DECONSTRUCT · Decomposition of Structured Tensors, Algorithms and Characterization

Decomposition of Structured Tensors, Algorithms and Characterization.

My work in the project DECONSTRUCT at INRIA begun with a collaboration with the hosting group GALAAD. Thanks to the experience of the group and my knowleges in the algebraic geometry concerning tensor decomposition, we were almost immediately able to write the first algorithm for the computation of the partially symmetric rank of all partially symmetric tensors over the complex numbers. This solved completely the part of the main objective of my project regarding algorithms for the computation of the rank of partially symmetric tensors. I presented this result during the Spring Semester 2011 “Algebraic geometry with a view towards applications” at Mittag-Leffler Institut where I was invited as visitor by A. Dickenstein, S. Di Rocco, R. Piene, K. Ranestad and B. Sturmfels. I started a new collaboration with K. Ranestad on a new concept of rank, and a collaboration with J. Hauenstein to write the first effective numerical algorithm for computing the rank and the border rank of real and complex symmetric tensors (this is still a work in progress). The interest showed by the European and the International communities on the topics of ranks and border ranks of structured tensors, gave a serious motivation to pay more and more attention to the main topic of my project, namely classify secant varieties of varieties parameterizing partially symmetric tensors and skew-symmetric tensors in terms of dimension and of rank. Important results that I obtained on the main topic of this project are related to the computation of the dimensions of secant varieties of Segre-Veronese varieties: in collaboration with E. Ballico and M.V.Catalisano we wrote a complete classification of the dimensions of secant varieties of Segre-Veronese varieties of two factors Pn × P1 embedded in bi-degree (a, b). After the very famous work of Alexander and Hirschowitz on the dimensions of secant varieties of Veronese varieties, [?] is the first paper that can give a complete classification of the dimensions of secant varieties of a given class of varieties parameterizing tensors. For what concerns the objective of the project to takle the problems over the field of real numbers, I started a collaboration with Prof. G. Ottaviani to compute the typical ranks of real ternary and quaternary cubics. This is still a work in progress since we are planning to cover the case of ternary quartics in collaboration with G. Blekhermann. I also wrote a Preprint together with E. Ballico on the relation between the complex and the real ranks of a real symmetric tensor. Regarding my work in connections with the applications, beside the first quoted pqper on the decomposition of all partially symmetric tensors which is related with problems in Signal Processing, another work that it is worth mentioning is in collaboration with I. Carusotto. Here we use the algebraic geometry tools for the decomposition polynomials to applications to quantum and atomic physics.

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