89,669 research outputs found
Compressed Sensing Using Binary Matrices of Nearly Optimal Dimensions
In this paper, we study the problem of compressed sensing using binary
measurement matrices and -norm minimization (basis pursuit) as the
recovery algorithm. We derive new upper and lower bounds on the number of
measurements to achieve robust sparse recovery with binary matrices. We
establish sufficient conditions for a column-regular binary matrix to satisfy
the robust null space property (RNSP) and show that the associated sufficient
conditions % sparsity bounds for robust sparse recovery obtained using the RNSP
are better by a factor of compared to the
sufficient conditions obtained using the restricted isometry property (RIP).
Next we derive universal \textit{lower} bounds on the number of measurements
that any binary matrix needs to have in order to satisfy the weaker sufficient
condition based on the RNSP and show that bipartite graphs of girth six are
optimal. Then we display two classes of binary matrices, namely parity check
matrices of array codes and Euler squares, which have girth six and are nearly
optimal in the sense of almost satisfying the lower bound. In principle,
randomly generated Gaussian measurement matrices are "order-optimal". So we
compare the phase transition behavior of the basis pursuit formulation using
binary array codes and Gaussian matrices and show that (i) there is essentially
no difference between the phase transition boundaries in the two cases and (ii)
the CPU time of basis pursuit with binary matrices is hundreds of times faster
than with Gaussian matrices and the storage requirements are less. Therefore it
is suggested that binary matrices are a viable alternative to Gaussian matrices
for compressed sensing using basis pursuit. \end{abstract}Comment: 28 pages, 3 figures, 5 table
Structured random measurements in signal processing
Compressed sensing and its extensions have recently triggered interest in
randomized signal acquisition. A key finding is that random measurements
provide sparse signal reconstruction guarantees for efficient and stable
algorithms with a minimal number of samples. While this was first shown for
(unstructured) Gaussian random measurement matrices, applications require
certain structure of the measurements leading to structured random measurement
matrices. Near optimal recovery guarantees for such structured measurements
have been developed over the past years in a variety of contexts. This article
surveys the theory in three scenarios: compressed sensing (sparse recovery),
low rank matrix recovery, and phaseless estimation. The random measurement
matrices to be considered include random partial Fourier matrices, partial
random circulant matrices (subsampled convolutions), matrix completion, and
phase estimation from magnitudes of Fourier type measurements. The article
concludes with a brief discussion of the mathematical techniques for the
analysis of such structured random measurements.Comment: 22 pages, 2 figure
On Convergence of Approximate Message Passing
Approximate message passing is an iterative algorithm for compressed sensing
and related applications. A solid theory about the performance and convergence
of the algorithm exists for measurement matrices having iid entries of zero
mean. However, it was observed by several authors that for more general
matrices the algorithm often encounters convergence problems. In this paper we
identify the reason of the non-convergence for measurement matrices with iid
entries and non-zero mean in the context of Bayes optimal inference. Finally we
demonstrate numerically that when the iterative update is changed from parallel
to sequential the convergence is restored.Comment: 5 pages, 3 figure
Nonuniform Sparse Recovery with Subgaussian Matrices
Compressive sensing predicts that sufficiently sparse vectors can be
recovered from highly incomplete information. Efficient recovery methods such
as -minimization find the sparsest solution to certain systems of
equations. Random matrices have become a popular choice for the measurement
matrix. Indeed, near-optimal uniform recovery results have been shown for such
matrices. In this note we focus on nonuniform recovery using Gaussian random
matrices and -minimization. We provide a condition on the number of
samples in terms of the sparsity and the signal length which guarantees that a
fixed sparse signal can be recovered with a random draw of the matrix using
-minimization. The constant 2 in the condition is optimal, and the
proof is rather short compared to a similar result due to Donoho and Tanner
Compressed Sensing of Approximately-Sparse Signals: Phase Transitions and Optimal Reconstruction
Compressed sensing is designed to measure sparse signals directly in a
compressed form. However, most signals of interest are only "approximately
sparse", i.e. even though the signal contains only a small fraction of relevant
(large) components the other components are not strictly equal to zero, but are
only close to zero. In this paper we model the approximately sparse signal with
a Gaussian distribution of small components, and we study its compressed
sensing with dense random matrices. We use replica calculations to determine
the mean-squared error of the Bayes-optimal reconstruction for such signals, as
a function of the variance of the small components, the density of large
components and the measurement rate. We then use the G-AMP algorithm and we
quantify the region of parameters for which this algorithm achieves optimality
(for large systems). Finally, we show that in the region where the GAMP for the
homogeneous measurement matrices is not optimal, a special "seeding" design of
a spatially-coupled measurement matrix allows to restore optimality.Comment: 8 pages, 10 figure
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