Basis Pursuit in sensor networks

Basis Pursuit (BP) finds a minimum ℓ₁-norm vector z that satisfies the underdetermined linear system Mz = b, where the matrix M and vector b are given. Lately, BP has attracted attention because of its application in compressed sensing, where it is used to reconstruct signals by finding the sparsest solutions of linear systems. In this paper, we propose a distributed algorithm to solve BP. This means no central node is used for the processing and no node has access to all the data: the rows of M and the vector b are distributed over a set of interconnected compute nodes. A typical scenario is a sensor network. The novelty of our method is in using an optimal first-order method to solve an augmented Lagrangian-based reformulation of BP. We implemented our algorithm in a computer cluster, and show that it can solve problems that are too large to be stored in and processed by a single node.