Discrete and Continuous-time Soft-Thresholding with Dynamic Inputs (bibtex)
by , and
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.
Reference:
Discrete and Continuous-time Soft-Thresholding with Dynamic InputsA. Balavoine, C.J. Rozell and J. Romberg. IEEE Transactions on Signal Processing, 63(12), pp. 3165–3176, June 2015.
Bibtex Entry:
@Article{balavoine.14c,
  author = 	 {Balavoine, A. and Rozell, C.J. and Romberg, J.},
  title = 	 {Discrete and Continuous-time Soft-Thresholding with Dynamic Inputs},
year = 2015,
  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.},
url = {http://arxiv.org/pdf/1405.1361v1.pdf},
journal = {IEEE Transactions on Signal Processing},
volume = {63},
number = {12},
month = jun,
pages = {3165--3176}
}
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