Cramer-Rao Bound for Sparse Signals Fitting the Low-Rank Model with Small Number of Parameters

In this paper, we consider signals with a low-rank covariance matrix which reside in a low-dimensional subspace and can be written in terms of a finite (small) number of parameters. Although such signals do not necessarily have a sparse representation in a finite basis, they possess a sparse structure which makes it possible to recover the signal from compressed measurements. We study the statistical performance bound for parameter estimation in the low-rank signal model from compressed measurements. Specifically, we derive the Cramer-Rao bound (CRB) for a generic low-rank model and we show that the number of compressed samples needs to be larger than the number of sources for the existence of an unbiased estimator with finite estimation variance. We further consider the applications to direction-of-arrival (DOA) and spectral estimation which fit into the low-rank signal model. We also investigate the effect of compression on the CRB by considering numerical examples of the DOA estimation scenario, and show how the CRB increases by increasing the compression or equivalently reducing the number of compressed samples.Comment: 14 pages, 1 figure, Submitted to IEEE Signal Processing Letters on December 201

Compressive sensor networks : fundamental limits and algorithms

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2009.Cataloged from PDF version of thesis.Includes bibliographical references (p. 85-92).Compressed sensing is a non-adaptive compression method that takes advantage of natural sparsity at the input and is fast gaining relevance to both researchers and engineers for its universality and applicability. First developed by Candis et al., the subject has seen a surge of high-quality results both in its theory and applications. This thesis extends compressed sensing ideas to sensor networks and other bandwidth-constrained communication systems. In particular, we explore the limits of performance of compressive sensor networks in relation to fundamental operations such as quantization and parameter estimation. Since compressed sensing is originally formulated as a real-valued problem, quantization of the measurements is a very natural extension. Although several researchers have proposed modified reconstruction methods that mitigate quantization noise for a fixed quantizer, the optimal design of such quantizers is still unknown. We propose to find the optimal quantizer in terms of minimizing quantization error by using recent results in functional scalar quantization. The best quantizer in this case is not the optimal design for the measurements themselves but rather is reweighted by a factor we call the sensitivity. Numerical results demonstrate a constant-factor improvement in the fixed-rate case. Parameter estimation is an important goal of many sensing systems since users often care about some function of the data rather than the data itself.(cont.) Thus, it is of interest to see how efficiently nodes using compressed sensing can estimate a parameter, and if the measurements scalings can be less restrictive than the bounds in the literature. We explore this problem for time difference and angle of arrival, two common methods for source geolocation. We first derive Cramer-Rao lower bounds for both parameters and show that a practical block-OMP estimator can be relatively efficient for signal reconstruction. However, there is a large gap between theory and practice for time difference or angle of arrival estimation, which demonstrates the CRB to be an optimistic lower bound for nonlinear estimation. We also find scaling laws 'for time difference estimation in the discrete case. This is strongly related to partial support recovery, and we derive some new sufficient conditions that show a very simple reconstruction algorithm can achieve substantially better scaling than full support recovery suggests is possible.by John Zheng Sun.S.M

Cram\'er-Rao Bound Optimized Subspace Reconstruction in Quantitative MRI

We extend the traditional framework for estimating subspace bases that maximize the preserved signal energy to additionally preserve the Cram\'er-Rao bound (CRB) of the biophysical parameters and, ultimately, improve accuracy and precision in the quantitative maps. To this end, we introduce an \textit{approximate compressed CRB} based on orthogonalized versions of the signal's derivatives with respect to the model parameters. This approximation permits singular value decomposition (SVD)-based minimization of both the CRB and signal losses during compression. Compared to the traditional SVD approach, the proposed method better preserves the CRB across all biophysical parameters with negligible cost to the preserved signal energy, leading to reduced bias and variance of the parameter estimates in simulation. In vivo, improved accuracy and precision are observed in two quantitative neuroimaging applications, permitting the use of smaller basis sizes in subspace reconstruction and offering significant computational savings

Conditional Posterior Cramer-Rao Lower Bound and Distributed Target Tracking in Sensor Networks

