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
Location: Unitobler, F023
104 seats, 126m^2
Date: Tuesday, 09/Jul/2019
10:00am - 12:00pmMS142: Algebraic geometry of low-rank matrix completion
Unitobler, F023 
 
10:00am - 12:00pm

Algebraic geometry of low-rank matrix completion

Chair(s): Carlos Améndola (TU Munich), Daniel Irving Bernstein (MIT)

In a matrix completion problem, one is presented with a subset of entries of a matrix and wishes to find values for the remaining entries so that the completed matrix has a particular property. For example, one may want the completed matrix to have low rank or to be positive semidefinite. Such problems abound in application areas ranging from recommender systems (e.g. the "Netflix problem"), to rigidity theory, to compressed sensing, to maximum likelihood estimation for graphical models. Matrix completion problems also motivate many questions that can be considered fundamental within algebraic geometry. For example, studying low-rank matrix completion motivates the question: which coordinate projections of a given determinantal variety are dominant? What changes when one restricts to the real part of this determinantal variety? This minisymposium aims to bring together researchers who study algebraic aspects of matrix completion, both from theoretical and applied perspectives.

 

(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)

 

Real geometry of matrix completion

Rainer Sinn
FU Berlin

I will discuss matrix completion problems from the point of view of real algebraic geometry. This means that coordinate projections of rank varieties of matrices are the central object of study. Important invariants are called generic rank, typical rank, and maximum likelihood threshold.

 

Low algebraic dimension matrix completion

Greg Ongie
U Chicago

In the low rank matrix completion (LRMC) problem, the low rank assumption means that the columns (or rows) of the matrix to be completed are points on a low-dimensional linear algebraic variety. We extend this thinking to cases where the columns are points on a low-dimensional nonlinear algebraic variety, a problem we call Low Algebraic Dimension Matrix Completion (LADMC). Matrices whose columns belong to a union of subspaces are an important special case. We propose a LADMC algorithm that leverages existing LRMC methods on a tensorized representation of the data. This approach succeeds in many cases where traditional LRMC is guaranteed to fail because the data are low-rank in the tensorized representation but not in the original representation. We also provide a formal mathematical justification for the success of our method. In particular, we show bounds of the rank of these data in the tensorized representation, and prove sampling requirements on the number of observed entries per column necessary and sufficient to guarantee uniqueness of the completion. We also provide experimental results showing that the new approach significantly outperforms existing state-of-the-art methods for matrix completion in many situations.

 

The tropical Cayley-Menger variety

Daniel Irving Bernstein
MIT

Varieties that arise in the algebraic study of matrix completion come embedded in a vector space whose coordinates are indexed by some graph. A recurring problem is to find combinatorial descriptions, in terms of these graphs, of the algebraic matroids underlying these varieties. Tropical geometry can be used to solve such problems. In this talk, I will show that the tropicalization of the Cayley-Menger variety of points in the plane has a simplicial complex structure that can be described in terms of rooted trees. Then, I will show how one can use this to obtain a new proof of Laman's theorem, a celebrated theorem from rigidity theory giving a combinatorial description of the algebraic matroid underlying the Cayley-Menger variety of points in the plane.

 

Unlabelled global rigidity and low-rank matrix completion

Louis Theran
University of St. Andrews

A graph $G$ with $n$ is said to be generically globally rigid in dimension $d$ if we can reconstruct an unknown, but generic, set of $n$ points in $d$-dimensional Euclidean space from the pairwise distance measurements indexed by the edges of $G$. Perhaps surprisingly, even when $G$ is an unknown generically globally rigid graph, it is still possible to reconstruct the original points from the distance measurements. Despite the fact that global rigidity is a special case of low-rank matrix completion, the analogous unlabelled matrix completion problem is mostly open. I’ll talk about both problems and some key differences. This talk is based on joint work with Shlomo Gortler and Dylan Thurston.

 
3:00pm - 5:00pmMS187, part 1: Signature tensors of paths
Unitobler, F023 
 
3:00pm - 5:00pm

Signature tensors of paths

Chair(s): Carlos Améndola (TU Munich), Joscha Diehl (MPI Leipzig), Francesco Galuppi (MPI Leipzig), Anna Seigal (UC Berkeley)

Given a path X in R^n, it is possible to naturally associate an infinite list of tensors, called the iterated-integral signature of X. These tensors were introduced in the 1950s by Kuo-Tsai Chen, who proved that every (smooth enough) path is uniquely determined by its signature. Over the years this topic became central in control theory, stochastic analysis and, lately, in time series analysis.

In applications the following inverse problem appears: given a finite collection of tensors, can we find a path that yields them as its signature? One usually introduces additional requirements, like minimal length, or a parameterized class of functions (say, piecewise linear). It then becomes crucial to know when there are only finitely many paths having a given signature that satisfy the constraints. This problem, called identifiability, can be tackled with an algebraic-geometric approach.

