Discrete and Continuous-Time Soft-Thresholding for Dynamic Signal Recovery

There exist many well-established techniques to recover sparse signals from compressed measurements with known performance guarantees in the static case. More recently, new methods have been proposed to tackle the recovery of time-varying signals, but few benefit from a theoretical analysis. In this paper, we give theoretical guarantees for the Iterative Soft-Thresholding Algorithm (ISTA) and its continuous-time analogue the Locally Competitive Algorithm (LCA) to perform this tracking in real time. ISTA is a well-known digital solver for static sparse recovery, whose iteration is a first-order discretization of the LCA differential equation. Our analysis is based on the Restricted Isometry Property (RIP) and shows that the outputs of both algorithms can track a time-varying signal while compressed measurements are streaming, even when no convergence criterion is imposed at each time step. The l2-distance between the target signal and the outputs of both discrete- and continuous-time solvers is shown to decay to a bound that is essentially optimal. Our analysis is supported by simulations on both synthetic and real data.