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.

/pdf/tensor-decompositions-and-applications-46vl2ih5gn.pdf

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

This paper addresses the problem of channel tracking and equalization in frequency-selective fading channels. Low-order autoregressive models approximate the channel taps, where each tap is a circular complex Gaussian random process with the typical U-shaped spectrum, and uncorrelated with each other. A Kalman filter tracks the time-varying channel, using the decisions of an adaptive minimum-mean-squared-error decision-feedback equalizer (DFE). The DFE is optimized for decision delay /spl Delta/>0, which exhibits performance advantages over decision delay /spl Delta/=0 for a wide range of channels. The DFE staggered decisions cause the Kalman filter to also track the channel with a delay. The receiver also uses a channel prediction module to bridge the time gap between the Kalman channel estimation and the channel estimates needed for the DFE adaptation. The proposed algorithm offers good tracking behavior thus allowing for reduction in the amount of training symbols needed to effectively track a time-varying frequency selective channel.

Channel estimation and equalization in fading

We consider distributed detection problems over adaptive networks, where dispersed agents learn continually from streaming data by means of local interactions. The requirement of adaptation allows the network of detectors to track drifts in the underlying hypothesis. The requirement of cooperation allows each agent to deliver a performance superior to what would be obtained if it were acting individually. The simultaneous requirements of adaptation and cooperation are achieved by employing diffusion algorithms with constant step-size $\mu$ . By conducting a refined asymptotic analysis based on the mathematical framework of exact asymptotics, we arrive at a revealing understanding of the universal behavior of distributed detection over adaptive networks : as functions of $1/\mu$ , the error (log-)probability curves corresponding to different agents stay nearly-parallel to each other (as already discovered in [1] and [2] ), however, these curves are ordered following a criterion reflecting the degree of connectivity of each agent. Depending on the combination weights, the more connected an agent is, the lower its error probability curve will be. The analysis provides explicit analytical formulas for the detection error probabilities and these expressions are also verified by means of extensive simulations. We further enlarge the reference setting from the case of doubly-stochastic combination matrices considered in [1] and [2] to the more general and demanding setting of right-stochastic combination matrices; this extension poses new and interesting questions in terms of the interplay between the network topology, the combination weights, and the inference performance. The potential of the proposed methods is illustrated by application of the results to canonical detection problems, to typical network topologies, for both doubly-stochastic and right-stochastic combination matrices. Interesting and somehow unexpected behaviors emerge, and the lesson learned is that connectivity matters .

Distributed Detection Over Adaptive Networks: Refined Asymptotics and the Role of Connectivity

Under uncertain channel conditions, local and global power control factors for amplify-and-forward relay processing and source-destination beamforming are jointly and iteratively designed based on a minimum mean-square-error (MMSE) criterion. The influence of imperfect channel information on system performance is examined by computer simulation. As a result, it is verified that the proposed power control methods can relieve the performance degradation arising from channel uncertainties.

Power Allocation for Beamforming Relay Networks under Channel Uncertainties

We develop array algorithms for H/sup /spl infin// filtering. These algorithms can be regarded as the Krein space generalizations of H/sup 2/ array algorithms, which are currently the preferred method fur implementing H/sup 2/ filters. The array algorithms considered include two main families: square-root array algorithms, which are typically numerically more stable than conventional ones, and fast array algorithms which, when the system is time-invariant, typically offer an order of magnitude reduction in the computational effort. Both have the interesting feature that one does not need to explicitly check for the positivity conditions required for the existence of H/sup /spl infin// filters, as these conditions are built into the algorithms themselves. However, since H/sup /spl infin// square-root algorithms predominantly use J-unitary transformations, rather than the unitary transformations required in the H/sup 2/ case, further investigation is needed to determine the numerical behavior of such algorithms.

Array algorithms for H/sup /spl infin// estimation

In Part I of this paper, also in this issue, we introduced a fairly general model for asynchronous events over adaptive networks including random topologies, random link failures, random data arrival times, and agents turning on and off randomly. We performed a stability analysis and established the notable fact that the network is still able to converge in the mean-square-error sense to the desired solution. Once stable behavior is guaranteed, it becomes important to evaluate how fast the iterates converge and how close they get to the optimal solution. This is a demanding task due to the various asynchronous events and due to the fact that agents influence each other. In this Part II, we carry out a detailed analysis of the mean-square-error performance of asynchronous strategies for solving distributed optimization and adaptation problems over networks. We derive analytical expressions for the mean-square convergence rate and the steady-state mean-square-deviation. The expressions reveal how the various parameters of the asynchronous behavior influence network performance. In the process, we establish the interesting conclusion that even under the influence of asynchronous events, all agents in the adaptive network can still reach an O(ν
 1 + γo'
) near-agreement with some γo
 '
 > 0 while approaching the desired solution within O(ν) accuracy, where ν is proportional to the small step-size parameter for adaptation.

Ali H. Sayed

Papers

Channel estimation and equalization in fading

Distributed Detection Over Adaptive Networks: Refined Asymptotics and the Role of Connectivity

Power Allocation for Beamforming Relay Networks under Channel Uncertainties

Array algorithms for H/sup /spl infin// estimation

Asynchronous Adaptation and Learning Over Networks—Part II: Performance Analysis