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Keynote Lecture

No Need to be Afraid of Multivariance in Multiway Arrays: Eigenvalue and Singular Value Related Issues

Professor Metin Demiralp
Istanbul Technical University
Informatics Institute
Istanbul, TURKEY
E-mail: metin.demiralp@gmail.com

Abstract: Especially in the last two decays there have been an increasing tendency to use the multiway arrays in the analyses of the systems given by big number of data. Multiway arrays are given by the elements depending on more than two indices. Hence they constitute somehow an extension to the usual matrices and vectors of ordinary linear algebra. This extension separates their analyses from matrices and vector and gathers under a different title “Multilinear Algebra”. Many scientists call them “Tensors” despite the fact that this term defines more restricted items. Hence, like the presenter of this talk, many other scientists intend not to use the term “Tensor”. Instead, the terms “Multiway Arrays” or “Multilinear Arrays” are widely used.
Each index of the general term of an array corresponds to a different direction or way. So, the vectors of the ordinary linear algebra are unidirectional or one way arrays while the matrices are bidirectional or two way arrays in this sense. Beyond these, the simplest multiway array whose elements have three independent indices is a tridirectional or three way array. Geometrically, vectors and matrices correspond to lines (very specific case is dot) and rectangles (or specifically squares) while the three way arrays can be described by the rectangular prisms (or in specific cases by cubes).
Like matrices, the multiway arrays can be considered as either just linear vector space elements or some operators mapping one space to other. In this sense, vectors correspond to just points in the relevant linear vector space whereas the matrices take a vector from a linear vector space and produces another vector in a different linear vector space. In the case of multilinear arrays each way can be considered either domain or range depending on the modelling needs. Each of the directions or ways corresponds to a different linear vector space even though certain matching structures appear. These linear vector spaces are joined together to get a single domain and a single range. Then the multiway array under consideration can be considered as an operator mapping from this domain. The number of the possible unions (set theoretical combinations) in these operations increases quite nonlinearly as the multivariance grows. For this reason, the author and his colleagues in his group defined specific entities from the multiway arrays and called them “Folvec (folded vector)”, and “Folmat (folded matrix)” by regrouping the indices of the multiway array elements. Folmats have two group indices, separated by semicolon, such that the indices at the right are considered as column indices while the left indices are for row. Hence, the row and column concepts in this case are not unidirectional but multidirectional entities.
The dubiosity or uncertainty in the domain definition of multiway array also makes it difficult to define eigenvalue problems since the eigenvectors are specific directions which remain unchanged under the action of the considered array while the scaling is charac- terized by the eigenvalue. The conserved direction definition needs many specifications in comparison with the eigenvalue problems of the matrices in ordinary linear algebra. Hence, some of the scientists focusing on this issue used the Rayleigh Quotient optimisa- tion and similar concepts to define the eigenvalue and eigenvectors. However this may not be the case when each way of the multiway array is emphasized on differently for different needs. Then explicit and more than one definitions to be used in different cases should be considered. The presentation focuses on these issues.

Brief Biography of the Speaker: Metin Demiralp was born in Türkiye (Turkey) on 4 May 1948. His education from elementary school to university was entirely in Turkey. He got his BS, MS degrees and PhD from the same institution, Ë™Istanbul Technical University. He was originally chemical engineer, however, through theoretical chemistry, applied mathematics, and computational science years he was mostly working on methodology for computational sciences and he is continuing to do so. He has a group (Group for Science and Methods of Computing) in Informatics Institute of Ë™Istanbul Technical University (he is the founder of this institute). He collaborated with the Prof. Herschel A. Rabitz’s group at Princeton University (NJ, USA) at summer and winter semester breaks during the period 1985-2003 after his 14 month long postdoctoral visit to the same group in 1979-1980. He was also (and still is) in collaboration with a neuroscience group at the Psychology Department in the University of Michigan at Ann Arbour in last three years (with certain publications in journals and proceedings).
Metin Demiralp has more than 100 papers in well known and prestigious scientific journals, and, more than 230 contributions together with various keynote, plenary, and, tutorial talks to the proceedings of various international conferences. He gave many invited talks in various prestigious scientific meetings and academic institutions. He has a good scientific reputation in his country and he was one of the principal members of Turkish Academy of Sciences since 1994. He has resigned on June 2012 because of the governmental decree changing the structure of the academy and putting politicial influence possibility by bringing a member assignation system. Metin Demiralp is also a member of European Mathematical Society. He has also two important awards of turkish scientific establishments.
The important recent foci in research areas of Metin Demiralp can be roughly listed as follows: Probabilistic Evolution Method in Explicit ODE Solutions and in Quantum and Liouville Mechanics, Fluctuation Expansions in Matrix Representations, High Dimensional Model Representations, Space Extension Methods, Data Processing via Multivariate Analytical Tools, Multivariate Numerical Integration via New Efficient Approaches, Matrix Decompositions, Multiway Array Decompositions, Enhanced Multivariate Product Representations, Quantum Optimal Control.

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