Dynamic Updating for $\ell_{1}$ Minimization

The theory of compressive sensing (CS) has shown us 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 $\ell_{1}$ norm of the solution against a data fidelity term. Tremendous progress has been made in recent years on algorithms for solving these $\ell_{1}$ 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 present a suite of dynamic algorithms for solving $\ell_{1}$ 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 $\ell_{1}$ 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 $\ell_{1}$ decoding problems, where an arbitrary signal is to be recovered from redundant coded measurements which have been corrupted by sparse errors.