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

Graph signal processing allows the generalization of DSP concepts to the graph domain. However, most works assume graph signals that are static with respect to time, which is a limitation even in comparison to classical DSP formulations where signals are generally sequences that evolve over time. Several earlier works on adaptive networks have addressed problems involving streaming data over graphs by developing effective learning strategies that are well-suited to dynamic data scenarios, in a manner that generalizes adaptive signal processing concepts to the graph domain. The objective of this paper is to blend concepts from adaptive networks and graph signal processing to propose new useful tools for adaptive graph signal processing.

A graph diffusion LMS strategy for adaptive graph signal processing

Graph filters, defined as polynomial functions of a graph-shift operator (GSO), play a key role in signal processing over graphs. In this work, we are interested in the adaptive and distributed estimation of graph filter coefficients from streaming graph signals. To this end, diffusion LMS strategies can be employed. However, most popular GSOs such as those based on the graph Laplacian matrix or the adjacency matrix are not energy preserving. This may result in a large eigenvalue spread and a slow convergence of the graph diffusion LMS. To address this issue and improve the transient performance, we introduce a graph diffusion LMS-Newton algorithm. We also propose a computationally efficient preconditioned diffusion strategy and we study its performance.

A Preconditioned Graph Diffusion LMS for Adaptive Graph Signal Processing

The emergence of wireless sensor networks has imposed many challenges on network design such as severe energy constraints, limited bandwidth and computing capabilities. This kind of networks necessitates network protocol architectures that are robust, energy-efficient, scalable, and easy for deployment. This paper proposes a robust energy-aware clustering architecture (REACA) for large-scale wireless sensor networks. We analyze the performance of the REACA network in terms of quality-of-service, asymptotic throughput capacity, and power consumption. In particular, we study how the throughput capacity scales with the number of nodes and the number of clusters. We show that by exploiting traffic locality, clustering can achieve performance improvement both in capacity and in power consumption over general-purpose ad hoc networks. We also explore the fundamental trade-off between throughput capacity and power consumption for single-hop and multi-hop routing schemes in cluster-based networks. The protocol architecture and performance analysis developed in this paper provide useful insights for practical design and deployment of large-scale wireless sensor network.

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REACA: An Efficient Protocol Architecture for Large Scale Sensor Networks (Corrected)*

In this paper, we consider learning dictionary models over a network of agents, where each agent is only in charge of a portion of the dictionary elements. This formulation is relevant in big data scenarios where multiple large dictionary models may be spread over different spatial locations and it is not feasible to aggregate all dictionaries in one location due to communication and privacy considerations. We first show that the dual function of the inference problem is an aggregation of individual cost functions associated with different agents, which can then be minimized efficiently by means of diffusion strategies. The collaborative inference step generates local error measures that are used by the agents to update their dictionaries without the need to share these dictionaries or even the coefficient models for the training data. This is a useful property that leads to an efficient distributed procedure for learning dictionaries over large networks.

Online dictionary learning over distributed models

We study the problem of distributed estimation, where a set of nodes are required to collectively estimate some parameter of interest from their measurements. Distributed implementations avoid the use of a fusion center and distribute the processing and communication across the entire network. Among distributed solutions, diffusion algorithms have been shown to achieve good performance, increased robustness and are amenable for ad-hoc implementation. In this work we focus on hierarchical diffusion algorithms, where we allow different nodes to have different responsibilities, as opposed to our previous work where every node performed exactly the same type of operations. Our results are general in the sense that they apply to any diffusion algorithm. We illustrate the concept using diffusion LMS, provide performance analysis for hierarchical collaboration and present simulation results showing improved performance over non-hierarchical methods.

Ali H. Sayed

Papers

A graph diffusion LMS strategy for adaptive graph signal processing

A Preconditioned Graph Diffusion LMS for Adaptive Graph Signal Processing

REACA: An Efficient Protocol Architecture for Large Scale Sensor Networks (Corrected)*

Online dictionary learning over distributed models

Hierarchical diffusion algorithms for distributed estimation