8,479 research outputs found
Compressed sensing with sparse, structured matrices
In the context of the compressed sensing problem, we propose a new ensemble
of sparse random matrices which allow one (i) to acquire and compress a
{\rho}0-sparse signal of length N in a time linear in N and (ii) to perfectly
recover the original signal, compressed at a rate {\alpha}, by using a message
passing algorithm (Expectation Maximization Belief Propagation) that runs in a
time linear in N. In the large N limit, the scheme proposed here closely
approaches the theoretical bound {\rho}0 = {\alpha}, and so it is both optimal
and efficient (linear time complexity). More generally, we show that several
ensembles of dense random matrices can be converted into ensembles of sparse
random matrices, having the same thresholds, but much lower computational
complexity.Comment: 7 pages, 6 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
Modulated Unit-Norm Tight Frames for Compressed Sensing
In this paper, we propose a compressed sensing (CS) framework that consists
of three parts: a unit-norm tight frame (UTF), a random diagonal matrix and a
column-wise orthonormal matrix. We prove that this structure satisfies the
restricted isometry property (RIP) with high probability if the number of
measurements for -sparse signals of length
and if the column-wise orthonormal matrix is bounded. Some existing structured
sensing models can be studied under this framework, which then gives tighter
bounds on the required number of measurements to satisfy the RIP. More
importantly, we propose several structured sensing models by appealing to this
unified framework, such as a general sensing model with arbitrary/determinisic
subsamplers, a fast and efficient block compressed sensing scheme, and
structured sensing matrices with deterministic phase modulations, all of which
can lead to improvements on practical applications. In particular, one of the
constructions is applied to simplify the transceiver design of CS-based channel
estimation for orthogonal frequency division multiplexing (OFDM) systems.Comment: submitted to IEEE Transactions on Signal Processin
Granger Causality for Compressively Sensed Sparse Signals
Compressed sensing is a scheme that allows for sparse signals to be acquired,
transmitted and stored using far fewer measurements than done by conventional
means employing Nyquist sampling theorem. Since many naturally occurring
signals are sparse (in some domain), compressed sensing has rapidly seen
popularity in a number of applied physics and engineering applications,
particularly in designing signal and image acquisition strategies, e.g.,
magnetic resonance imaging, quantum state tomography, scanning tunneling
microscopy, analog to digital conversion technologies. Contemporaneously,
causal inference has become an important tool for the analysis and
understanding of processes and their interactions in many disciplines of
science, especially those dealing with complex systems. Direct causal analysis
for compressively sensed data is required to avoid the task of reconstructing
the compressed data. Also, for some sparse signals, such as for sparse temporal
data, it may be difficult to discover causal relations directly using available
data-driven/ model-free causality estimation techniques. In this work, we
provide a mathematical proof that structured compressed sensing matrices,
specifically Circulant and Toeplitz, preserve causal relationships in the
compressed signal domain, as measured by Granger Causality. We then verify this
theorem on a number of bivariate and multivariate coupled sparse signal
simulations which are compressed using these matrices. We also demonstrate a
real world application of network causal connectivity estimation from sparse
neural spike train recordings from rat prefrontal cortex.Comment: Submitted to IEEE Transactions on Neural Networks and Learning
System
Quantized Compressed Sensing for Partial Random Circulant Matrices
We provide the first analysis of a non-trivial quantization scheme for
compressed sensing measurements arising from structured measurements.
Specifically, our analysis studies compressed sensing matrices consisting of
rows selected at random, without replacement, from a circulant matrix generated
by a random subgaussian vector. We quantize the measurements using stable,
possibly one-bit, Sigma-Delta schemes, and use a reconstruction method based on
convex optimization. We show that the part of the reconstruction error due to
quantization decays polynomially in the number of measurements. This is in line
with analogous results on Sigma-Delta quantization associated with random
Gaussian or subgaussian matrices, and significantly better than results
associated with the widely assumed memoryless scalar quantization. Moreover, we
prove that our approach is stable and robust; i.e., the reconstruction error
degrades gracefully in the presence of non-quantization noise and when the
underlying signal is not strictly sparse. The analysis relies on results
concerning subgaussian chaos processes as well as a variation of McDiarmid's
inequality.Comment: 15 page
Achievable Angles Between two Compressed Sparse Vectors Under Norm/Distance Constraints Imposed by the Restricted Isometry Property: A Plane Geometry Approach
The angle between two compressed sparse vectors subject to the norm/distance
constraints imposed by the restricted isometry property (RIP) of the sensing
matrix plays a crucial role in the studies of many compressive sensing (CS)
problems. Assuming that (i) u and v are two sparse vectors separated by an
angle thetha, and (ii) the sensing matrix Phi satisfies RIP, this paper is
aimed at analytically characterizing the achievable angles between Phi*u and
Phi*v. Motivated by geometric interpretations of RIP and with the aid of the
well-known law of cosines, we propose a plane geometry based formulation for
the study of the considered problem. It is shown that all the RIP-induced
norm/distance constraints on Phi*u and Phi*v can be jointly depicted via a
simple geometric diagram in the two-dimensional plane. This allows for a joint
analysis of all the considered algebraic constraints from a geometric
perspective. By conducting plane geometry analyses based on the constructed
diagram, closed-form formulae for the maximal and minimal achievable angles are
derived. Computer simulations confirm that the proposed solution is tighter
than an existing algebraic-based estimate derived using the polarization
identity. The obtained results are used to derive a tighter restricted isometry
constant of structured sensing matrices of a certain kind, to wit, those in the
form of a product of an orthogonal projection matrix and a random sensing
matrix. Follow-up applications to three CS problems, namely, compressed-domain
interference cancellation, RIP-based analysis of the orthogonal matching
pursuit algorithm, and the study of democratic nature of random sensing
matrices are investigated.Comment: submitted to IEEE Trans. Information Theor
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