A Simple Homotopy Algorithm for Compressive Sensing

In this paper, we consider the problem of recovering the s largest elements of an arbitrary vector from noisy measurements. Inspired by previous work, we develop an homotopy algorithm which solves the l1-regularized least square problem for a sequence of decreasing values of the regularization parameter. Compared to the previous method, our algorithm is more efficient in the sense it only updates the solution once for each intermediate problem, and more practical in the sense it has a simple stopping criterion by checking the sparsity of the intermediate solution. Theoretical analysis reveals that our method enjoys a linear convergence rate in reducing the recovery error. Furthermore, our guarantee for recovering the top s elements of the target vector is tighter than previous results, and that for recovering the target vector itself matches the state of the art in compressive sensing.