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[show abstract]
[hide abstract]
ABSTRACT: It is well known that $\ell_1$ minimization can be used to recover
sufficiently sparse unknown signals from compressed linear measurements. In
fact, exact thresholds on the sparsity, as a function of the ratio between the
system dimensions, so that with high probability almost all sparse signals can
be recovered from i.i.d. Gaussian measurements, have been computed and are
referred to as "weak thresholds" \cite{D}. In this paper, we introduce a
reweighted $\ell_1$ recovery algorithm composed of two steps: a standard
$\ell_1$ minimization step to identify a set of entries where the signal is
likely to reside, and a weighted $\ell_1$ minimization step where entries
outside this set are penalized. For signals where the non-sparse component
entries are independent and identically drawn from certain classes of
distributions, (including most well known continuous distributions), we prove a
\emph{strict} improvement in the weak recovery threshold. Our analysis suggests
that the level of improvement in the weak threshold depends on the behavior of
the distribution at the origin. Numerical simulations verify the distribution
dependence of the threshold improvement very well, and suggest that in the case
of i.i.d. Gaussian nonzero entries, the improvement can be quite
impressive---over 20% in the example we consider.
11/2011;
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[show abstract]
[hide abstract]
ABSTRACT: We introduce a new class of measurement matrices for compressed sensing,
using low order summaries over binary sequences of a given length. We prove
recovery guarantees for three reconstruction algorithms using the proposed
measurements, including $\ell_1$ minimization and two combinatorial methods. In
particular, one of the algorithms recovers $k$-sparse vectors of length $N$ in
sublinear time $\text{poly}(k\log{N})$, and requires at most
$\Omega(k\log{N}\log\log{N})$ measurements. The empirical oversampling constant
of the algorithm is significantly better than existing sublinear recovery
algorithms such as Chaining Pursuit and Sudocodes. In particular, for $10^3\leq
N\leq 10^8$ and $k=100$, the oversampling factor is between 3 to 8. We provide
preliminary insight into how the proposed constructions, and the fast recovery
scheme can be used in a number of practical applications such as market basket
analysis, and real time compressed sensing implementation.
02/2011;
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IEEE Transactions on Signal Processing. 01/2011; 59:1985-2001.
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CoRR. 01/2011; abs/1111.1396.
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IEEE Transactions on Signal Processing. 01/2011; 59:196-208.
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Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2011, May 22-27, 2011, Prague Congress Center, Prague, Czech Republic; 01/2011
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Proceedings of the Global Communications Conference, GLOBECOM 2011, 5-9 December 2011, Houston, Texas, USA; 01/2011
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Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2011, May 22-27, 2011, Prague Congress Center, Prague, Czech Republic; 01/2011
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[show abstract]
[hide abstract]
ABSTRACT: In this paper we introduce a nonuniform sparsity model and analyze the performance of an optimized weighted $\ell_1$ minimization over that sparsity model. In particular, we focus on a model where the entries of the unknown vector fall into two sets, with entries of each set having a specific probability of being nonzero. We propose a weighted $\ell_1$ minimization recovery algorithm and analyze its performance using a Grassmann angle approach. We compute explicitly the relationship between the system parameters-the weights, the number of measurements, the size of the two sets, the probabilities of being nonzero- so that when i.i.d. random Gaussian measurement matrices are used, the weighted $\ell_1$ minimization recovers a randomly selected signal drawn from the considered sparsity model with overwhelming probability as the problem dimension increases. This allows us to compute the optimal weights. We demonstrate through rigorous analysis and simulations that for the case when the support of the signal can be divided into two different subclasses with unequal sparsity fractions, the optimal weighted $\ell_1$ minimization outperforms the regular $\ell_1$ minimization substantially. We also generalize the results to an arbitrary number of classes.
09/2010;
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[show abstract]
[hide abstract]
ABSTRACT: It is well known that $\ell_1$ minimization can be used to recover sufficiently sparse unknown signals from compressed linear measurements. In fact, exact thresholds on the sparsity, as a function of the ratio between the system dimensions, so that with high probability almost all sparse signals can be recovered from iid Gaussian measurements, have been computed and are referred to as "weak thresholds" \cite{D}. In this paper, we introduce a reweighted $\ell_1$ recovery algorithm composed of two steps: a standard $\ell_1$ minimization step to identify a set of entries where the signal is likely to reside, and a weighted $\ell_1$ minimization step where entries outside this set are penalized. For signals where the non-sparse component has iid Gaussian entries, we prove a "strict" improvement in the weak recovery threshold. Simulations suggest that the improvement can be quite impressive-over 20% in the example we consider. Comment: accepted in ISIT 2010
04/2010;
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[show abstract]
[hide abstract]
ABSTRACT: The capacity region of multi-pair bidirectional relay networks, in which a
relay node facilitates the communication between multiple pairs of users, is
studied. This problem is first examined in the context of the linear shift
deterministic channel model. The capacity region of this network when the relay
is operating at either full-duplex mode or half-duplex mode for arbitrary
number of pairs is characterized. It is shown that the cut-set upper-bound is
tight and the capacity region is achieved by a so called divide-and-conquer
relaying strategy. The insights gained from the deterministic network are then
used for the Gaussian bidirectional relay network. The strategy in the
deterministic channel translates to a specific superposition of lattice codes
and random Gaussian codes at the source nodes and successive interference
cancelation at the receiving nodes for the Gaussian network. The achievable
rate of this scheme with two pairs is analyzed and it is shown that for all
channel gains it achieves to within 3 bits/sec/Hz per user of the cut-set
upper-bound. Hence, the capacity region of the two-pair bidirectional Gaussian
relay network to within 3 bits/sec/Hz per user is characterized.
