  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 L1 norm of the
solution against a data fidelity term. Tremendous progress has been made in
recent years on algorithms for solving these L1 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 L1
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
L1 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 L1 decoding problems, where an arbitrary signal is to be
recovered from redundant coded measurements which have been corrupted by sparse
errors.
