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Necessary and sufficient conditions on sparsity pattern recovery
, 2009
"... The paper considers the problem of detecting the sparsity pattern of a ksparse vector in R n from m random noisy measurements. A new necessary condition on the number of measurements for asymptotically reliable detection with maximum likelihood (ML) estimation and Gaussian measurement matrices is ..."
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Cited by 107 (13 self)
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The paper considers the problem of detecting the sparsity pattern of a ksparse vector in R n from m random noisy measurements. A new necessary condition on the number of measurements for asymptotically reliable detection with maximum likelihood (ML) estimation and Gaussian measurement matrices is derived. This necessary condition for ML detection is compared against a sufficient condition for simple maximum correlation (MC) or thresholding algorithms. The analysis shows that the gap between thresholding and ML can be described by a simple expression in terms of the total signaltonoise ratio (SNR), with the gap growing with increasing SNR. Thresholding is also compared against the more sophisticated lasso and orthogonal matching pursuit (OMP) methods. At high SNRs, it is shown that the gap between lasso and OMP over thresholding is described by the range of powers of the nonzero component values of the unknown signals. Specifically, the key benefit of lasso and OMP over thresholding is the ability of lasso and OMP to detect signals with relatively small components.
Asymptotic analysis of MAP estimation via the replica method and applications to compressed sensing
, 2009
"... The replica method is a nonrigorous but widelyaccepted technique from statistical physics used in the asymptotic analysis of large, random, nonlinear problems. This paper applies the replica method to nonGaussian maximum a posteriori (MAP) estimation. It is shown that with random linear measureme ..."
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Cited by 80 (10 self)
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The replica method is a nonrigorous but widelyaccepted technique from statistical physics used in the asymptotic analysis of large, random, nonlinear problems. This paper applies the replica method to nonGaussian maximum a posteriori (MAP) estimation. It is shown that with random linear measurements and Gaussian noise, the asymptotic behavior of the MAP estimate of anndimensional vector “decouples ” asnscalar MAP estimators. The result is a counterpart to Guo and Verdú’s replica analysis of minimum meansquared error estimation. The replica MAP analysis can be readily applied to many estimators used in compressed sensing, including basis pursuit, lasso, linear estimation with thresholding, and zero normregularized estimation. In the case of lasso estimation the scalar estimator reduces to a softthresholding operator, and for zero normregularized estimation it reduces to a hardthreshold. Among other benefits, the replica method provides a computationallytractable method for exactly computing various performance metrics including meansquared error and sparsity pattern recovery probability.
Random access compressed sensing for energyefficient underwater sensor networks
 IEEE Journal on Selected Areas in Communications
, 2011
"... Abstract—Inspired by the theory of compressed sensing and employing random channel access, we propose a distributed energyefficient sensor network scheme denoted by Random Access Compressed Sensing (RACS). The proposed scheme is suitable for longterm deployment of large underwater networks, in whi ..."
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Cited by 16 (1 self)
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Abstract—Inspired by the theory of compressed sensing and employing random channel access, we propose a distributed energyefficient sensor network scheme denoted by Random Access Compressed Sensing (RACS). The proposed scheme is suitable for longterm deployment of large underwater networks, in which saving energy and bandwidth is of crucial importance. During each frame, a randomly chosen subset of nodes participate in the sensing process, then share the channel using random access. Due to the nature of random access, packets may collide at the fusion center. To account for the packet loss that occurs due to collisions, the network design employs the concept of sufficient sensing probability. With this probability, sufficiently many data packets – as required for field reconstruction based on compressed sensing – are to be received. The RACS scheme prolongs network lifetime while employing a simple and distributed scheme which eliminates the need for scheduling. Index Terms—Sensor networks, compressed sensing, wireless communications, underwater acoustic networks, random access. I.
A sparsity detection framework for on–off random access channels
 in Proc. IEEE Int. Symp. Inform. Th., Seoul, Korea, Jun.–Jul. 2009
"... Abstract—This paper considers a simple on–off random multiple access channel (MAC), where n users communicate simultaneously to a single receiver. Each user is assigned a single codeword which it transmits with some probability λ over m degrees of freedom. The receiver must detect which users transm ..."
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Cited by 8 (3 self)
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Abstract—This paper considers a simple on–off random multiple access channel (MAC), where n users communicate simultaneously to a single receiver. Each user is assigned a single codeword which it transmits with some probability λ over m degrees of freedom. The receiver must detect which users transmitted. We show that detection for this random MAC is mathematically equivalent to a standard sparsity detection problem. Using new results in sparse estimation we are able to estimate the capacity of these channels and compare the achieved performance of various detection algorithms. The analysis provides insight into the roles of power control and multiuser detection. I.
Orthogonal matching pursuit from noisy measurements: a new analysis
 in TwentyThird Annual Conference on Neural Information Processing Systems
, 2009
"... A wellknown analysis of Tropp and Gilbert shows that orthogonal matching pursuit (OMP) can recover a ksparse ndimensional real vector from m = 4k log(n) noisefree linear measurements obtained through a random Gaussian measurement matrix with a probability that approaches one as n → ∞. This work ..."