On the other hand, by fixing a class of paths (polynomial, piecewise linear, lattice paths, ..), one can look at the variety carved out by the signatures of those paths inside the tensor algebra. Besides identifiability, the geometry of these signature varieties can give a lot of information on paths of that class. One important class is that of rough paths. Apart from applications to stochastic analysis, its signature variety has a strong geometric significance and it exhibits surprising similarities with the classical Veronese variety.

In time series analysis, it is often necessary to extract features that are invariant under some group action of the ambient space. The signature of iterated signals is a general way of feature extraction; one can think of it as a kind of nonlinear Fourier transform. Understanding its invariant elements relates to classical invariant theory but poses new algebraic questions owing to the particularities of iterated integrals.

Recent developments in these aspects will be explored in this minisymposium.

 

(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)

 

Varieties of signature tensors

Carlos Améndola
TU Munich

The signature of a parametric curve is a sequence of tensors whose entries are iterated integrals. This construction is central to the theory of rough paths in stochastic analysis. It is here examined through the lens of algebraic geometry. We introduce varieties of signature tensors for both deterministic paths and random paths. For the former, we focus on piecewise linear paths, on polynomial paths, and on varieties derived from free nilpotent Lie groups. For the latter, we focus on Brownian motion and its mixtures. Joint work with Peter Fritz and Bernd Sturmfels.

 

Learning paths from signature tensors

Max Pfeffer
MPI Leipzig

We aim to recover paths from their third order signature tensors. For this, we apply methods from tensor decomposition, algebraic geometry and numerical optimization to the group action of matrix congruence. Given a signature tensor in the orbit of another tensor, we compute a matrix which transforms one to the other. We establish identifiability results, both exact and numerical, for piecewise linear paths, polynomial paths, and generic dictionaries. Numerical optimization is applied for recovery from inexact data. We also compute the shortest path with a given signature tensor.

 

Signatures of paths: an algebraic perspective

Laura Colmenarejo
MPI Leipzig

Coming from stochastic analysis, the signature of a path is the collection of all the iterated integrals of the path. It can be seen in terms of tensors or as formal power series in words, which make them more relevant in other areas such as algebraic geometry or combinatorics. In this talk, I would like to look at the signatures of paths from an algebra perspective. For that, we will look at the work done by C. Améndola, P. Friz, and B. Sturmfels about the variety defined by the signature of piecewise linear paths, as well as the work done by F. Galuppi about the variety of rough paths. As a continuation, I would like to present our work on the signature varieties of two very different classes of paths: rough paths and axis-parallel paths. This is joint work with F. Galuppi and M. Michalek.

 

Signatures of paths transformed by polynomial maps

Rosa Preiss
TU Berlin

We characterize the signature of piecewise continuously differentiable paths transformed by a polynomial map in terms of the signature of the original path. For this aim, we define recursively an algebra homomorphism between two shuffle algebras on words. This homomorphism does not depend on the path and behaves well with respect to composition and homogeneous maps. Joint work with Laura Colmenarejo.

 

Date: Wednesday, 10/Jul/2019
10:00am - 12:00pmMS163: Theory and methods for tensor decomposition
Unitobler, F023 
 
10:00am - 12:00pm

Theory and methods for tensor decomposition

Chair(s): Tamara Kolda (Sandia National Laboratories), Elina Robeva (MIT)

Tensors are a ubiquitous data structure with applications in numerous fields, including machine learning and big data. Decomposing a tensor is important for understanding the structure of the data it represents. Furthermore, there are different ways to decompose tensors, each of which poses its own theoretical and computational challenges and has its own applications. In our minisymposium, we will bring together researchers from different communities to share their recent research discoveries in the theory, methods, and applications of tensor decomposition.

 

(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)

 

A nearly optimal algorithm to decompose binary forms

Elias Tsigaridas
Inria Paris

Symmetric tensor decomposition is equivalent to Waring’s problem for homogeneous polynomials; that is, to write a homogeneous polynomial in n variables of degree D as a sum of D-th powers of linear forms, using the minimal number of summands. We focus on decomposing binary forms, a problem that corresponds to the decomposition of symmetric tensors of dimension 2 and order D. We present the first quasi-linear algorithm to decompose binary forms. It computes a symbolic decomposition in O(M(D)log(D)) arithmetic operations, where M(D) is the complexity of multiplying two polynomials of degree D. We also bound the algebraic degree of the problem by min(rank, D − rank + 1) and show that this bound is tight.

