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 show that several applications considered in the context of H/sup /spl infin//-filtering and game theory, risk sensitive control and estimation, follow as special cases of the Krein space Kalman filter. We show that these problems can be cast into the problem of calculating the stationary points of certain second order forms: and that by considering appropriate state space models and error Gramians, we can use the Krein space Kalman filter to recursively compute these stationary points and to study their properties. >

Recursive linear estimation in Krein spaces. II. Applications

Diffusion LMS was originally conceived for online distributed parameter estimation in single-task environments where agents pursue a common objective. However, estimating distinct but correlated objects (multitask problems) is useful in many applications. To address multitask problems with combine-then-adapt diffusion LMS strategies, we derive an unsupervised strategy that allows each node to continuously select the neighboring nodes with which it should exchange information to improve its estimation accuracy. Simulation experiments illustrate the efficiency of this clustering strategy. In particular, nodes do not know which other nodes share similar objectives.

Adaptive clustering for multitask diffusion networks

In this paper we exploit the one-to-one correspondences between the recursive least-squares (RLS) and Kalman variables to formulate extended forms of the RLS algorithm. Two particular forms are considered, one pertaining to a system identification problem and the other to the tracking of a chirped sinusoid in additive noise. For both applications, experiments are presented that demonstrate the tracking optimality of the extended RLS algorithms, compared with the standard RLS and least-mean-squares (LMS) algorithms.

Tracking of linear time-variant systems

In this work we consider the problem of learning an Erdős-Renyi graph over a diffusion network when: i) data from only a limited subset of nodes are available (partial observation); ii) and the inferential goal is to discover the graph of interconnections linking the accessible nodes (local structure learning). We propose three matrix estimators, namely, the Granger, the one-lag correlation, and the residual estimators, which, when followed by a universal clustering algorithm, are shown to retrieve the true subgraph in the limit of large network sizes. Remarkably, it is seen that a fundamental role is played by the uniform concentration of node degrees, rather than by sparsity.

Graph Learning with Partial Observations: Role of Degree Concentration

This paper examines the mean-square error performance of diffusion stochastic algorithms under a generalized coordinate-descent scheme. In this setting, the adaptation step by each agent is limited to a random subset of the coordinates of its stochastic gradient vector. The selection of coordinates varies randomly from iteration to iteration and from agent to agent across the network. Such schemes are useful in reducing computational complexity at each iteration in power-intensive large data applications. They are also useful in modeling situations where some partial gradient information may be missing at random. Interestingly, the results show that the steady-state performance of the learning strategy is not always degraded, while the convergence rate suffers some degradation. The results provide yet another indication of the resilience and robustness of adaptive distributed strategies.

Ali H. Sayed

Papers

Recursive linear estimation in Krein spaces. II. Applications

Adaptive clustering for multitask diffusion networks

Tracking of linear time-variant systems

Graph Learning with Partial Observations: Role of Degree Concentration

Coordinate-Descent Diffusion Learning by Networked Agents