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

We consider the problem of distributed estimation in adaptive networks where a collection of nodes are required to estimate in a collaborative manner some parameter of interest from their measurements. The centralized solution to the problem uses a fusion center, thus requiring a large amount of energy for communication. We propose a diffusion recursive least-squares algorithm where nodes need to communicate only with their closest neighbors. The algorithm has no topology constraints, and requires no transmission or inversion of matrices, therefore saving in communications and complexity. We show that the algorithm is stable and analyze its performance comparing it to the centralized global solution.

A diffusion rls scheme for distributed estimation over adaptive networks

We study the problem of distributed state-space estimation, where a set of nodes are required to estimate the state of a nonlinear state-space system based on their observations. We extend our previous work on distributed Kalman filtering to the nonlinear case, and propose algorithms for Extended and Unscented Kalman filtering. The resulting algorithms are robust to node and link failure, scalable, and fully distributed, in the sense that no fusion center is required, and nodes communicate with their neighbors only. We apply the algorithms to the problem of estimating the position of every node in an ad-hoc network, also known as wireless localization. Simulation results illustrate the performance of the proposed algorithms.

Distributed nonlinear Kalman filtering with applications to wireless localization

This paper provides a detailed analysis that shows how to stabilize the {\em generalized} Schur algorithm, which is a fast procedure for the Cholesky factorization of positive-definite structured matrices $R$ that satisfy displacement equations of the form $R-FRF^T=GJG^T$, where $J$ is a $2\times 2$ signature matrix, $F$ is a stable lower-triangular matrix, and $G$ is a generator matrix. In particular, two new schemes for carrying out the required hyperbolic rotations are introduced and special care is taken to ensure that the entries of a Blaschke matrix are computed to high relative accuracy. Also, a condition on the smallest eigenvalue of the matrix, along with several computational enhancements, is introduced in order to avoid possible breakdowns of the algorithm by assuring the positive-definiteness of the successive Schur complements. We use a perturbation analysis to indicate the best accuracy that can be expected from {\em any} finite-precision algorithm that uses the generator matrix as the input data. We then show that the modified Schur algorithm proposed in this work essentially achieves this bound when coupled with a scheme to control the generator growth. The analysis further clarifies when pivoting strategies may be helpful and includes illustrative numerical examples. For all practical purposes, the major conclusion of the analysis is that the modified Schur algorithm is backward stable for a large class of structured matrices.

Stabilizing the Generalized Schur Algorithm

This note addresses the problem of robust multiobjective filtering for discrete time-delay systems with mixed stochastic and deterministic uncertainties, in addition to unmodeled nonlinearities. A procedure is developed for the design of linear and exponentially stable filters with a bounded error variance, exponential rate of decay, and robust performance for the error system.

Multiobjective filter design for uncertain stochastic time-delay systems

We consider distributed multitask learning problems over a network of agents where each agent is interested in estimating its own parameter vector, also called task, and where the tasks at neighboring agents are related according to a set of linear equality constraints. Each agent possesses its own convex cost function of its parameter vector and a set of linear equality constraints involving its own parameter vector and the parameter vectors of its neighboring agents. We propose an adaptive stochastic algorithm based on the projection gradient method and diffusion strategies in order to allow the network to optimize the individual costs subject to all constraints. Although the derivation is carried out for linear equality constraints, the technique can be applied to other forms of convex constraints. We conduct a detailed mean-square-error analysis of the proposed algorithm and derive closed-form expressions to predict its learning behavior. We provide simulations to illustrate the theoretical findings. Finally, the algorithm is employed for solving two problems in a distributed manner: A minimum-cost flow problem over a network and a space–time varying field reconstruction problem.

Ali H. Sayed

Papers

A diffusion rls scheme for distributed estimation over adaptive networks

Distributed nonlinear Kalman filtering with applications to wireless localization

Stabilizing the Generalized Schur Algorithm

Multiobjective filter design for uncertain stochastic time-delay systems

Diffusion LMS for Multitask Problems With Local Linear Equality Constraints