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

This paper examines the learning mechanism of adaptive agents over weakly connected graphs and reveals an interesting behavior on how information flows through such topologies. The results clarify how asymmetries in the exchange of data can mask local information at certain agents and make them totally dependent on other agents. A leader-follower relationship develops with the performance of some agents being fully determined by the performance of other agents that are outside their domain of influence. This scenario can arise, for example, due to intruder attacks by malicious agents or as the result of failures by some critical links. The findings in this paper help explain why strong-connectivity of the network topology, adaptation of the combination weights, and clustering of agents are important ingredients to equalize the learning abilities of all agents against such disturbances. The results also clarify how weak-connectivity can be helpful in reducing the effect of outlier data on learning performance.

Information Exchange and Learning Dynamics Over Weakly Connected Adaptive Networks

In biological systems, animals exhibit organized behavior that arises from localized interactions. The interaction is implemented through information exchange, either directly or indirectly. Adaptive networks, consisting of a collection of nodes with learning abilities that interact with each other to solve distributed inference problems in real-time, are well-suited to model these kinds of behavior. Usually the information exchange between two nodes is imperfect and the data from neighbors are noisy. In this paper, we examine the effect of noisy communication links on network performance and derive an optimal strategy for adjusting the combination weights.

Adaptive Networks with Noisy Links

Graphs provide a powerful framework to represent high-dimensional but structured data, and to make inferences about relationships between subsets of the data. In this work we consider graph signals that evolve dynamically according to a heat diffusion process and are subject to persistent perturbations. We develop an online algorithm that is able to learn the underlying graph structure from observations of the signal evolution. The algorithm is adaptive in nature and in particular able to respond to changes in the graph structure and the perturbation statistics.

Online graph learning from sequential data

Mobile adaptive networks consist of a collection of nodes with learning and motion abilities that interact with each other locally in order to solve distributed processing and distributed inference problems in real-time. In this paper, we develop adaptation algorithms that exhibit self-organization properties and apply them to the model of cooperative hunting among predators. The results help provide an explanation for the agile adjustment of network patterns in the interaction between fish schools and predators.

Cooperative prey herding based on diffusion adaptation

This paper proposes two modifications of the FxLMS algorithm with improved convergence behaviour albeit at the same computational cost of 2M operations per time step as the original FxLMS update. The paper further introduces a generalized FxLMS recursion and establishes that the various recursions are in fact of filtered-error form. An optimal choice of the step-size parameter in order to guarantee faster convergence, and conditions for robustness, are also derived. Several simulation results are included to illustrate the discussions.

Ali H. Sayed

Papers

Information Exchange and Learning Dynamics Over Weakly Connected Adaptive Networks

Adaptive Networks with Noisy Links

Online graph learning from sequential data

Cooperative prey herding based on diffusion adaptation

Modified FxLMS algorithms with improved convergence performance