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 studies the tracking performance of adaptive filters operating in the presence of two sources of nonstationarities: carrier frequency offsets and random channel variations. Both impairments are common in digital communications due to mismatches between transmitter and receiver carrier generators and channel fading. The paper derives expressions for the mean-square error and shows how filter performance is degraded under such nonstationary conditions. Selections of step sizes for optimal tracking performance are derived, different adaptive algorithms are compared, and supporting simulation results are provided.

Ability of adaptive filters to track carrier offsets and channel nonstationarities

The paper develops a leakage-based adaptive algorithm, referred to as circular-leaky, which in addition to solving the drift problem of the classical least mean squares (LMS) adaptive algorithm, it also avoids the bias problem that is created by the standard leaky LMS solution. These two desirable properties of unbiased and bounded estimates are guaranteed by circular-leaky at essentially the same computational cost as LMS. The derivation in the paper relies on results from averaging theory and from Lyapunov stability theory, and the analysis shows that the above properties hold not only in infinite-precision but also in finite-precision arithmetic. The paper further extends the results to a so-called switching-/spl sigma/ algorithm, which is a leakage-based solution used in adaptive control.

/pdf/unbiased-and-stable-leakage-based-adaptive-filters-3w9nq83s6q.pdf

Unbiased and stable leakage-based adaptive filters

We consider the problem of fixed-point distributed Kalman smoothing, where a set of nodes are required to estimate the initial condition of a certain process based on their measurements of the evolution of the process. Specifically, we consider linear state-space models where the Kalman smoother gives us the MMSE estimate of the initial state of the system. We propose distributed diffusion solutions where nodes communicate with their neighbors and information is propagated through the network via a diffusion process. Hierarchical cooperation schemes are also described.

Diffusion mechanisms for fixed-point distributed Kalman smoothing

Recursive Least-Squares Adaptive Filters

This paper examines the close interplay between cooperation and adaptation for distributed detection schemes over fully decentralized networks. The combined attributes of cooperation and adaptation are necessary to enable networks of detectors to continually learn from streaming data and to continually track drifts in the state of nature when deciding in favor of one hypothesis or another. The results in this paper establish a fundamental scaling law for the steady-state probabilities of miss detection and false alarm in the slow adaptation regime, when the agents interact with each other according to distributed strategies that employ small constant step-sizes. The latter are critical to enable continuous adaptation and learning. This paper establishes three key results. First, it is shown that the output of the collaborative process at each agent has a steady-state distribution. Second, it is shown that this distribution is asymptotically Gaussian in the slow adaptation regime of small step-sizes. Third, by carrying out a detailed large deviations analysis, closed-form expressions are derived for the decaying rates of the false-alarm and miss-detection probabilities. Interesting insights are gained from these expressions. In particular, it is verified that as the step-size $\mu $ decreases, the error probabilities are driven to zero exponentially fast as functions of $1/\mu $ , and that the exponents governing the decay increase linearly in the number of agents. It is also verified that the scaling laws governing the errors of detection and the errors of estimation over the network behave very differently, with the former having exponential decay proportional to $1/\mu $ , while the latter scales linearly with decay proportional to $\mu $ . Moreover, and interestingly, it is shown that the cooperative strategy allows each agent to reach the same detection performance, in terms of detection error exponents, of a centralized stochastic-gradient solution. The results of this paper are illustrated by applying them to canonical distributed detection problems.

Ali H. Sayed

Papers

Ability of adaptive filters to track carrier offsets and channel nonstationarities

Unbiased and stable leakage-based adaptive filters

Diffusion mechanisms for fixed-point distributed Kalman smoothing

Recursive Least-Squares Adaptive Filters

Diffusion-Based Adaptive Distributed Detection: Steady-State Performance in the Slow Adaptation Regime