1,903 research outputs found
Approximate Sparsity Pattern Recovery: Information-Theoretic Lower Bounds
Recovery of the sparsity pattern (or support) of an unknown sparse vector
from a small number of noisy linear measurements is an important problem in
compressed sensing. In this paper, the high-dimensional setting is considered.
It is shown that if the measurement rate and per-sample signal-to-noise ratio
(SNR) are finite constants independent of the length of the vector, then the
optimal sparsity pattern estimate will have a constant fraction of errors.
Lower bounds on the measurement rate needed to attain a desired fraction of
errors are given in terms of the SNR and various key parameters of the unknown
vector. The tightness of the bounds in a scaling sense, as a function of the
SNR and the fraction of errors, is established by comparison with existing
achievable bounds. Near optimality is shown for a wide variety of practically
motivated signal models
"Compressed" Compressed Sensing
The field of compressed sensing has shown that a sparse but otherwise
arbitrary vector can be recovered exactly from a small number of randomly
constructed linear projections (or samples). The question addressed in this
paper is whether an even smaller number of samples is sufficient when there
exists prior knowledge about the distribution of the unknown vector, or when
only partial recovery is needed. An information-theoretic lower bound with
connections to free probability theory and an upper bound corresponding to a
computationally simple thresholding estimator are derived. It is shown that in
certain cases (e.g. discrete valued vectors or large distortions) the number of
samples can be decreased. Interestingly though, it is also shown that in many
cases no reduction is possible
The Sampling Rate-Distortion Tradeoff for Sparsity Pattern Recovery in Compressed Sensing
Recovery of the sparsity pattern (or support) of an unknown sparse vector
from a limited number of noisy linear measurements is an important problem in
compressed sensing. In the high-dimensional setting, it is known that recovery
with a vanishing fraction of errors is impossible if the measurement rate and
the per-sample signal-to-noise ratio (SNR) are finite constants, independent of
the vector length. In this paper, it is shown that recovery with an arbitrarily
small but constant fraction of errors is, however, possible, and that in some
cases computationally simple estimators are near-optimal. Bounds on the
measurement rate needed to attain a desired fraction of errors are given in
terms of the SNR and various key parameters of the unknown vector for several
different recovery algorithms. The tightness of the bounds, in a scaling sense,
as a function of the SNR and the fraction of errors, is established by
comparison with existing information-theoretic necessary bounds. Near
optimality is shown for a wide variety of practically motivated signal models
Signal Estimation with Additive Error Metrics in Compressed Sensing
Compressed sensing typically deals with the estimation of a system input from
its noise-corrupted linear measurements, where the number of measurements is
smaller than the number of input components. The performance of the estimation
process is usually quantified by some standard error metric such as squared
error or support set error. In this correspondence, we consider a noisy
compressed sensing problem with any arbitrary error metric. We propose a
simple, fast, and highly general algorithm that estimates the original signal
by minimizing the error metric defined by the user. We verify that our
algorithm is optimal owing to the decoupling principle, and we describe a
general method to compute the fundamental information-theoretic performance
limit for any error metric. We provide two example metrics --- minimum mean
absolute error and minimum mean support error --- and give the theoretical
performance limits for these two cases. Experimental results show that our
algorithm outperforms methods such as relaxed belief propagation (relaxed BP)
and compressive sampling matching pursuit (CoSaMP), and reaches the suggested
theoretical limits for our two example metrics.Comment: to appear in IEEE Trans. Inf. Theor
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