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

In this work, a distributed multi-agent optimization problem is studied where different subsets of agents are coupled with each other through affine constraints. Moreover, each agent is only aware of its own contribution to the constraints and only knows which neighboring agents share constraints with it. An effective distributed first-order algorithm is developed, which requires sharing dual variables only and takes advantage of the constraint sparsity. The algorithm is shown to converge to the exact minimizer under sufficiently small constant step sizes. A simulation is given to illustrate the effect of the constraint structure and advantages of the proposed algorithm.

Dual Coupled Diffusion for Distributed Optimization with Affine Constraints

We consider an estimation network of many distributed sensors, where each senor takes a noisy measurement of some unknown parameter. Due to energy limitation, the network selects only a subset of sensors for data fusion as long as the distortion is tolerable. In this paper, we present a sampling framework based on linear minimum variance unbiased estimation. The framework enables the system to achieve a desired estimation fidelity level and to improve the network lifetime. Simulations illustrate the effectiveness of the proposed sampling schemes.

Innovations-Based Sampling Over Spatially-Correlated Sensors

Clock timing jitters refer to random perturbations in the sampling time in analog-to-digital converters (ADCs). The perturbations are caused by circuit imperfections in the sampling clock. This paper analyzes the effect of sampling clock jitter on the acquired samples. The paper proposes two methods to estimate the jitter for superheterodyne receiver architectures and cognitive radio architectures at high sampling rates. The paper also proposes a method to compensate for the jitter. The methods are tested and validated via computer simulations and theoretical analysis.

Sampling clock jitter estimation and compensation in ADC circuits

Detection of fading overlapping multipath components

We examine the behavior of multiagent networks where information-sharing is subject to a positive communications cost over the edges linking the agents. We consider a general mean-square-error formulation, where all agents are interested in estimating the same target vector. We first show that in the absence of any incentives to cooperate, the optimal strategy for the agents is to behave in a selfish manner with each agent seeking the optimal solution independently of the other agents. Pareto inefficiency arises as a result of the fact that agents are not using historical data to predict the behavior of their neighbors and to know whether they will reciprocate and participate in sharing information. Motivated by this observation, we develop a reputation protocol to summarize the opponent’s past actions into a reputation score, which can then be used to form a belief about the opponent’s subsequent actions. The reputation protocol entices agents to cooperate and turns their optimal strategy into an action-choosing strategy that enhances the overall social benefit of the network. In particular, we show that when the communications cost becomes large, the expected social benefit of the proposed protocol outperforms the social benefit that is obtained by cooperative agents that always share data. We perform a detailed mean-square-error analysis of the evolution of the network over three domains: 1) far field; 2) near-field; and 3) middle-field, and show that the network behavior is stable for sufficiently small step-sizes. The various theoretical results are illustrated by numerical simulations.

Ali H. Sayed

Papers

Dual Coupled Diffusion for Distributed Optimization with Affine Constraints

Innovations-Based Sampling Over Spatially-Correlated Sensors

Sampling clock jitter estimation and compensation in ADC circuits

Detection of fading overlapping multipath components

Information-Sharing Over Adaptive Networks With Self-Interested Agents