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

An efficient recursive solution is presented for the one-sided unconstrained tangential interpolation problem. The method relies on the triangular factorization of a certain structured matrix that is implicitly defined by the interpolation data. The recursive procedure admits a physical interpretation in terms of discretized transmission lines. In this framework the generating system is constructed as a cascade of first-order sections. Singular steps occur only when the input data is contradictory, i.e., only when the interpolation problem does not have a solution. Various pivoting schemes can be used to improve numerical accuracy or to impose additional constraints on the interpolants. The algorithm also provides coprime factorizations for all rational interpolants and can be used to solve polynomial interpolation problems such as the general Hermite matrix interpolation problem. A recursive method is proposed to compute a column-reduced generating system that can be used to solve the minimal tangential interpolation problem.

A recursive method for solving unconstrained tangential interpolation problems

System and methods that cancel artifacts of stimulation signals from neural signals are disclosed. In several embodiments, the systems and methods determine a threshold value for the neural signal in the absence of artifacts. The threshold value can then be used to detect an artifact in received neural signals. In a number of embodiments, a template can be used to cancel an artifact from a neural signal in response to the neural signal being greater than the threshold value.

Systems and methods for reducing noise caused by stimulation artifacts in neural signals received by neuro-modulation devices

This work develops diffusion algorithms for adaptation over networks that endow nodes with both cooperation abilities and temporal processing abilities. Each node is allowed to share information locally with its neighbors. At the same time, each node filters past data and uses them to enhance the collaborative process. In this manner, the resulting algorithms consist of three stages: adaptation, spatial processing, and temporal processing. The order of these operations can be inter-changed leading to a total of six variations. The results indicate that the version that performs adaptation prior to the steps of spatial cooperation and temporal processing leads to best performance.

Spatio-temporal diffusion mechanisms for adaptation over networks

In diffusion social learning over weakly-connected graphs, it has been shown recently that influential agents end up shaping the beliefs of non-influential agents. This paper analyzes this control mechanism more closely and addresses two main questions. First, the article examines how much freedom influential agents have in controlling the beliefs of the receiving agents. That is, the derivations clarify whether receiving agents can be driven to arbitrary beliefs and whether the network structure limits the scope of control by the influential agents. Second, even if there is a limit to what influential agents can accomplish, this article develops mechanisms by which these agents can lead receiving agents to adopt certain beliefs. These questions raise interesting possibilities about belief control over networked agents. Once addressed, one ends up with design procedures that allow influential agents to drive other agents to endorse particular beliefs regardless of their local observations or convictions. The theoretical findings are illustrated by means of several examples.

Belief Control Strategies for Interactions over Weak Graphs

A technique based on linear precoding is introduced for broadcasting on linear networks. The precoding allows the different message components of a broadcast message to be separated and decoded at the desired sink nodes, thus providing a systematic design methodology for broadcasting over a given network with a given linear network code. To achieve a good throughput, however, the network code itself must also be chosen judiciously. Motivated by several recent results on random network codes, we propose a combination of precoding and random linear network codes. This approach does not require a centralized coordination for network code design. One of the advantages of this approach is that by simply changing the precoding matrix (together with associated decoding strategies), different broadcast objectives can be achieved without tampering with the network code, therefore one can manage the network operation by controlling the origin and destination nodes of the network and without manipulating the network interior. Together, random network codes and linear precodings provide a simple yet powerful methodology for broadcast over linear networks.

Ali H. Sayed

Papers

A recursive method for solving unconstrained tangential interpolation problems

Systems and methods for reducing noise caused by stimulation artifacts in neural signals received by neuro-modulation devices

Spatio-temporal diffusion mechanisms for adaptation over networks

Belief Control Strategies for Interactions over Weak Graphs

Precoding for broadcasting with linear network codes