1,709 research outputs found
Optimally Tuned Iterative Reconstruction Algorithms for Compressed Sensing
We conducted an extensive computational experiment, lasting multiple
CPU-years, to optimally select parameters for two important classes of
algorithms for finding sparse solutions of underdetermined systems of linear
equations. We make the optimally tuned implementations available at {\tt
sparselab.stanford.edu}; they run `out of the box' with no user tuning: it is
not necessary to select thresholds or know the likely degree of sparsity. Our
class of algorithms includes iterative hard and soft thresholding with or
without relaxation, as well as CoSaMP, subspace pursuit and some natural
extensions. As a result, our optimally tuned algorithms dominate such
proposals. Our notion of optimality is defined in terms of phase transitions,
i.e. we maximize the number of nonzeros at which the algorithm can successfully
operate. We show that the phase transition is a well-defined quantity with our
suite of random underdetermined linear systems. Our tuning gives the highest
transition possible within each class of algorithms.Comment: 12 pages, 14 figure
Message Passing Algorithms for Compressed Sensing
Compressed sensing aims to undersample certain high-dimensional signals, yet
accurately reconstruct them by exploiting signal characteristics. Accurate
reconstruction is possible when the object to be recovered is sufficiently
sparse in a known basis. Currently, the best known sparsity-undersampling
tradeoff is achieved when reconstructing by convex optimization -- which is
expensive in important large-scale applications. Fast iterative thresholding
algorithms have been intensively studied as alternatives to convex optimization
for large-scale problems. Unfortunately known fast algorithms offer
substantially worse sparsity-undersampling tradeoffs than convex optimization.
We introduce a simple costless modification to iterative thresholding making
the sparsity-undersampling tradeoff of the new algorithms equivalent to that of
the corresponding convex optimization procedures. The new
iterative-thresholding algorithms are inspired by belief propagation in
graphical models. Our empirical measurements of the sparsity-undersampling
tradeoff for the new algorithms agree with theoretical calculations. We show
that a state evolution formalism correctly derives the true
sparsity-undersampling tradeoff. There is a surprising agreement between
earlier calculations based on random convex polytopes and this new, apparently
very different theoretical formalism.Comment: 6 pages paper + 9 pages supplementary information, 13 eps figure.
Submitted to Proc. Natl. Acad. Sci. US
Orthonormal Expansion l1-Minimization Algorithms for Compressed Sensing
Compressed sensing aims at reconstructing sparse signals from significantly
reduced number of samples, and a popular reconstruction approach is
-norm minimization. In this correspondence, a method called orthonormal
expansion is presented to reformulate the basis pursuit problem for noiseless
compressed sensing. Two algorithms are proposed based on convex optimization:
one exactly solves the problem and the other is a relaxed version of the first
one. The latter can be considered as a modified iterative soft thresholding
algorithm and is easy to implement. Numerical simulation shows that, in dealing
with noise-free measurements of sparse signals, the relaxed version is
accurate, fast and competitive to the recent state-of-the-art algorithms. Its
practical application is demonstrated in a more general case where signals of
interest are approximately sparse and measurements are contaminated with noise.Comment: 7 pages, 2 figures, 1 tabl
Sparse Vector Distributions and Recovery from Compressed Sensing
It is well known that the performance of sparse vector recovery algorithms
from compressive measurements can depend on the distribution underlying the
non-zero elements of a sparse vector. However, the extent of these effects has
yet to be explored, and formally presented. In this paper, I empirically
investigate this dependence for seven distributions and fifteen recovery
algorithms. The two morals of this work are: 1) any judgement of the recovery
performance of one algorithm over that of another must be prefaced by the
conditions for which this is observed to be true, including sparse vector
distributions, and the criterion for exact recovery; and 2) a recovery
algorithm must be selected carefully based on what distribution one expects to
underlie the sensed sparse signal.Comment: Originally submitted to IEEE Signal Processing Letters in March 2011,
but rejected June 2011. Revised, expanded, and submitted July 2011 to EURASIP
Journal special issue on sparse signal processin
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