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ABSTRACT: We investigate the behavior of the mean-square error (MSE) of low-rank and sparse matrix decomposition, in particular the special case of the robust principal component analysis (RPCA), and its generalization matrix completion and correction (MCC). We derive a constrained Cramér-Rao bound (CRB) for any locally unbiased estimator of the low-rank matrix and of the sparse matrix. We analyze the typical behavior of the constrained CRB for MCC where a subset of entries of the underlying matrix are randomly observed, some of which are grossly corrupted. We obtain approximated constrained CRBs by using a concentration of measure argument. We design an alternating minimization procedure to compute the maximum-likelihood estimator (MLE) for the low-rank matrix and the sparse matrix, assuming knowledge of the rank and the sparsity level. For relatively small rank and sparsity level, we demonstrate numerically that the performance of the MLE approaches the constrained CRB when the signal-to-noise-ratio is high. We discuss the implications of these bounds and compare them with the empirical performance of the accelerated proximal gradient algorithm as well as other existing bounds in the literature.
IEEE Transactions on Signal Processing 11/2011; · 2.63 Impact Factor
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ABSTRACT: We propose a multiobjective optimization (MOO) technique to design an orthogonal-frequency-division multiplexing (OFDM) radar signal for detecting a moving target in the presence of multipath reflections. We employ an OFDM signal to increase the frequency diversity of the system, as different scattering centers of a target resonate variably at different frequencies. Moreover, the multipath propagation increases the spatial diversity by providing extra “looks” at the target. First, we develop a parametric OFDM radar model by reformulating the target-detection problem as the task of sparse-signal spectrum estimation. At a particular range cell, we exploit the sparsity of multiple paths and the knowledge of the environment to estimate the paths along which the target responses are received. Then, to estimate the sparse vector, we employ a collection of multiple small Dantzig selectors (DS) that utilizes more prior structures of the sparse vector. We use the ℓ<sub>1</sub>-constrained minimal singular value (ℓ<sub>1</sub>-CMSV) of the measurement matrix to analytically evaluate the reconstruction performance and demonstrate that our decomposed DS performs better than the standard DS. In addition, we propose a constrained MOO-based algorithm to optimally design the spectral parameters of the OFDM waveform for the next coherent processing interval by simultaneously optimizing two objective functions: minimizing the upper bound on the estimation error to improve the efficiency of sparse-recovery and maximizing the squared Mahalanobis-distance to increase the performance of the underlying detection problem. We provide a few numerical examples to illustrate the performance characteristics of the sparse recovery and demonstrate the achieved performance improvement due to adaptive OFDM waveform design.
IEEE Transactions on Signal Processing 03/2011; · 2.63 Impact Factor
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ABSTRACT: We propose a multi-objective optimization (MOO) technique to design an orthogonal frequency division multiplexing (OFDM) radar signal for detecting a moving target in the presence of multipath reflections. We employ an OFDM signal to increase the frequency diversity of the system. Moreover, the multipath propagation increase the spatial diversity by providing extra “looks” at the target. First, we develop a parametric measurement model by reformulating the target detection problem as a sparse estimation method. At a particular range cell, we exploit the sparsity of multiple paths and the knowledge of the environment to estimate along which path the target responses are received. Then, we formulate a constrained MOO algorithm, to optimally design the spectral parameters of the OFDM waveform, by simultaneously optimizing two objective functions: minimizing the upper bound on the estimation error to improve the efficiency of sparse-recovery and maximizing the squared Mahalanobis-distance to increase the performance of the underlying detection problem. We present numerical examples to demonstrate the performance improvement due to adaptive waveform design.
Signals, Systems and Computers (ASILOMAR), 2010 Conference Record of the Forty Fourth Asilomar Conference on; 12/2010
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ABSTRACT: The ℓ<sub>1</sub>-constrained minimal singular value (ℓ<sub>1</sub>-CMSV) of the sensing matrix is shown to determine, in a concise and tight manner, the recovery performance of ℓ<sub>1</sub>-based algorithms such as Basis Pursuit, the Dantzig selector, and the LASSO estimator. Several random measurement ensembles are shown to have ℓ<sub>1</sub>-CMSVs bounded away from zero with high probability, as long as the number of measurements is relatively large. Three algorithms based on projected gradient method and interior point algorithm are developed to compute ℓ<sub>1</sub>-CMSV. A lower bound of the ℓ<sub>1</sub>-CMSV is also available by solving a semi-definite programming problem.
