Conference Agenda

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Session Overview
Session
MS127, part 2: The algebra and geometry of tensors 2: structured tensors
Time:
Saturday, 13/Jul/2019:
10:00am - 12:00pm

Location: Unitobler, F023
104 seats, 126m^2

Presentations
10:00am - 12:00pm

The algebra and geometry of tensors 2: structured tensors

Chair(s): Elena Angelini (Università degli studi di Siena), Enrico Carlini (Politecnico di Torino), Alessandro Oneto (Humboldt Fundation, and Otto-von-Guericke-Universität Magdeburg)

Tensors are ubiquitous in mathematics and science. Tensor decompositions and approximations are an important tool in artificial intelligence, chemometrics, complexity theory, signal processing, statistics, and quantum information theory. Often, due to the nature of the problem under investigation, it might be natural to consider tensors equipped with additional structures or might be useful to consider tensor decompositions which respect particular structures. Among many interesting constructions, we might think of: symmetric, partially-symmetric and skew-symmetric tensors; tensor networks; Hadamard products of tensors or non-negative ranks. This minisymposium focuses on how exploiting these additional structures from algebraic and geometric perspectives recently gave new tools to study these special classes of tensors and decompositions. This is a sister minisymposium to "The algebra and geometry of tensors 1: general tensors" organized by Y. Qi and N. Vannieuwenhoven.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

The monic rank

Jan Draisma
Universität Bern

I will introduce the monic rank of a vector relative to an affine-hyperplane section of an irreducible Zariski-closed affine cone X; and describe an algorithmic technique based on classical invariant theory to determine, in certain concrete situations, the maximal monic rank. Using this, we prove that each univariate complex polynomial of degree 6,9,12 is the sum of 3 cubes of polynomials of degrees 2,3,4, respectively, and similarly that each univariate octic is a sum of 4 fourth powers of quadrics---special cases of a question by Boris Shapiro. I will also raise the question whether for cones X over equivariantly embedded projective homogeneous varieties, and the hyperplane corresponding to a highest weight vector, the maximal (ordinary) rank and maximal monic rank coincide. This is true in several concrete examples. If true in general, it would yield sharper lower bounds to the maximal (ordinary) rank.

Based on joint work with Arthur Bik, Alessandro Oneto, and Emanuele Ventura.

 

The average condition number of tensor rank decomposition is infinite

Nick Vannieuwenhoven
KU Leuven

Tensor rank decomposition is the problem of computing a set of rank-1 tensors whose sum is a given tensor. We are interested in quantifying the sensitivity of real rank-1 summands when moving the tensor infinitesimally on the semialgebraic set of tensors of bounded real rank. For this purpose, the standard approach in numerical analysis consists of computing the condition number of this problem. If the condition number is infinite, then the problem is said to be ill-posed. In this talk, we present the condition number of tensor rank decomposition. For most ranks, we compute its average value over the semialgebraic set of real tensors of bounded rank, relative to a natural choice of probability distribution. The results show that the condition number blows up too fast in a neighborhood of ill-posed problems to result in a finite average value.

This is joint work with Carlos Beltrán and Paul Breiding.

 

Symmetry groups of tensors

Emanuele Ventura
Texas A&M

To analyze the complexity of the matrix multiplication tensor, Strassen introduced a class of tensors that vastly generalize it, the tight tensors. Tight tensors are essentially tensors with a ”good” positive dimensional symmetry group. Besides the motivation from algebraic complexity, the study of symmetry groups of vectors in a representation of an algebraic group is a classical topic in algebraic geometry and invariant theory. It is then natural to investigate tensors with large symmetry groups, under a genericity assumption (1-generic).

In this talk, we discuss some combinatorial consequences of tightness, and sketch the geometry behind the classification of 1-generic tensors with maximal symmetry groups.

This is based on joint works with A. Conner, F. Gesmundo, JM Landsberg, and Y. Wang.

 

On the rank preserving property of linear sections and its applications in tensors

Yang Qi
University of Chicago

This talk is motivated by several questions on tensor ranks arising from signal processing and complexity theory. In the talk, we will first translate these conjectures into the geometric language, and reduce the problems to the study of a particular property of a linear section of an irreducible nondegenerate projective variety, namely the rank preserving property. Then we will introduce several useful tools and show some results obtained via these tools.

This talk is based on a joint work with Lek-Heng Lim.