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

There is a fundamental tradeoff between power consumption, data transmission rates, and congestion levels in a wireless network. These three elements influence the performance of rate and power control strategies, and they need to be coordinated judiciously. This paper proposes dynamic rate and power control algorithms for distributed wireless networks that also account for the congestion levels in a network. The design is pursued by formulating state-space models with and without uncertain dynamics and by determining control signals that help meet certain performance criteria (such as robustness and desired levels of signal-to-interference ratio). Simulation results illustrate the performance of the proposed control schemes.

Joint rate and power control algorithms for wireless networks

This paper formulates and solves a robust criterion for least-squares designs in the presence of uncertain data. Compared with earlier studies, the proposed criterion incorporates simultaneously both regularization and weighting and applies to a large class of uncertainties. The solution method is based on reducing a vector optimization problem to an equivalent scalar minimization problem of a provably unimodal cost function, thus achieving considerable reduction in computational complexity.

A Regularized Robust Design Criterion for Uncertain Data

We study the problem of distributed least-squares estimation over ad hoc adaptive networks, where the nodes have a common objective to estimate and track a parameter vector. We consider the case where there is stationary additive colored noise on both the regressors and the output response, which results in biased local least-squares estimators. Assuming that the noise covariance can be estimated (or is known a priori), we first propose a bias-compensated recursive least-squares algorithm (BC-RLS). However, this bias compensation increases the variance or the mean-square deviation (MSD) of the local estimators, and errors in the noise covariance estimates may still result in residual bias. We demonstrate that the MSD and residual bias can then be significantly reduced by applying diffusion adaptation, i.e., by letting nodes combine their local estimates with those of their neighbors. We derive a necessary and sufficient condition for mean-square stability of the algorithm, under some mild assumptions. Furthermore, we derive closed-form expressions for its steady-state mean and mean-square performance. Simulation results are provided, which agree well with the theoretical results. We also consider some special cases where the mean-square stability improvement of diffusion BC-RLS over BC-RLS can be mathematically verified.

Diffusion Bias-Compensated RLS Estimation Over Adaptive Networks

This paper examines a stochastic formulation of the generalized Nash equilibrium problem where agents are subject to randomness in the environment of unknown statistical distribution. We focus on fully distributed online learning by agents and employ penalized individual cost functions to deal with coupled constraints. Three stochastic gradient strategies are developed with constant step-sizes. We allow the agents to use heterogeneous step-sizes and show that the penalty solution is able to approach the Nash equilibrium in a stable manner within $O(\mu _\text{max})$ , for small step-size value $\mu _\text{max}$ and sufficiently large penalty parameters. The operation of the algorithm is illustrated by considering the network Cournot competition problem.

https://ieeexplore.ieee.org/ielaam/78/7933060/7903731-aam.pdf

Distributed Learning for Stochastic Generalized Nash Equilibrium Problems

Spectrum sensing is a key enabling functionality in cognitive radio (CR) networks, where the CRs act as secondary users that opportunistically access free frequency bands. Due to the effects of channel fading, individual CRs may not be able to reliably detect the existence of a primary radio, who is a licensed user for the particular band. In this paper, we present optimal cooperation strategies for spectrum sensing to combat the effects of destructive channels and malfunctioning devices. Our approach conducts spectrum sensing based on the linear combination of local test statistics from individual secondary users. We propose two optimization schemes to control the combining weights, and compare their performance. Our first approach is to optimize the probability distribution function of the global test statistics at the fusion center. For the second scheme, we maximize the global detection sensitivity under constraints on the false alarm probability. Simulation results illustrate the significant cooperative gain achieved by the proposed strategies.

Ali H. Sayed

Papers

Joint rate and power control algorithms for wireless networks

A Regularized Robust Design Criterion for Uncertain Data

Diffusion Bias-Compensated RLS Estimation Over Adaptive Networks

Distributed Learning for Stochastic Generalized Nash Equilibrium Problems

An Optimal Strategy for Cooperative Spectrum Sensing in Cognitive Radio Networks