Signals, Systems and Computers (ASILOMAR), 2010 Conference Record of the Forty Fourth Asilomar Conference on; 12/2010
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ABSTRACT: We develop a near-field compressive sensing (CS) estimation scheme for localizing scattering objects in vacuum. The potential of CS for localizing sparse targets was demonstrated in previous work. We extend the standard far-field approach to near-field scenarios by employing the electric field integral equation to capture the mutual interference among targets. We show that the advanced modeling improves the capability to resolve closely spaced targets. We compare the performance of our algorithm with the performances of CS applied to point targets and beamforming. In this paper, we consider two-dimensional (2D) scatterers. However, the results and conclusions can be extended to three-dimensional (3D) problems.
Communication, Control, and Computing (Allerton), 2010 48th Annual Allerton Conference on; 11/2010
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ABSTRACT: The stability of low-rank matrix reconstruction is investigated in this paper. The ℓ<sub>*</sub>-constrained minimal singular value (ℓ<sub>*</sub>-CMSV) of the measurement operator is shown to determine the recovery performance of nuclear norm minimization based algorithms. Compared with the stability results using the matrix restricted isometry constant, the performance bounds established using ℓ<sub>*</sub>-CMSV are more concise and tight, and their derivations are less complex. Several random measurement ensembles are shown to have ℓ<sub>*</sub>-CMSVs bounded away from zero with high probability, as long as the number of measurements is relatively large.
Communication, Control, and Computing (Allerton), 2010 48th Annual Allerton Conference on; 11/2010
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ABSTRACT: We analyze the performance of a direction-of-arrival (DOA) estimation scheme based on overcomplete signal representation in this paper. We formulate the problem as a support recovery problem with joint sparsity constraints and analyze it in a hypothesis testing framework. We derive both upper and lower bounds on the probability of error by using Chernoff bound and Fano's inequality, respectively. The lower bound implies that the minimal number of samples necessary for accurate DOA estimation is proportional to the logarithm of the discretization level for arbitrary isotropic sensor arrays. We apply the upper bound to study the effect of noise. For uniform linear array (ULA) with only one source, the upper bound exponent indicates that the optimal overcomplete representation is achieved by uniform partition of the wave number space instead of the DOA space.
Acoustics Speech and Signal Processing (ICASSP), 2010 IEEE International Conference on; 04/2010 · 4.63 Impact Factor
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ABSTRACT: The performance of estimating the common support for jointly sparse signals based on their projections onto lower-dimensional space is analyzed. Support recovery is formulated as a multiple-hypothesis testing problem. Both upper and lower bounds on the probability of error are derived for general measurement matrices, by using the Chernoff bound and Fano's inequality, respectively. The upper bound shows that the performance is determined by a quantity measuring the measurement matrix incoherence, while the lower bound reveals the importance of the total measurement gain. The lower bound is applied to derive the minimal number of samples needed for accurate direction-of-arrival (DOA) estimation for a sparse representation based algorithm. When applied to Gaussian measurement ensembles, these bounds give necessary and sufficient conditions for a vanishing probability of error for majority realizations of the measurement matrix. Our results offer surprising insights into sparse signal recovery. For example, as far as support recovery is concerned, the well-known bound in Compressive Sensing with the Gaussian measurement matrix is generally not sufficient unless the noise level is low. Our study provides an alternative performance measure, one that is natural and important in practice, for signal recovery in Compressive Sensing and other application areas exploiting signal sparsity.
IEEE Transactions on Information Theory 04/2010; · 3.01 Impact Factor
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ABSTRACT: In this paper, we analyze the performance of estimating the common support for jointly sparse signals based on their projections onto lower-dimensional space. We formulate support recovery as a multiple-hypothesis testing problem and derive both upper and lower bounds on the probability of error for general measurement matrices, by using Chernoff bound and Fano's inequality, respectively. When applied to Gaussian measurement ensembles, these bounds give necessary and sufficient conditions to guarantee a vanishing probability of error for majority realizations of the measurement matrix. Our results offer surprising insights into sparse signal reconstruction based on their projections. For example, as far as support recovery is concerned, the well-known bound in compressive sensing is generally not sufficient if the Gaussian ensemble is used. Our study provides an alternative performance measure, one that is natural and important in practice, for signal recovery in com-pressive sensing as well as other application areas taking advantage of signal sparsity.
Communication, Control, and Computing, 2009. Allerton 2009. 47th Annual Allerton Conference on; 11/2009
