bidual-based relaxation is different from the SDP bidual-based relaxation derived by [3], due to the different (albeit equivalent) reformulations of the primal problem.…”

“…We note that there are other works on using Lagrangian biduality to derive relaxations of sparsity minimization problems [3,7]. The work most related to this paper is that of [3], which derives a semi-definite program (SDP) as the bidual for the problem P 0 .…”

“…The work most related to this paper is that of [3], which derives a semi-definite program (SDP) as the bidual for the problem P 0 . In contrast, we derive the well-known 1 -minimization problem to be the bidual for the problem P 0 , which is a linear program and easier to solve than an SDP.…”

A fast algorithm for the semi-definite relaxation of the state estimation problem in power grids

“…Extension of this work to the case of unknown noise variance have been studied in [23], [16], [3], [35], [38]. Interpretation using Lagrange duality and improved reconstruction performance were studied in [15]. Extension of compressed sensing ideas to low rank matrix and tensor estimation and completion was achieved rapidly after the vector case [11], [13], [9], [18], [42], [37], [17], [39] and the case of unknown noise variance was studied in [26].…”

Electrical Faults Signals Restoring Based on Compressed Sensing Techniques

“…However, this approach was done for an audio signal. Both Chretien [12] and Tropp [13] provided an extensive and comprehensive report about matching pursuit. The first one used MP to decompose a signal in a linear expansion of waveforms that were selected from a redundant dictionary of functions, which was simply the resulting matrix after applying a transform to the input signal.…”