Dynamic updating for sparse time varying signals

Many signal processing applications revolve around finding a sparse solution to a (often underdetermined) system of linear equations. Recent results in compressive sensing (CS) have shown that when the signal we are trying to acquire is sparse and the measurements are incoherent, the signal can be reconstructed reliably from an incomplete set of measurements. However, the signal recovery is an involved process, usually requiring the solution of an lscr₁ minimization program. In this paper we discuss the problem of estimating a time-varying sparse signal from a series of linear measurements. We propose an efficient way to dynamically update the solution to two types of lscr₁ problems when the underlying signal changes. The proposed dynamic update scheme is based on homotopy continuation, which systematically breaks down the solution update into a small number of linear steps. The computational cost for each step is just a few matrix-vector multiplications.