Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

 
Session Overview
Session
MS124, part 1: The algebra and geometry of tensors 1: general tensors
Time:
Thursday, 11/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)

 

The distance function from a real algebraic variety

Giorgio Ottaviani
Università di Firenze

The distance function from a real algebraic variety X is a algebraic function, its degree is twice the Euclidean distance degree of X. Its constant term describes the points at zero Euclidean distance; while on real numbers these are just the points of X, there are additional points with complex entries. When X is projective, the constant term vanishes on the variety dual to X^vee cap Q, where Q is the isotropic quadric. The leading term of the distance function is a scalar when X is transversal to Q, according to a Whitney stratification of X. The important case when X is the variety of rank one tensors is exposed in another talk by Sodomaco, who is coauthor of the above results.

 

Algorithms for rank, tangential and cactus decompositions of polynomials

Alessandra Bernardi
University of Trento

I will present a review of the famous algorithm for symmetric tensor decomposition due to J. Brachat, P. Comon, B. Mourrain and E. Tsigaridas. I will also present a generalization to decompositions of polynomials involving points on the tangential variety of a Veronese variety. I will conclude by showing how the same technique allows to compute the cactus rank and decomposition of any polynomial. This is the outcome of a joint work together with D. Taufer.

 

Pencil-based algorithms for tensor rank decomposition are not stable

Paul Breiding
Max-Planck-Institute for Mathematics in the Sciences

I will discuss the existence of an open set of n1× n2× n3 tensors of rank r on which a popular and efficient class of algorithms for computing tensor rank decompositionsis numerically unstable. Algorithm of this class are based on a reduction to a linear matrix pencil, typically followed by a generalized eigendecomposition. The analysis shows that the instability is caused by the fact that the condition number of the tensor rank decomposition can be much larger for n1×n2×2 tensors than for the n1×n2×n3 input tensor. Joint work with Carlos Beltran and Nick Vannieuwenhoven.

 

Identifiability of a general polynomial

Francesco Galuppi
Max-Planck-Institute for Mathematics in the Sciences

The study of tensor decompositions is a wonderful topic with powerful applications and a lovely geometric interpretation. One of the most interesting scenarios is identifiability, that is the existence of a unique decomposition for the tensor. Identifiability was conjectured to be a very rare phenomenon. In this joint work with Massimiliano Mella we look at it from a geometric viewpoint and we use birational techniques to completely classify all pairs (n,d) such that the general degree d polynomial in n+1 variables admits a unique decomposition.