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Cited by 8 (2 self)
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A wellknown analysis of Tropp and Gilbert shows that orthogonal matching pursuit (OMP) can recover a ksparse ndimensional real vector from m = 4k log(n) noisefree linear measurements obtained through a random Gaussian measurement matrix with a probability that approaches one as n → ∞. This work strengthens this result by showing that a lower number of measurements, m = 2k log(n − k), is in fact sufficient for asymptotic recovery. More generally, when the sparsity level satisfies kmin ≤ k ≤ kmax but is unknown, m = 2kmax log(n − kmin) measurements is sufficient. Furthermore, this number of measurements is also sufficient for detection of the sparsity pattern (support) of the vector with measurement errors provided the signaltonoise ratio (SNR) scales to infinity. The scaling m = 2k log(n − k) exactly matches the number of measurements required by the more complex lasso method for signal recovery in a similar SNR scaling. 1
ReducedDimension Multiuser Detection
"... Abstract—We present a new framework for reduceddimension multiuser detection (RDMUD) that trades off complexity for biterrorrate (BER) performance. This approach can significantly reduce the number of matched filter branches required by classic multiuser detection designs. We show that the RDMUD ..."
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Cited by 5 (3 self)
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Abstract—We present a new framework for reduceddimension multiuser detection (RDMUD) that trades off complexity for biterrorrate (BER) performance. This approach can significantly reduce the number of matched filter branches required by classic multiuser detection designs. We show that the RDMUD can perform similarly to the linear MUD detector when M is sufficiently large relative to N and K, where N and K are the number of total and active users, respectively. We also study the inherent RDMUD tradeoff between complexity (the number of correlating signals) and BER performance. This leads to a new notion of approximate sufficient statistics, whereby sufficient statistics are approximated to reduce complexity at the expense of some BER performance loss. 1 I.
Orthogonal matching pursuit: a brownian motion analysis
 IEEE Trans. Signal Processing
, 2012
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Perturbation Analysis of Orthogonal Matching Pursuit
, 2011
"... Orthogonal Matching Pursuit (OMP) is a canonical greedy pursuit algorithm for sparse approximation. Previous studies of OMP have considered the recovery of a sparse signal x through and y = x + b, where is a matrix with more columns than rows and b denotes the measurement noise. In this paper, bas ..."
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Cited by 5 (3 self)
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Orthogonal Matching Pursuit (OMP) is a canonical greedy pursuit algorithm for sparse approximation. Previous studies of OMP have considered the recovery of a sparse signal x through and y = x + b, where is a matrix with more columns than rows and b denotes the measurement noise. In this paper, based on Restricted Isometry Property (RIP), the performance of OMP is analyzed under general perturbations, which means both y and are perturbed. Though the exact recovery of an almost sparse signal x is no longer feasible, the main contribution reveals that the support set of the best kterm approximation of x can be recovered under reasonable conditions. The error bound between x and the estimation of OMP is also derived. By constructing an example it is also demonstrated that the sufficient conditions for support recovery of the best kterm approximation of x are rather tight. When x is strongdecaying, it is proved that the sufficient conditions for support recovery of the best kterm approximation of x can be relaxed, and the support can even be recovered in the order of the entries’ magnitude. Our results are also compared in detail with some related previous ones.
Compressive Demodulation of Mutually Interfering Signals
 IEEE Trans. on Information Theory
, 2013
"... The challenge of Multiuser Detection (MUD) is that of demodulating mutually interfering signals given that at any time instant the number of active users is typically small. The promise of compressed sensing is the demodulation of sparse superpositions of signature waveforms from very few measure ..."
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Cited by 5 (4 self)
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The challenge of Multiuser Detection (MUD) is that of demodulating mutually interfering signals given that at any time instant the number of active users is typically small. The promise of compressed sensing is the demodulation of sparse superpositions of signature waveforms from very few measurements. This paper considers signature waveforms that are are drawn from a Gabor frame. It describes a MUD architecture that uses subsampling to convert analog input to a digital signal, and then uses iterative matching pursuit to recover the active users. Compressive demodulation requires K logN samples to recover K active users whereas standard MUD requires N samples. The paper provides theoretical performance guarantees and consistent numerical simulations.
Multiuser Detection in Asynchronous On–Off Random Access Channels Using Lasso
"... Abstract—This paper considers on–off random access channels where users transmit either a one or a zero to a base station. Such channels represent an abstraction of control channels used for scheduling requests in thirdgeneration cellular systems and uplinks in wireless sensor networks deployed for ..."
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Cited by 4 (0 self)
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Abstract—This paper considers on–off random access channels where users transmit either a one or a zero to a base station. Such channels represent an abstraction of control channels used for scheduling requests in thirdgeneration cellular systems and uplinks in wireless sensor networks deployed for target detection. This paper introduces a novel convexoptimizationbased scheme for multiuser detection (MUD) in asynchronous on–off random access channels that does not require knowledge of the delays or the instantaneous received signaltonoise ratios of the individual users at the base station. For any fixed number of temporal signal space dimensions N and maximum delay τ in the system, the proposed scheme can accommodate M � exp(O(N 1/3)) total users and k � N/logM active users in the system—a significant improvement over thek ≤ M � N scaling suggested by the use of classical matchedfilteringbased approaches to MUD employing orthogonal signaling. Furthermore, the computational complexity of the proposed scheme differs from that of a similar oraclebased scheme with perfect knowledge of the user delays by at most a factor oflog(N+τ). Finally, the results presented in here are nonasymptotic, in contrast to related previous work for synchronous channels that only guarantees that the probability of MUD error at the base station goes to zero asymptotically in M. I.