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

Fast RLS Laguerre adaptive filtering

In previous work, we have proposed an adaptive sigma delta modulator with improved dynamic range. The modulator adapts the step-size of the quantizer from estimates of the quantizer input. In this paper, we conduct a stability analysis of the new modulator.

Stability analysis of an adaptive structure for sigma delta modulation

We propose a probabilistic routing algorithm for wireless ad-hoc networks. The focus is on applications that can tolerate end-to-end delays and those for which the final destination for the packets is a small region. The nodes are assumed stationary and establish connections with a master node according to a priority scheme that relates to their distances from the master node. Simulation results illustrate the performance of the proposed algorithm.

A probabilistic power-aware routing algorithm for wireless ad-hoc networks

This paper provides a time-domain feedback analysis of the perceptron learning algorithm. It studies the robustness performance of the algorithm in the presence of uncertainties that might be due to noisy perturbations in the reference signal or to modeling mismatch. In particular, bounds are established on the step-size parameter in order to guarantee that the resulting algorithm will behave as a robust filter in the sense of H/sup /spl infin//-theory. The paper also establishes that an intrinsic feedback structure can be associated with the training scheme. The feedback configuration is motivated via energy arguments and is shown to consist of two major blocks: a time-variant lossless (i.e. energy preserving) feedforward path and a time-variant feedback path. The stability of the feedback structure is then analyzed via the small gain theorem and choices for the step-size parameter in order to guarantee faster convergence are further derived by appealing to the mean-value theorem. Simulation results are included to validate the findings.

A feedback analysis of perceptron learning for neural networks

Part I of this paper considered optimization problems over networks where agents have individual objectives to meet, or individual parameter vectors to estimate, subject to subspace constraints that require the objectives across the network to lie in low-dimensional subspaces. Starting from the centralized projected gradient descent, an iterative and distributed solution was proposed that responds to streaming data and employs stochastic approximations in place of actual gradient vectors, which are generally unavailable. We examined the second-order stability of the learning algorithm and we showed that, for small step-sizes $\mu$, the proposed strategy leads to small estimation errors on the order of $\mu$. This Part II examines steady-state performance. The results reveal explicitly the influence of the gradient noise, data characteristics, and subspace constraints, on the network performance. The results also show that in the small step-size regime, the iterates generated by the distributed algorithm achieve the centralized steady-state performance.

Ali H. Sayed

Papers

Fast RLS Laguerre adaptive filtering

Stability analysis of an adaptive structure for sigma delta modulation

A probabilistic power-aware routing algorithm for wireless ad-hoc networks

A feedback analysis of perceptron learning for neural networks

Adaptation and learning over networks under subspace constraints -- Part II: Performance Analysis