Discrete and Continuous-time Soft-Thresholding with Dynamic Inputs

TL;DR

This paper analyzes the real-time tracking capabilities of ISTA and LCA algorithms for dynamic sparse signals, demonstrating their effectiveness and optimality in streaming measurement scenarios through theoretical analysis and simulations.

Contribution

It provides the first theoretical analysis of ISTA and LCA for tracking time-varying signals in streaming compressed measurements.

Findings

Both algorithms can effectively track dynamic signals without convergence at each step.

The L2-distance between the signal and algorithm output decays to an essentially optimal bound.

Simulations confirm theoretical results on synthetic and real data.

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.