135,569 research outputs found
Compressed Sensing and Parallel Acquisition
Parallel acquisition systems arise in various applications in order to
moderate problems caused by insufficient measurements in single-sensor systems.
These systems allow simultaneous data acquisition in multiple sensors, thus
alleviating such problems by providing more overall measurements. In this work
we consider the combination of compressed sensing with parallel acquisition. We
establish the theoretical improvements of such systems by providing recovery
guarantees for which, subject to appropriate conditions, the number of
measurements required per sensor decreases linearly with the total number of
sensors. Throughout, we consider two different sampling scenarios -- distinct
(corresponding to independent sampling in each sensor) and identical
(corresponding to dependent sampling between sensors) -- and a general
mathematical framework that allows for a wide range of sensing matrices (e.g.,
subgaussian random matrices, subsampled isometries, random convolutions and
random Toeplitz matrices). We also consider not just the standard sparse signal
model, but also the so-called sparse in levels signal model. This model
includes both sparse and distributed signals and clustered sparse signals. As
our results show, optimal recovery guarantees for both distinct and identical
sampling are possible under much broader conditions on the so-called sensor
profile matrices (which characterize environmental conditions between a source
and the sensors) for the sparse in levels model than for the sparse model. To
verify our recovery guarantees we provide numerical results showing phase
transitions for a number of different multi-sensor environments.Comment: 43 pages, 4 figure
Improving A*OMP: Theoretical and Empirical Analyses With a Novel Dynamic Cost Model
Best-first search has been recently utilized for compressed sensing (CS) by
the A* orthogonal matching pursuit (A*OMP) algorithm. In this work, we
concentrate on theoretical and empirical analyses of A*OMP. We present a
restricted isometry property (RIP) based general condition for exact recovery
of sparse signals via A*OMP. In addition, we develop online guarantees which
promise improved recovery performance with the residue-based termination
instead of the sparsity-based one. We demonstrate the recovery capabilities of
A*OMP with extensive recovery simulations using the adaptive-multiplicative
(AMul) cost model, which effectively compensates for the path length
differences in the search tree. The presented results, involving phase
transitions for different nonzero element distributions as well as recovery
rates and average error, reveal not only the superior recovery accuracy of
A*OMP, but also the improvements with the residue-based termination and the
AMul cost model. Comparison of the run times indicate the speed up by the AMul
cost model. We also demonstrate a hybrid of OMP and A?OMP to accelerate the
search further. Finally, we run A*OMP on a sparse image to illustrate its
recovery performance for more realistic coefcient distributions
Exploring Algorithmic Limits of Matrix Rank Minimization under Affine Constraints
Many applications require recovering a matrix of minimal rank within an
affine constraint set, with matrix completion a notable special case. Because
the problem is NP-hard in general, it is common to replace the matrix rank with
the nuclear norm, which acts as a convenient convex surrogate. While elegant
theoretical conditions elucidate when this replacement is likely to be
successful, they are highly restrictive and convex algorithms fail when the
ambient rank is too high or when the constraint set is poorly structured.
Non-convex alternatives fare somewhat better when carefully tuned; however,
convergence to locally optimal solutions remains a continuing source of
failure. Against this backdrop we derive a deceptively simple and
parameter-free probabilistic PCA-like algorithm that is capable, over a wide
battery of empirical tests, of successful recovery even at the theoretical
limit where the number of measurements equal the degrees of freedom in the
unknown low-rank matrix. Somewhat surprisingly, this is possible even when the
affine constraint set is highly ill-conditioned. While proving general recovery
guarantees remains evasive for non-convex algorithms, Bayesian-inspired or
otherwise, we nonetheless show conditions whereby the underlying cost function
has a unique stationary point located at the global optimum; no existing cost
function we are aware of satisfies this same property. We conclude with a
simple computer vision application involving image rectification and a standard
collaborative filtering benchmark
Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit
This paper demonstrates theoretically and empirically that a greedy algorithm called Orthogonal Matching Pursuit (OMP) can reliably recover a signal with nonzero entries in dimension given random linear measurements of that signal. This is a massive improvement over previous results, which require measurements. The new results for OMP are comparable with recent results for another approach called Basis Pursuit (BP). In some settings, the OMP algorithm is faster and easier to implement, so it is an attractive alternative to BP for signal recovery problems
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