 

On convergence of matrix and tensor approximate diagonalization algorithms by unitary transformations

Konstantin Usevich1, Jianze Li2, Pierre Comon3
1CNRS and University of Lorraine, 2No affiliation, 3CNRS, Université Grenoble Alpes

Jacobi-type methods are commonly used in signal processing for approximate diagonalization of complex matrices and tensors by unitary transformations. In this paper, we propose a gradient-based Jacobi algorithm and prove several convergence results for this algorithm. We establish global convergence rates for the norm of the gradient and prove local linear convergence under mild conditions.The convergence results also apply to the case of approximate orthogonal diagonalisation of real-valued tensors.

 

Non-linear singular value decomposition

Mariya Ishteva1, Philippe Dreesen2
1Free University Brussels, 2Vrije Universiteit Brussel (VUB)

In data mining, machine learning, and signal processing, among others, many tasks such as dimensionality reduction, feature extraction, and classification are often based on the singular value decomposition (SVD). As a result, the usage and computation of the SVD have been extensively studied and well understood. However, as current models take into account the non-linearity of the world around us, non-linear generalizations of the SVD are needed. We present our ideas on this topic. In particular, we aim at decomposing nonlinear multivariate vector functions with the following three goals in mind: 1. to provide an interpretation of the underlying processes or phenomena, 2. to simplify the model by reducing the number of parameters, 3. and to preserve its descriptive power. We use tensor techniques to achieve these goals and briefly discuss the potential of this approach for inverting nonlinear functions and curve fitting.

 

A symmetrization approach to hypermatrix {SVD}

Edinah Gnang
Johns Hopkins University

We describe how to derive the third order hypermatrix SVD from the spectral decomposition of third order hypermatrices resulting from the product of transposes of a given third order hypermatrix.

 
3:00pm - 5:00pmMS187, part 2: Signature tensors of paths
Unitobler, F023 
 
3:00pm - 5:00pm

Signature tensors of paths

Chair(s): Carlos Améndola (TU Munich), Joscha Diehl (MPI Leipzig), Francesco Galuppi (MPI Leipzig), Anna Seigal (UC Berkeley)

Given a path X in R^n, it is possible to naturally associate an infinite list of tensors, called the iterated-integral signature of X. These tensors were introduced in the 1950s by Kuo-Tsai Chen, who proved that every (smooth enough) path is uniquely determined by its signature. Over the years this topic became central in control theory, stochastic analysis and, lately, in time series analysis.

In applications the following inverse problem appears: given a finite collection of tensors, can we find a path that yields them as its signature? One usually introduces additional requirements, like minimal length, or a parameterized class of functions (say, piecewise linear). It then becomes crucial to know when there are only finitely many paths having a given signature that satisfy the constraints. This problem, called identifiability, can be tackled with an algebraic-geometric approach.

On the other hand, by fixing a class of paths (polynomial, piecewise linear, lattice paths, ..), one can look at the variety carved out by the signatures of those paths inside the tensor algebra. Besides identifiability, the geometry of these signature varieties can give a lot of information on paths of that class. One important class is that of rough paths. Apart from applications to stochastic analysis, its signature variety has a strong geometric significance and it exhibits surprising similarities with the classical Veronese variety.

In time series analysis, it is often necessary to extract features that are invariant under some group action of the ambient space. The signature of iterated signals is a general way of feature extraction; one can think of it as a kind of nonlinear Fourier transform. Understanding its invariant elements relates to classical invariant theory but poses new algebraic questions owing to the particularities of iterated integrals.

Recent developments in these aspects will be explored in this minisymposium.

 

(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)

 

Invariants of the iterated-integral signature

Joscha Diehl
MPI Leipzig

Recently the iterated-integral signature, known from stochastic analysis, has found applications in statistics and machine learning as a method for extracting features of time series. In many situations, there is a group acting on data that one wants to "mod out". One example is the accelerator data coming from a mobile phone. The orientation of the phone in a user's pocket is unknown. One then usually tries to calculate features that are invariant to the action of SO(3). I describe how such invariant features can be found in the signature. This is joint work with Jeremy Reizenstein (University of Warwick).

 

The areas of areas problem

Jeremy Reizenstein
University of Warwick

When introducing the iterated-integral signature of a path, we often give the following fact as an illuminating example: The information that level 2 adds beyond the total displacement given by level 1, is the "signed area" of each two-dimensional projection of the path. Given any two one-dimensional paths on the same interval, we can construct another as the cumulative signed area. It is natural to ask about all the paths we can get starting with all projections of a path and iteratively taking signed area. In particular the collection of their final values, which we call the areas-of-areas. What signature elements do they correspond to? Do they contain the same information as the signature (yes). How might we find a minimum subset of them which we can take which determine the signature?

