2,192 research outputs found
Discrete and Continuous-time Soft-Thresholding with Dynamic Inputs
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.Comment: 18 pages, 7 figures, journa
Adaptive Non-uniform Compressive Sampling for Time-varying Signals
In this paper, adaptive non-uniform compressive sampling (ANCS) of
time-varying signals, which are sparse in a proper basis, is introduced. ANCS
employs the measurements of previous time steps to distribute the sensing
energy among coefficients more intelligently. To this aim, a Bayesian inference
method is proposed that does not require any prior knowledge of importance
levels of coefficients or sparsity of the signal. Our numerical simulations
show that ANCS is able to achieve the desired non-uniform recovery of the
signal. Moreover, if the signal is sparse in canonical basis, ANCS can reduce
the number of required measurements significantly.Comment: 6 pages, 8 figures, Conference on Information Sciences and Systems
(CISS 2017) Baltimore, Marylan
Sequential Compressed Sensing
Compressed sensing allows perfect recovery of sparse signals (or signals
sparse in some basis) using only a small number of random measurements.
Existing results in compressed sensing literature have focused on
characterizing the achievable performance by bounding the number of samples
required for a given level of signal sparsity. However, using these bounds to
minimize the number of samples requires a-priori knowledge of the sparsity of
the unknown signal, or the decay structure for near-sparse signals.
Furthermore, there are some popular recovery methods for which no such bounds
are known.
In this paper, we investigate an alternative scenario where observations are
available in sequence. For any recovery method, this means that there is now a
sequence of candidate reconstructions. We propose a method to estimate the
reconstruction error directly from the samples themselves, for every candidate
in this sequence. This estimate is universal in the sense that it is based only
on the measurement ensemble, and not on the recovery method or any assumed
level of sparsity of the unknown signal. With these estimates, one can now stop
observations as soon as there is reasonable certainty of either exact or
sufficiently accurate reconstruction. They also provide a way to obtain
"run-time" guarantees for recovery methods that otherwise lack a-priori
performance bounds.
We investigate both continuous (e.g. Gaussian) and discrete (e.g. Bernoulli)
random measurement ensembles, both for exactly sparse and general near-sparse
signals, and with both noisy and noiseless measurements.Comment: to appear in IEEE transactions on Special Topics in Signal Processin
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