Sequential Bayesian estimation is the process of recursively estimating the state of a dynamical system observed in the presence of noise. Posterior Cramer-Rao lower bound (PCRLB) sets a performance limit onany Bayesian estimator for the given dynamical system. The PCRLBdoes not fully utilize the existing measurement information to give anindication of the mean squared error (MSE) of the estimator in the future. In many practical applications, we are more concerned with the value of the bound in the future than in the past. PCRLB is an offline bound, because it averages out the very useful measurement information, which makes it an off-line bound determined only by the system dynamical model, system measurement model and the prior knowledge of the system state at the initial time. This dissertation studies the sequential Bayesian estimation problem and then introduces the notation of conditional PCRLB, which utilizes the existing measurement information up to the current time, and sets the limit on the MSE of any Bayesian estimators at the next time step. This work has two emphases: firstly, we give the mathematically rigorous formulation of the conditional PCRLB as well as the approximate recursive version of conditional PCRLB for nonlinear, possibly non-Gaussian dynamical systems. Secondly, we apply particle filter techniques to compute the numerical values of the conditional PCRLB approximately, which overcomes the integration problems introduced by nonlinear/non-Gaussian systems. Further, we explore several possible applications of the proposed bound to find algorithms that provide improved performance. The primary problem of interest is the sensor selection problem for target tracking in sensor networks. Comparisons are also made between the performance of sensor selection algorithm based on the proposed bound and the existing approaches, such as information driven, nearest neighbor, and PCRLB with renewal strategy, to demonstrate the superior performances of the proposed approach. This dissertation also presents a bandwidth-efficient algorithm for tracking a target in sensor networks using distributed particle filters. This algorithm distributes the computation burden for target tracking over the sensor nodes. Each sensor node transmits a compressed local tracking result to the fusion center by a modified expectationmaximization (EM) algorithm to save the communication bandwidth. The fusion center incorporates the compressed tracking results to give the estimate of the target state. Finally, the target tracking problem in heterogeneous sensor networks is investigated extensively. Extended Kalman Filter and particle filter techniques are implemented and compared for tracking a maneuvering

Saving phase: Injectivity and stability for phase retrieval

Recent advances in convex optimization have led to new strides in the phase retrieval problem over finite-dimensional vector spaces. However, certain fundamental questions remain: What sorts of measurement vectors uniquely determine every signal up to a global phase factor, and how many are needed to do so? Furthermore, which measurement ensembles lend stability? This paper presents several results that address each of these questions. We begin by characterizing injectivity, and we identify that the complement property is indeed a necessary condition in the complex case. We then pose a conjecture that 4M-4 generic measurement vectors are both necessary and sufficient for injectivity in M dimensions, and we prove this conjecture in the special cases where M=2,3. Next, we shift our attention to stability, both in the worst and average cases. Here, we characterize worst-case stability in the real case by introducing a numerical version of the complement property. This new property bears some resemblance to the restricted isometry property of compressed sensing and can be used to derive a sharp lower Lipschitz bound on the intensity measurement mapping. Localized frames are shown to lack this property (suggesting instability), whereas Gaussian random measurements are shown to satisfy this property with high probability. We conclude by presenting results that use a stochastic noise model in both the real and complex cases, and we leverage Cramer-Rao lower bounds to identify stability with stronger versions of the injectivity characterizations.Comment: 22 page

Off-the-Grid Line Spectrum Denoising and Estimation with Multiple Measurement Vectors

Compressed Sensing suggests that the required number of samples for reconstructing a signal can be greatly reduced if it is sparse in a known discrete basis, yet many real-world signals are sparse in a continuous dictionary. One example is the spectrally-sparse signal, which is composed of a small number of spectral atoms with arbitrary frequencies on the unit interval. In this paper we study the problem of line spectrum denoising and estimation with an ensemble of spectrally-sparse signals composed of the same set of continuous-valued frequencies from their partial and noisy observations. Two approaches are developed based on atomic norm minimization and structured covariance estimation, both of which can be solved efficiently via semidefinite programming. The first approach aims to estimate and denoise the set of signals from their partial and noisy observations via atomic norm minimization, and recover the frequencies via examining the dual polynomial of the convex program. We characterize the optimality condition of the proposed algorithm and derive the expected convergence rate for denoising, demonstrating the benefit of including multiple measurement vectors. The second approach aims to recover the population covariance matrix from the partially observed sample covariance matrix by motivating its low-rank Toeplitz structure without recovering the signal ensemble. Performance guarantee is derived with a finite number of measurement vectors. The frequencies can be recovered via conventional spectrum estimation methods such as MUSIC from the estimated covariance matrix. Finally, numerical examples are provided to validate the favorable performance of the proposed algorithms, with comparisons against several existing approaches.Comment: 14 pages, 10 figure