Basis pursuit with sequential measurements and time varying signals

Recovery of sparse signals from linear measurements arises in several signal processing applications. Basis pursuit is a standard convex optimization program, often used to perform this task. In this paper we present two algorithms to dynamically update the solution of basis pursuit as (1) new measurements are sequentially added or (2) the underlying signal changes slightly. The goal is to avoid solving the (computationally expensive) optimization routine every time a small change occurs in the measurements. Our proposed update algorithms are based on homotopy principles, which iteratively update the solution by moving from an already solved problem towards the desired problem. Each homotopy step involves only a few matrix-vector multiplications. Simulation results show that the number of homotopy steps required for the update is comparable to the sparsity of the underlying signals.