Conference Agenda

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Session Overview
Session
MS124, part 2: The algebra and geometry of tensors 1: general tensors
Time:
Thursday, 11/Jul/2019:
3:00pm - 5:00pm

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

Presentations
3:00pm - 5:00pm

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)

 

On apolar subsets

Enrico Carlini
Politecnico di Torino

Minimal apolar subsets have a well understood meaning: their cardinality tells us the Waring rank. However, the geometry of apolar subsets (even not minimal) can give us relevant information. In this talk, after a general review, we focus on apolar star configurations and their connection with generic forms.

 

On the identifiability of ternary forms beyond the Kruskal's bound

Elena Angelini
Universita di Siena

I will describe a new method to determine the minimality and identifiability of a Waring decomposition A of a specific ternary form T, even beyond the range of applicability of Kruskal's criterion. This method is based on the study of the Hilbert function and Cayley-Bacharach of A. As an application, we will see the cases of ternary optics and nonics. (joint works with Luca Chiantini).

 

Variants of Comon's problem via simultaneous ranks

Alessandro Oneto
Universitat Politècnica de Catalunya

The rank of a tensor is the smallest length of an additive decomposition as sum of decomposable tensors. Whenever the tensor has symmetries, it can be useful to consider additive decompositions whose summands respect the same symmetries. A symmetric tensor can be regarded as an element of the space of partially symmetric tensors for different choices of partial symmetries and one can ask what are the relations among the different (partially symmetric) ranks which arise in this way. This was the object of a famous question raised by Comon, who asked whether the tensor rank of a symmetric tensor equals its symmetric rank. This problem received a great deal of attention in the last few years. Affirmative answers were derived under certain assumptions, but recently Shitov provided an example where Comon’s question has negative answer. In a joint work with Fulvio Gesmundo and Emanuele Ventura (arXiv:1810.07679), we approached a partially symmetricversion of Comon’s question investigating relations among the partially symmetric ranks of a symmetric tensor. In particular, by exploiting algebraic tools as apolarity theory, we show how the study of the simultaneous symmetric rank of partial derivatives of the homogeneous polynomial associated to the symmetric tensor can be used to prove equalities among different partially symmetric ranks. In this way, we try to understand to what extent the symmetries of a tensor affect its rank. In this communication, after a brief introduction of the topic, I will present the main tools and results of our work.

 

Complex best r-term approximations almost always exist in finite dimensions

Lek-Heng Lim
University of Chicago

We show that in finite-dimensional nonlinear approximations, the best r-term approximant of a function almost always exists over complex numbers but that the same is not true over the reals. Our result extends to functions that possess special properties like symmetry or skew-symmetry under permutations of arguments. For the case where we use separable functions for approximations, the problem becomes that of best rank-r tensor approximations. We show that over the complex numbers, any tensor almost always has a unique best rank-r approximation. This extends to other notions of tensor ranks such as symmetric rank and alternating rank, to best r-block-terms approximations, and to best approximations by tensor networks. When applied to sparse-plus-low-rank approximations, we obtain that for any given r and k, a general tensor has a unique best approximation by a sum of a rank-r tensor and a k-sparse tensor with a fixed sparsity pattern. The existential (but not the uniqueness) part of our result also applies to best approximations by a sum of a rank-r tensor and a k-sparse tensor with no fixed sparsity pattern, as well as to tensor completion problems.