19,488 research outputs found
Efficient Matrix Sensing Using Rank-1 Gaussian Measurements
Abstract. In this paper, we study the problem of low-rank matrix sensing where the goal is to reconstruct a matrix exactly using a small number of linear measurements. Existing methods for the problem either rely on measurement operators such as random element-wise sampling which cannot recover arbitrary low-rank matrices or require the measurement operator to satisfy the Restricted Isometry Property (RIP). However, RIP based linear operators are generally full rank and require large computation/storage cost for both measurement (encoding) as well as reconstruction (decoding). In this paper, we propose simple rank-one Gaussian measurement operators for matrix sensing that are significantly less expensive in terms of memory and computation for both encoding and decoding. Moreover, we show that the matrix can be reconstructed exactly using a simple alternating minimization method as well as a nuclear-norm minimization method. Finally, we demonstrate the effectiveness of the measurement scheme vis-a-vis existing RIP based methods
Info-Greedy sequential adaptive compressed sensing
We present an information-theoretic framework for sequential adaptive
compressed sensing, Info-Greedy Sensing, where measurements are chosen to
maximize the extracted information conditioned on the previous measurements. We
show that the widely used bisection approach is Info-Greedy for a family of
-sparse signals by connecting compressed sensing and blackbox complexity of
sequential query algorithms, and present Info-Greedy algorithms for Gaussian
and Gaussian Mixture Model (GMM) signals, as well as ways to design sparse
Info-Greedy measurements. Numerical examples demonstrate the good performance
of the proposed algorithms using simulated and real data: Info-Greedy Sensing
shows significant improvement over random projection for signals with sparse
and low-rank covariance matrices, and adaptivity brings robustness when there
is a mismatch between the assumed and the true distributions.Comment: Preliminary results presented at Allerton Conference 2014. To appear
in IEEE Journal Selected Topics on Signal Processin
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
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
Consistent Basis Pursuit for Signal and Matrix Estimates in Quantized Compressed Sensing
This paper focuses on the estimation of low-complexity signals when they are
observed through uniformly quantized compressive observations. Among such
signals, we consider 1-D sparse vectors, low-rank matrices, or compressible
signals that are well approximated by one of these two models. In this context,
we prove the estimation efficiency of a variant of Basis Pursuit Denoise,
called Consistent Basis Pursuit (CoBP), enforcing consistency between the
observations and the re-observed estimate, while promoting its low-complexity
nature. We show that the reconstruction error of CoBP decays like
when all parameters but are fixed. Our proof is connected to recent bounds
on the proximity of vectors or matrices when (i) those belong to a set of small
intrinsic "dimension", as measured by the Gaussian mean width, and (ii) they
share the same quantized (dithered) random projections. By solving CoBP with a
proximal algorithm, we provide some extensive numerical observations that
confirm the theoretical bound as is increased, displaying even faster error
decay than predicted. The same phenomenon is observed in the special, yet
important case of 1-bit CS.Comment: Keywords: Quantized compressed sensing, quantization, consistency,
error decay, low-rank, sparsity. 10 pages, 3 figures. Note abbout this
version: title change, typo corrections, clarification of the context, adding
a comparison with BPD
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