5,387 research outputs found
Sparsity Order Estimation from a Single Compressed Observation Vector
We investigate the problem of estimating the unknown degree of sparsity from
compressive measurements without the need to carry out a sparse recovery step.
While the sparsity order can be directly inferred from the effective rank of
the observation matrix in the multiple snapshot case, this appears to be
impossible in the more challenging single snapshot case. We show that specially
designed measurement matrices allow to rearrange the measurement vector into a
matrix such that its effective rank coincides with the effective sparsity
order. In fact, we prove that matrices which are composed of a Khatri-Rao
product of smaller matrices generate measurements that allow to infer the
sparsity order. Moreover, if some samples are used more than once, one of the
matrices needs to be Vandermonde. These structural constraints reduce the
degrees of freedom in choosing the measurement matrix which may incur in a
degradation in the achievable coherence. We thus also address suitable choices
of the measurement matrices. In particular, we analyze Khatri-Rao and
Vandermonde matrices in terms of their coherence and provide a new design for
Vandermonde matrices that achieves a low coherence
Deterministic Constructions of Binary Measurement Matrices from Finite Geometry
Deterministic constructions of measurement matrices in compressed sensing
(CS) are considered in this paper. The constructions are inspired by the recent
discovery of Dimakis, Smarandache and Vontobel which says that parity-check
matrices of good low-density parity-check (LDPC) codes can be used as
{provably} good measurement matrices for compressed sensing under
-minimization. The performance of the proposed binary measurement
matrices is mainly theoretically analyzed with the help of the analyzing
methods and results from (finite geometry) LDPC codes. Particularly, several
lower bounds of the spark (i.e., the smallest number of columns that are
linearly dependent, which totally characterizes the recovery performance of
-minimization) of general binary matrices and finite geometry matrices
are obtained and they improve the previously known results in most cases.
Simulation results show that the proposed matrices perform comparably to,
sometimes even better than, the corresponding Gaussian random matrices.
Moreover, the proposed matrices are sparse, binary, and most of them have
cyclic or quasi-cyclic structure, which will make the hardware realization
convenient and easy.Comment: 12 pages, 11 figure
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
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