**Keynote Lecture
Folding and Unfolding Related Issues, Especially Decompositions, in Data Processing**

**Professor Metin Demiralp**

Principal Member of Turkish Academy of Sciences

Istanbul Technical University

Informatics Institute

Turkey

E-mail: metin.demiralp@be.itu.edu.tr

**Abstract: **Many branches in science and engineering deal with data composed of huge number of elements. Neuroscience, signal processing and similar issues are amongst these types of applications where each data vector contains hundreds of thousands or millions of elements. These types of data vectors can be partitioned to sets having rather small number of elements at the expense of dimensionality increase. Thus certain arrays having more than two indices appear after this partitioning. Their processing is generally based on the expressing of those arrays in terms of rather simple arrays which can be processed more easily. This is somehow decomposition of the arrays to rather simple arrays. This issue is one of the core topics of multilinear algebra. There have been many attempts to get efficient decomposition methods by using folding and unfolding operations.

The general tendency is to use the tensor concept to consider the multilinear algebraic entities. However, a multiindex entity need not be adjacently considered to a tensor even though a tensor can be characterized by a multiindex array depending on the coordinate system in which the considered tensor is represented. We prefer to use the folded arrays to this end. Folded arrays (folarrs) and especially their specific forms, folded vectors (folvecs) and folded matrices (folmats) are very harmonious to the conceptual structure of the ordinary linear algebra. Thus the decomposition of folmats becomes the basic issue.

One way is the use of spectral representation for the decompositions of the folmats. To this end the eigenvalue problems of the folmats should be brought to the scene. On the other hand a complete analogy to the singular value decomposition of ordinary matrices is possible for the singular value decompositions of the folmats. What we need is to consider a folmat as a transforming agent from a specific type folmat to another type folmat even though type conservation is possible.

Spectral decomposition, singular value decomposition, reductive array decomposition, high dimensional model representation, enhanced multivariance product representation will be the main foci for folmat decompositions in the presentation.

**Brief Biography of the Speaker: **Metin Demiralp was born in Turkiye (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, ÿIstanbul 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 ÿIstanbul 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 90 papers in well known and prestigious scientific journals, and, more than 200 contributions 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 is one of the principal members of Turkish Academy of Sciences since 1994. He 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.