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

We pose and solve oblique estimation problems with and without state-space structure. In particular, we derive an oblique Kalman filter and indicate connections with instrumental-variable (IV) methods and higher-order spectra (HOS) analysis.

http://ieeexplore.ieee.org/iel3/3920/11352/00531232.pdf

Oblique state-space estimation algorithms

A Robustness Analysis of Gauss-Newton Recursive Methods

Structured matrices and moment problems

J. P. Burg gave a now-famous solution of the problem of finding the maximum-entropy extension of a section {c o, c 1,…, c n } of a scalar (stationary) covariance sequence, with a nice interpretation in terms of the reflection coefficients of an associated discrete transmission line: namely, that the maximum-entropy extension is obtained by adding trivial sections with zero reflection coefficients. This work inspired various extensions and generalizations, mostly without any associated physical interpretation. In this paper we shall use the generalized transmission lines associated with the basic triangular matrix factorization algorithms of the Displacement Structure Theory to obtain several interesting maximum entropy extension results. One is the perhaps surprising fact that Burg’s zero-reflection-coefficient extension solution may not always hold when the coefficients {c i } are matrix-valued.

Recursive Construction of Multichannel Transmission Lines with a Maximum Entropy Property

We pursue a time-domain feedback analysis of adaptive schemes with nonlinear update relations. We consider commonly used algorithms in blind equalization and study their performance in a purely deterministic framework. The derivation employs insights from system theory and feedback analysis, and it clarifies the combined effects of the step-size parameter and the nature of the nonlinear functional on the convergence and robustness performance of adaptive schemes.

Ali H. Sayed

Papers

Oblique state-space estimation algorithms

A Robustness Analysis of Gauss-Newton Recursive Methods

Structured matrices and moment problems

Recursive Construction of Multichannel Transmission Lines with a Maximum Entropy Property

Robustness and convergence of adaptive schemes in blind equalization