Conference Agenda

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Session Overview
Session
MS127, part 1: The algebra and geometry of tensors 2: structured tensors
Time:
Friday, 12/Jul/2019:
3:00pm - 5:00pm

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

Presentations
3:00pm - 5:00pm

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 (Barcelona Graduate School of Mathematics)

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)

 

Projective geometry and tensor identifiability

Massimiliano Mella
Università di Ferrara

A tensor rank-1 decomposition of a tensor T, lying in a given tensor space is an additive decomposition with rank one tensors. In many instances, for both pure and applied mathematics, it is interesting to understand when such a decomposition is unique, in a suitable sense. This problem translates very efficiently into geometric statements and can be attached via old and new techniques in projective geometry. In the talk, as an application, I will present some results concerning identifiability of tensors and partially symmetric tensors obtained via birational geometry techniques.

 

A bound for the Waring rank of the determinant via syzygies

Zach Teitler
Boise State University

The Waring rank of the 3x3 generic determinant is known to be greater than or equal to 14, and less than or equal to 20. Proofs of the lower bound of 14 were given in terms of geometric singularities or the Hilbert function of the apolar ideal. We improve the lower bound to 15 by considering higher syzygies in the minimal graded free resolution of the apolar ideal of the determinant.

This is joint work with Mats Boij.

 

On the identifiability of ternary forms

Luca Chiantini
Università degli studi di Siena

I will discuss a method which in principle can determine the uniqueness (and the minimality) of any given Waring decomposition of a ternary form of any degree. The method is based on an algebraic and geometric study of the set of points representing the decomposition, and from this point of view it can be seen as an extension of the Kruskal criterion for the identifiability of tensors. In addition, the method is sensitive of the coefficients of the elementary tensors in the given decomposition. Thus, it can distinguish between identifiable and not identifiable forms in the span of a given set of powers.

 

Real Waring Rank Geometry of Quaternary Forms

Hyunsuk Moon
National Institute for Mathematical Sciences

We studied the real ternary forms whose real rank equals the generic complex rank, and we characterize the semialgebraic set of sums of powers representations with that rank in the previous paper [1]. The semialgebraic set is called the Space of Sums of Powers, which is naturally included in the Variety of Sums of Powers. In this talk, we will go further for quaternary forms.

For quadrics, we find a simple way to characterize the Space of Sums of Powers, and characterize the behaviors of representations. For cubics, we developed an algorithm to obtain the unique Waring rank decomposition and using this, we determined which quaternary cubics has a real rank decomposition. For other cases with degrees bigger than 4, we identify some of components of the real rank boundary. And also we will present some problems related to this topic.

[1] 1. Michalek, M., Moon, H., Sturmfels, B.,Ventura, E., “Real Rank Geometry of Ternary Forms”, Annali di Matematica Pura ed Applicata, June 2017, Volume 196, Issue 3, pp. 1025-1054