 

Persistence paths and signature features in topological data analysis

Ilya Chevyrev
Oxford University

Persistent homology is a tool used to analyse topological features of data. In this talk, I will describe a new feature map for barcodes that arise in persistent homology computation. The main idea is to first realize each barcode as a path in a convenient vector space, and to then compute its path signature which takes values in the tensor algebra of that vector space. The composition of these two operations — barcode to path, path to tensor series — results in a feature map that has several desirable properties for statistical learning, such as universality and characteristicness, and achieves high performance on several classification benchmarks.

 

Character groups of Hopf algebras and their applications

Alexander Schmeding
TU Berlin

Character groups of Hopf algebras arise naturally in a variety of applications. For example, they appear in numerical analysis, control theory and the theory of rough paths and stochastic analysis. In the talk we will review the geometry and some main examples of these (infinite-dimensional) groups. Then we will report on some recent progress for character groups associated to so called combinatorial Hopf algebras. In the combinatorial setting, certain subgroups of the character group are closely connected to locally convergent Taylor series like expansions which are of interest in the applications mentioned above.

 

Date: Thursday, 11/Jul/2019
10:00am - 12:00pmMS124, part 1: The algebra and geometry of tensors 1: general tensors
Unitobler, F023 
 
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.

 
3:00pm - 5:00pmMS124, part 2: The algebra and geometry of tensors 1: general tensors
Unitobler, F023 
 
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)

 

Bounds on the rank general and special results

Enrico Carlini
Politecnico di Torino

We will see how algebra and geometry can lead to intetesting bounds on the rank with some special focus on the symmetric case.

 

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.

 

Date: Friday, 12/Jul/2019
10:00am - 12:00pmMS124, part 3: The algebra and geometry of tensors 1: general tensors
Unitobler, F023 
 
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.

 
3:00pm - 5:00pmMS127, part 1: The algebra and geometry of tensors 2: structured tensors
Unitobler, F023 
 
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

 

Date: Saturday, 13/Jul/2019
10:00am - 12:00pmMS127, part 2: The algebra and geometry of tensors 2: structured tensors
Unitobler, F023 
 
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.

 
3:00pm - 5:00pmMS127, part 3: The algebra and geometry of tensors 2: structured tensors
Unitobler, F023 
 
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)

 

Varieties of tensor decompositions and multi secants to curves and surfaces

Kristian Ranestad
University of Oslo

For a point p and a variety X in a projective space, the X -rank of p is the minimal n such that p lies on an n-secant (n-1)-space to X. I consider the variety V(p,X) of such n-secants, will recall the cases when X is a rational or elliptic curve and explain some surface cases. A very nice result of Iliev and Manivel for tensors C^3 x C^3 have applications on these issues.

 

Varieties of Hankel matrices and their secant varieties

Hirotachi Abo
University of Idaho

The maximal minors of the generic k x n Hankel matrix (also known as the catalacticant matrix) defines a rational map from the projective n-space to the Grassmann variety of (k-1)-planes in the projective n-space. The image of the projective n-space under the rational map is called the Hankel variety of (k-1)-planes, which is birationally equivalent to the projective n-space if n is sufficiently large compared with k. This talk concerns higher secant varieties of the Hankel variety of (k-1)-planes. The main focus of the talk is set on the defectivity of the Hankel variety of lines.

 

Tensor decomposition, sparse representation and moment varieties

Bernard Mourrain
INRIA

Tensor decomposition problems appear in many areas such as Signal Processing, Quantum Information Theory, Algebraic Statistics, Biology, Complexity Analysis, etc as a way to recover hidden structures from data. The decomposition is a representation of the tensor as a weighted sum of a minimal number of indecomposable terms. This problem can be seen as a sparse recovery problem from sequences of moments. We will develop this analogy and present an algebraic approach to address the decomposition problem, via duality and Hankel operators. We will analyze the varieties of moments associated to low rank decompositions, investigate their defining equations and some of their properties that can be exploited in the decomposition problem. Links with the Hilbert scheme of points will be presented. Examples exploiting these properties will illustrate the approach.

 

The Distance Function from the Variety of Rank One Partially-Symmetric Tensors

Luca Sodomaco
Università di Firenze

Let X be a Segre-Veronese product of projective spaces and denote with X* its dual variety. In this talk, we outline the main properties of the ``Euclidean Distance polynomial'' (ED polynomial) of X*, as a remarkable example of a more general theory on ED polynomials developed in a recent work with Ottaviani. Given a tensor T, the roots of the ED polynomial of X* at T correspond to the singular values of T. Moreover, we describe the variety of tensors that fail to have the expected number of singular vector tuples, counted with multiplicity. This variety is, in general, a non-reduced hypersurface and its equation is, up to scalars, the leading coefficient of the ED polynomial of X*.