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Estimation of the covariance matrix and its inverse, the precision matrix, in high-dimensional situations is of great interest in many applications. In this paper, we focus on the estimation of a class of sparse precision matrices which are assumed to be approximately inversely closed for the case that the dimensionality p can be much larger than the sample size n, which is fundamentally different from the classical case that p < n. Different in nature from state-of-the-art methods that are based on penalized likelihood maximization or constrained error minimization, based on the truncated Neumann series representation, we propose a computationally efficient precision matrix estimator that has a computational complexity of O(p^3). We prove that the proposed estimator is consistent in probability and in L^2 under the spectral norm. Moreover, its convergence is shown to be rate-optimal in the sense of minimax risk. We further prove that the proposed estimator is model selection consistent by establishing a convergence result under the entry-wise \infty-norm. Simulations demonstrate the encouraging finite sample size performance and computational advantage of the proposed estimator. The proposed estimator is also applied to a real breast cancer data and shown to outperform existing precision matrix estimators. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of direction of arrival (DOA) estimation in the presence of sensor gain-phase errors. Under some mild assumptions, we propose a new DOA estimation method based on the eigendecomposition of a covariance matrix which is constructed by the dot product of the array output vector and its conjugate. By combining the new DOA estimation with the conventional gain-phase error estimation, a method is proposed to simultaneously estimate the DOA and gain-phase errors without joint iteration. Theoretical analysis shows that the proposed method performs independently of phase errors and thus behaves well regardless of phase errors. However, the resolution capability of the proposed method is lower than that of the method in [A. J. Weiss and B. Friedlander, “Eigenstructure methods for direction finding with sensor gain and phase uncertainties,” Circuits Systems Signal Process., vol. 9, no. 3, pp. 271–300, 1990], named as the WF method. In order to improve the resolution capability and maintain phase error independence, a combined strategy is developed using the proposed and WF methods. The advantage of the proposed methods is that they are independent of phase errors, leading to the cancellation of phase error calibration during the operation life of an array. Moreover, the proposed methods avoid the problem of suboptimal convergence which occurs in the WF method. The drawbacks of the proposed methods are their high computational complexity and their requirement for the condition that at least two signals are spatially far from each other, and they are not applicable to a linear array. Simulation results verify the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

We consider the problem of sensor selection for event detection in wireless sensor networks (WSNs). We want to choose a subset of p out of n sensors that yields the best detection performance. As the sensor selection optimality criteria, we propose the Kullback–Leibler and Chernoff distances between the distributions of the selected measurements under the two hypothesis. We formulate the maxmin robust sensor selection problem to cope with the uncertainties in distribution means. We prove that the sensor selection problem is NP hard, for both Kullback–Leibler and Chernoff criteria. To (sub)optimally solve the sensor selection problem, we propose an algorithm of affordable complexity. Extensive numerical simulations on moderate size problem instances (when the optimum by exhaustive search is feasible to compute) demonstrate the algorithm's near optimality in a very large portion of problem instances. For larger problems, extensive simulations demonstrate that our algorithm outperforms random searches, once an upper bound on computational time is set. We corroborate numerically the validity of the Kullback–Leibler and Chernoff sensor selection criteria, by showing that they lead to sensor selections nearly optimal both in the Neyman–Pearson and Bayes sense. [ABSTRACT FROM PUBLISHER]

The adaptive estimation of a time-varying parameter vector in a linear Gaussian model is considered where we a priori know that the parameter vector belongs to a known arbitrary subset. We consider a family of efficient recursive estimators for this problem: the recursive constrained maximum likelihood (ML) estimator, the recursive affine minimax, and the recursive minimum mean square error (MMSE) estimator. We show that all three estimators can be substantially simplified by using the recursive weighted least squares (RWLS) algorithm in a first step as the RWLS computes the sufficient statistic for this estimation problem. The recursive constrained ML needs to solve an optimization problem in the second step for the case that the RWLS solution does not fulfill the constraint. In case of affine minimax, we have to solve an optimization problem and to perform an affine transform. The MMSE estimator needs to calculate the mean of a truncated Gaussian density in the second step which is done by Monte Carlo integration. A simple rejection scheme is used to take general constraints for the parameter vector into account. An example shows the superior performance of our proposed estimators in comparison to many other estimators. [ABSTRACT FROM PUBLISHER]

In this paper, a new algorithm for approximate joint diagonalization (AJD) of positive-definite Hermitian matrices is presented. The AJD matrix, which is assumed to be square non-unitary, is derived via minimization of a quasi-maximum likelihood (QML) objective function. This objective function coincides asymptotically with the maximum likelihood (ML) objective function, hence enabling the proposed algorithm to asymptotically approach the ML estimation performance. In the proposed method, the rows of the AJD matrix are obtained independently, in an iterative manner. This feature enables direct estimation of full row-rank rectangular AJD sub-matrices. Under some mild assumptions, convergence of the proposed algorithm is asymptotically guarantied, such that the error norm corresponding to each row of the AJD matrix reduces significantly after the first iteration, and the convergence is almost Q-super linear. This property results rapid convergence, which leads to low computational load in the proposed method. The performance of the proposed algorithm is evaluated and compared to other state-of-the-art algorithms for AJD and its practical use is demonstrated in the blind source separation and blind source extraction problems. The results imply that under the assumptions of high signal-to-noise ratio and large amount of matrices, the proposed algorithm is computationally efficient with performance similar to state-of-the-art algorithms for AJD. [ABSTRACT FROM PUBLISHER]