There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or $N$-way array. Decompositions of higher-order tensors (i.e., $N$-way arrays with $N \geq 3$) have applications in psycho-metrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, and elsewhere. Two particular tensor decompositions can be considered to be higher-order extensions of the matrix singular value decomposition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rank-one tensors, and the Tucker decomposition is a higher-order form of principal component analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The N-way Toolbox, Tensor Toolbox, and Multilinear Engine are examples of software packages for working with tensors.

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Tensor Decompositions and Applications

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Statistics for Spatial Data.

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Table Of Integrals Series And Products

Preface. Acknowledgments. Notation. Symbols. Optimal Estimation. Linear Estimation. Constrained Linear Estimation. Steepest-Descent Algorithms. Stochastic-Gradient Algorithms. Steady-State Performance of Adaptive Filters. Tracking Performance of Adaptive Filters. Finite Precision Effects. Transient Performance of Adaptive Filters. Block Adaptive Filters. The Least-Squares Criterion. Recursive Least-Squares. RLS Array Algorithms. Fast Fixed-Order Filters. Lattice Filters. Laguerre Adaptive Filters. Robust Adaptive Filters. Bibliography. Author Index. Subject Index. AC

Fundamentals of adaptive filtering

Adaptive Filters Adaptive filtering is a topic of immense practical and theoretical value, having applications in areas ranging from digital and wireless communications to biomedical systems. Now, preserving the style and main features of the earlier award-winning publication, Fundamentals of Adaptive Filtering (2005 Terman Award), the author offers readers and instructors a concentrated, systematic, and up-to-date treatment of the subject in this valuable new book. Adaptive Filters allows readers to gain a gradual and solid introduction to the subject, its applications to a variety of topical problems, existing limitations, and extensions of current theories. The book consists of eleven partseach part containing a series of focused lectures and ending with bibliographic comments, problems, and computer projects with MATLAB solutions available to all readers. Additional features include: Numerous tables, figures, and projects Special focus on geometric constructions, physical intuition, linear-algebraic concepts, and vector notation Background material on random variables, linear algebra, and complex gradients collected in three introductory chapters Complete solutions manual available for instructors MATLAB solutions available for all computer projects Adaptive Filters offers a fresh, focused look at the subject in a manner that will entice students, challenge experts, and appeal to practitioners and instructors.

Adaptive Filters

Wireless location refers to the geographic coordinates of a mobile subscriber in cellular or wireless local area network (WLAN) environments. Wireless location finding has emerged as an essential public safety feature of cellular systems in response to an order issued by the Federal Communications Commission (FCC) in 1996. The FCC mandate aims to solve a serious public safety problem caused by the fact that, at present, a large proportion of all 911 calls originate from mobile phones, the location of which cannot be determined with the existing technology. However, many difficulties intrinsic to the wireless environment make meeting the FCC objective challenging. These challenges include channel fading, low signal-to-noise ratios (SNRs), multiuser interference, and multipath conditions. In addition to emergency services, there are many other applications for wireless location technology, including monitoring and tracking for security reasons, location sensitive billing, fraud protection, asset tracking, fleet management, intelligent transportation systems, mobile yellow pages, and even cellular system design and management. This article provides an overview of wireless location challenges and techniques with a special focus on network-based technologies and applications.

Network-based wireless location: challenges faced in developing techniques for accurate wireless location information

We consider the problem of distributed estimation, where a set of nodes is required to collectively estimate some parameter of interest from noisy measurements. The problem is useful in several contexts including wireless and sensor networks, where scalability, robustness, and low power consumption are desirable features. Diffusion cooperation schemes have been shown to provide good performance, robustness to node and link failure, and are amenable to distributed implementations. In this work we focus on diffusion-based adaptive solutions of the LMS type. We motivate and propose new versions of the diffusion LMS algorithm that outperform previous solutions. We provide performance and convergence analysis of the proposed algorithms, together with simulation results comparing with existing techniques. We also discuss optimization schemes to design the diffusion LMS weights.

Diffusion LMS Strategies for Distributed Estimation

Cognitive radio technology has been proposed to improve spectrum efficiency by having the cognitive radios act as secondary users to opportunistically access under-utilized frequency bands. Spectrum sensing, as a key enabling functionality in cognitive radio networks, needs to reliably detect signals from licensed primary radios to avoid harmful interference. However, due to the effects of channel fading/shadowing, individual cognitive radios may not be able to reliably detect the existence of a primary radio. In this paper, we propose an optimal linear cooperation framework for spectrum sensing in order to accurately detect the weak primary signal. Within this framework, spectrum sensing is based on the linear combination of local statistics from individual cognitive radios. Our objective is to minimize the interference to the primary radio while meeting the requirement of opportunistic spectrum utilization. We formulate the sensing problem as a nonlinear optimization problem. By exploiting the inherent structures in the problem formulation, we develop efficient algorithms to solve for the optimal solutions. To further reduce the computational complexity and obtain solutions for more general cases, we finally propose a heuristic approach, where we instead optimize a modified deflection coefficient that characterizes the probability distribution function of the global test statistics at the fusion center. Simulation results illustrate significant cooperative gain achieved by the proposed strategies. The insights obtained in this paper are useful for the design of optimal spectrum sensing in cognitive radio networks.

Ali H. Sayed

Papers

Fundamentals of adaptive filtering

Adaptive Filters

Network-based wireless location: challenges faced in developing techniques for accurate wireless location information

Diffusion LMS Strategies for Distributed Estimation

Optimal Linear Cooperation for Spectrum Sensing in Cognitive Radio Networks