01/2010;
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CoRR. 01/2010; abs/1004.0402.
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IEEE International Symposium on Information Theory, ISIT 2010, June 13-18, 2010, Austin, Texas, USA, Proceedings; 01/2010
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CoRR. 01/2010; abs/1009.3525.
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CoRR. 01/2010; abs/1001.4271.
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Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2010, 14-19 March 2010, Sheraton Dallas Hotel, Dallas, Texas, USA; 01/2010
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[show abstract]
[hide abstract]
ABSTRACT: It is now well understood that $\ell_1$ minimization algorithm is able to recover sparse signals from incomplete measurements [2], [1], [3] and sharp recoverable sparsity thresholds have also been obtained for the $\ell_1$ minimization algorithm. However, even though iterative reweighted $\ell_1$ minimization algorithms or related algorithms have been empirically observed to boost the recoverable sparsity thresholds for certain types of signals, no rigorous theoretical results have been established to prove this fact. In this paper, we try to provide a theoretical foundation for analyzing the iterative reweighted $\ell_1$ algorithms. In particular, we show that for a nontrivial class of signals, the iterative reweighted $\ell_1$ minimization can indeed deliver recoverable sparsity thresholds larger than that given in [1], [3]. Our results are based on a high-dimensional geometrical analysis (Grassmann angle analysis) of the null-space characterization for $\ell_1$ minimization and weighted $\ell_1$ minimization algorithms.
05/2009;
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[show abstract]
[hide abstract]
ABSTRACT: We investigate the sparse recovery problem of reconstructing a high-dimensional non-negative sparse vector from lower dimensional linear measurements. While much work has focused on dense measurement matrices, sparse measurement schemes are crucial in applications, such as DNA microarrays and sensor networks, where dense measurements are not practically feasible. One possible construction uses the adjacency matrices of expander graphs, which often leads to recovery algorithms much more efficient than $\ell_1$ minimization. However, to date, constructions based on expanders have required very high expansion coefficients which can potentially make the construction of such graphs difficult and the size of the recoverable sets small. In this paper, we construct sparse measurement matrices for the recovery of non-negative vectors, using perturbations of the adjacency matrix of an expander graph with much smaller expansion coefficient. We present a necessary and sufficient condition for $\ell_1$ optimization to successfully recover the unknown vector and obtain expressions for the recovery threshold. For certain classes of measurement matrices, this necessary and sufficient condition is further equivalent to the existence of a "unique" vector in the constraint set, which opens the door to alternative algorithms to $\ell_1$ minimization. We further show that the minimal expansion we use is necessary for any graph for which sparse recovery is possible and that therefore our construction is tight. We finally present a novel recovery algorithm that exploits expansion and is much faster than $\ell_1$ optimization. Finally, we demonstrate through theoretical bounds, as well as simulation, that our method is robust to noise and approximate sparsity.
03/2009;
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[show abstract]
[hide abstract]
ABSTRACT: In this paper we study the compressed sensing problem of recovering a sparse signal from a system of underdetermined linear equations when we have prior information about the probability of each entry of the unknown signal being nonzero. In particular, we focus on a model where the entries of the unknown vector fall into two sets, each with a different probability of being nonzero. We propose a weighted $\ell_1$ minimization recovery algorithm and analyze its performance using a Grassman angle approach. We compute explicitly the relationship between the system parameters (the weights, the number of measurements, the size of the two sets, the probabilities of being non-zero) so that an iid random Gaussian measurement matrix along with weighted $\ell_1$ minimization recovers almost all such sparse signals with overwhelming probability as the problem dimension increases. This allows us to compute the optimal weights. We also provide simulations to demonstrate the advantages of the method over conventional $\ell_1$ optimization.
02/2009;
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CoRR. 01/2009; abs/0904.0994.
