Enhanced joint sparsity via iterative support detection

Joint sparsity has attracted considerable attention in recent years in many fields including sparse signal recovery in compressive sensing, statistics, and machine learning. Traditional convex models with joint sparsity suffer from the suboptimal performance though enjoying tractable computation. In this paper, we propose a new non-convex joint sparsity model, and develop a corresponding multi-stage adaptive convex relaxation algorithm. This method extends the idea of iterative support detection (ISD) from the single vector estimation to the multi-vector estimation by considering the joint sparsity prior. We provide some preliminary theoretical analysis including convergence analysis and a sufficient recovery condition. Numerical experiments from both compressive sensing and multi-task feature learning show the better performance of the proposed method in comparison with several state-of-the-art alternatives. Moreover, we demonstrate that the extension of ISD from the single vector to multi-vector estimation is not trivial. While ISD does not well reconstruct the single channel sparse Bernoulli signal, it does achieve significantly improved performance when recovering the multi-channel sparse Bernoulli signal thanks to its ability of natural incorporation of the joint sparsity structure.