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

Describe a straightforward recursive and global approach for the solution of rational interpolation problems. The derivation is based on a simple and well known matrix identity, namely, the Schur reduction procedure, and exploits connections with structured matrices. The authors use the interpolation data to construct a convenient structure and then apply a recursive triangularization procedure. This leads to a transmission-line cascade of first-order J-lossless sections that makes evident the interpolation property. State-space descriptions for each section and for the entire cascade are given. >

Recursive solutions to rational interpolation problems

Although the least mean fourth (LMF) and the least mean mixed norm (LMMN) adaptive algorithms are recommended for highly nonstationary environments, their tracking capabilities are not yet fully understood. This is mainly due to the fact that both algorithms involve nonlinear update equations for the weight error vector. We present a new approach to the tracking analysis of the LMF and LMMN algorithms, which bypasses the need for working directly with the weight error vector, and is based on a fundamental energy-preserving relation. By studying the energy flow through the system in steady-state, we derive expressions for the steady-state excess mean square error (EMSE) for both algorithms. We also derive optimal parameter values that minimize the EMSE in each case, and support our conclusions by simulations.

Tracking analysis of the LMF and LMMN adaptive algorithms

This extraordinary three-volume work, written in an engaging and rigorous style by a world authority in the field, provides an accessible, comprehensive introduction to the full spectrum of mathematical and statistical techniques underpinning contemporary methods in data-driven learning and inference. This first volume, Foundations, introduces core topics in inference and learning, such as matrix theory, linear algebra, random variables, convex optimization and stochastic optimization, and prepares students for studying their practical application in later volumes. A consistent structure and pedagogy is employed throughout this volume to reinforce student understanding, with over 600 end-of-chapter problems (including solutions for instructors), 100 figures, 180 solved examples, datasets and downloadable Matlab code. Supported by sister volumes Inference and Learning, and unique in its scale and depth, this textbook sequence is ideal for early-career researchers and graduate students across many courses in signal processing, machine learning, statistical analysis, data science and inference.

Inference and Learning from Data

We study the asymptotic learning rates under linear and log-linear combination rules of belief vectors in a distributed hypothesis testing problem. We show that under both combination strategies, agents are able to learn the truth exponentially fast, with a faster rate under log-linear fusion. We examine the gap between the rates in terms of network connectivity and information diversity. We also provide closed-form expressions for special cases involving federated architectures and exchangeable networks.

On the Arithmetic and Geometric Fusion of Beliefs for Distributed Inference

Conventional derivations of fast fixed-order RLS filters rely on the shift structure that is characteristic of regressors in a tapped-delay line implementation. In this paper, we study adaptive Laguerre networks, where shift structure no longer holds. We show that fast fixed-order updates are still possible.

Ali H. Sayed

Papers

Recursive solutions to rational interpolation problems

Tracking analysis of the LMF and LMMN adaptive algorithms

Inference and Learning from Data

On the Arithmetic and Geometric Fusion of Beliefs for Distributed Inference

Extended fast fixed order RLS adaptive filters