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 describes a multi-relay strategy for wireless networks and examines the influence of imperfect channel information on system performance. A modified relay scheme is proposed to compensate for such imperfections.

/pdf/multi-relay-strategy-for-imperfect-channel-information-in-m9z3se0mpg.pdf

Multi-relay strategy for imperfect channel information in sensor networks

This paper considers the problem of adaptive filtering in the presence of uncertainties in the regression data. A recursive procedure is derived that is based on solving local optimization problems that attemps to alleviate the worst-case effect of data uncertainties on filter performance. The resulting procedure turns out to have similarities with leakage-based adaptive filters.

An adaptive filter robust to data uncertainties

The study of matrices with a displacement structure is mainly concerned with recursions for the so-called generator matrices. The recursion usually involves free parameters, which can be chosen in several ways so as to simplify the resulting algorithm. In this correspondence we present a choice for the parameters that is motivated by a maximum-entropy formulation, This choice further motivates the introduction of the so-called generalized reflection coefficients which are, in general, different from the better known Schur coefficients.

Displacement structure and maximum entropy

We examine the problem of learning a set of parameters from a distributed dataset. We assume the datasets are collected by agents over a distributed ad-hoc network, and that the communication of the actual raw data is prohibitive due to either privacy constraints or communication constraints. We propose a distributed algorithm for online learning that is proved to guarantee a bounded excess risk and the bound can be made arbitrary small for sufficiently small step-sizes. We apply our framework to the expert advice problem where nodes learn the weights for the trained experts distributively.

Distributed Learning via Diffusion Adaptation with Application to Ensemble Learning

Federated learning is a useful framework for centralized learning from distributed data under practical considerations of heterogeneity, asynchrony, and privacy. Federated architectures are frequently deployed in deep learning settings, which generally give rise to non-convex optimization problems. Nevertheless, most existing analysis are either limited to convex loss functions, or only establish first-order stationarity, despite the fact that saddle-points, which are first-order stationary, are known to pose bottlenecks in deep learning. We draw on recent results on the second-order optimality of stochastic gradient algorithms in centralized and decentralized settings, and establish second-order guarantees for a class of federated learning algorithms.

Ali H. Sayed

Papers

Multi-relay strategy for imperfect channel information in sensor networks

An adaptive filter robust to data uncertainties

Displacement structure and maximum entropy

Distributed Learning via Diffusion Adaptation with Application to Ensemble Learning

Second-Order Guarantees in Federated Learning