Dynamic Updating for ` 1 Minimization

The theory of compressive sensing (CS) suggests that under certain conditions, a sparse signal can be recovered from a small number of linear incoherent measurements. An effective class of reconstruction algorithms involve solving a convex optimization program that balances the 1 norm of the solution against a data fidelity term. Tremendous progress has been made in recent years on algorithms for solving these 1 minimization programs. These algorithms, however, are for the most part static: they focus on finding the solution for a fixed set of measurements. In this paper, we will discuss dynamic algorithms for solving 1 minimization programs for streaming sets of measurements. We consider cases where the underlying signal changes slightly between measurements, and where new measurements of a fixed signal are sequentially added to the system. We develop algorithms to quickly update the solution of several different types of 1 optimization problems whenever these changes occur, thus avoiding having to solve a new optimization problem from scratch. Our proposed schemes are based on homotopy continuation, which breaks down the solution update in a systematic and efficient way into a small number of linear steps. Each step consists of a low-rank update and a small number of matrix-vector multiplications – very much like recursive least squares. Our investigation also includes dynamic updating schemes for 1 decoding problems, where an arbitrary signal is to be recovered from redundant coded measurements which have been corrupted by sparse errors. Manuscript submitted to IEEE Journal of selected topics in signal processing, Special issue on compressive sensing.