Conference Agenda

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Session Overview
Session
MS124, part 3: The algebra and geometry of tensors 1: general tensors
Time:
Friday, 12/Jul/2019:
10:00am - 12:00pm

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

Presentations
10:00am - 12:00pm

The algebra and geometry of tensors 1: general tensors

Chair(s): Yang Qi (University of Chicago, United States of America), Nick Vannieuwenhoven (KU Leuven)

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. These topics raise challenging computational problems, but also the theory behind them is far from fully understood. Algebraic geometry has already played an important role in the study of tensors. It has shed light on: the ill-posedness of tensor approximation problems, the generic number of decompositions of a rank-r tensor, the number and structure of tensor eigen- and singular tuples, the number and structure of the critical points of tensor approximation problems, and on the sensitivity of tensor decompositions among many others. This minisymposium focuses on recent developments on the geometry of tensors and their decompositions, their applications, and mathematical tools for studying them, and is a sister minisymposium to "The algebra and geometry of tensors 2: structured tensors" organized by E. Angelini, E. Carlini, and A. Oneto.

 

(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)

 

Apolarity for border rank

Jarosław Buczyński
Polish Academy of Sciences

For a fixed polynomial or tensor F, the standard apolarity lemma provides a correspondence between the set of points realising the Waring rank or tensor rank of F and the apolar ideal of F. Thus it is very useful for calculating the rank of the polynomial or tensor. I will present a non-saturated version of apolarity, which instead is useful for calculating the border rank of T. The general theory is going to be illustrated by several classes of examples. Based on a joint work with Weronika Buczyńska.

 

Symmetric tensor decompositions on varieties

Ke Ye
Chinese Academy of Sciences

Interesting tensors always have rich geometric structure, which can be helpful when we decompose these tensors. For instance, the signal separation problem in signal process corresponds to the so-called Vandermonde decomposition of a tensor. In this talk, we will introduce the problem of decomposing a tensor on a given algebraic variety. We will first discuss some basic properties and then we will present an algorithm to decompose a symmetric tensor such that each summand lies on a given algebraic variety. In particular, we will see how our proposed algorithm can be applied to study the Vandermonde decomposition of a tensor. If time permits, we will exhibit some numerical examples. This talk is based on a joint work with Jiawang Nie and Lihong Zhi.

 

Tensors under the congruence action

Anna Seigal
UC Berkeley

Matrix congruence extends naturally to the setting of tensors. We apply methods from tensor decomposition, algebraic geometry and numerical optimization to this group action. Given a tensor in the orbit of another tensor, we compute a matrix which transforms one to the other. Our primary application is an inverse problem from stochastic analysis: the recovery of paths from their third order signature tensors. This talk is based on joint work with Max Pfeffer and Bernd Sturmfels.

 

Rank additivity for small three-way tensors

Filip Rupniewski
Polish Academy of Sciences

I will present some result from the joint work with J. Buczyński and E. Postinghel, where we investigate the problem of the additivity of the tensor rank. That is for two independent tensors we study if the rank of their direct sum is equal to the sum of their individual ranks. A positive answer to this problem was previously known as Strassen's conjecture until recent counterexamples were proposed by Shitov (2017). The latter are not very explicit, and they are only known to exist asymptotically for very large tensor spaces. We proved that for some small three-way tensors the additivity holds. For instance, if the rank of one of the tensors is at most 6. In addition we also treat some cases of the additivity of border rank of such tensors.