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arXiv · 1405.1361

Discrete and Continuous-time Soft-Thresholding with Dynamic Inputs

Aurele Balavoine·Christopher J. Rozell·Justin Romberg
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Abstract

There exist many well-established techniques to recover sparse signals from compressed measurements with known performance guarantees in the static case. However, only a few methods have been proposed to tackle the recovery of time-varying signals, and even fewer benefit from a theoretical analysis. In this paper, we study the capacity of 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 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 analyses is supported by simulations on both synthetic and real data.

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Aurele Balavoine, Christopher J. Rozell, Justin Romberg. 2014-05-06. Discrete and Continuous-time Soft-Thresholding with Dynamic Inputs. https://doi.org/10.1109/tsp.2015.2420535

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