106 research outputs found
New Bounds for Restricted Isometry Constants
In this paper we show that if the restricted isometry constant of
the compressed sensing matrix satisfies then -sparse
signals are guaranteed to be recovered exactly via minimization when
no noise is present and -sparse signals can be estimated stably in the noisy
case. It is also shown that the bound cannot be substantively improved. An
explicitly example is constructed in which ,
but it is impossible to recover certain -sparse signals
Limits on Sparse Data Acquisition: RIC Analysis of Finite Gaussian Matrices
One of the key issues in the acquisition of sparse data by means of
compressed sensing (CS) is the design of the measurement matrix. Gaussian
matrices have been proven to be information-theoretically optimal in terms of
minimizing the required number of measurements for sparse recovery. In this
paper we provide a new approach for the analysis of the restricted isometry
constant (RIC) of finite dimensional Gaussian measurement matrices. The
proposed method relies on the exact distributions of the extreme eigenvalues
for Wishart matrices. First, we derive the probability that the restricted
isometry property is satisfied for a given sufficient recovery condition on the
RIC, and propose a probabilistic framework to study both the symmetric and
asymmetric RICs. Then, we analyze the recovery of compressible signals in noise
through the statistical characterization of stability and robustness. The
presented framework determines limits on various sparse recovery algorithms for
finite size problems. In particular, it provides a tight lower bound on the
maximum sparsity order of the acquired data allowing signal recovery with a
given target probability. Also, we derive simple approximations for the RICs
based on the Tracy-Widom distribution.Comment: 11 pages, 6 figures, accepted for publication in IEEE transactions on
information theor
Sparse approximation property and stable recovery of sparse signals from noisy measurements
In this paper, we introduce a sparse approximation property of order for
a measurement matrix :
where is the best -sparse approximation of the vector
in , is the -sparse approximation error of the
vector in , and and are positive constants. The
sparse approximation property for a measurement matrix can be thought of as a
weaker version of its restricted isometry property and a stronger version of
its null space property. In this paper, we show that the sparse approximation
property is an appropriate condition on a measurement matrix to consider stable
recovery of any compressible signal from its noisy measurements. In particular,
we show that any compressible signalcan be stably recovered from its noisy
measurements via solving an -minimization problem if the measurement
matrix has the sparse approximation property with , and
conversely the measurement matrix has the sparse approximation property with
if any compressible signal can be stably recovered from
its noisy measurements via solving an -minimization problem.Comment: To appear in IEEE Trans. Signal Processing, 201
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