Intro

The TensorDec laboratory wants to be a recipient for activities, both of research and formation. It aims to connect master and graduate students to the different aspects of the topic of tensor decompositions:

  • a great literature in pure algebra and geometry has been dedicated to problems inspired by questions on tensor decompositions;
  • the practical search for decompositions of given tensors presents very challenging computational problems which require tools from computational mathematics and complexity theory;
  • often data are stored in the form of tensors and, therefore, the study of tensor decompositions has a very wide spectrum of applications in real world problems.

Who we are

Current members



Former PhD students


Former Master students

  • Daniele Taufer
    currently post-doc at CISPA Helmholtz Center, Saarbrücken, Germany.

  • Martina Iannacito
    currently PhD student at INRIA (team HIEPACS), Bordeaux, France.

Recent Publications

(since 2020)


Title Authors Reference Year
High order singular value decomposition for plant diversity estimation A. Bernardi, M. Iannacito, D. Rocchini Bollettino dell'Unione Matematica Italiana 2021
Strict inclusions of high rank loci E. Ballico, A. Bernardi, E. Ventura J. of Symbolic Computation 2021
On the strength of general polynomials A. Bik, A. Oneto Linear and Multilinear Algebra 2021
Space curves, X-ranks and cuspidal projections E. Ballico Annali dell'Università di Ferrara 2021
Identifiability of rank-3 tensors E. Ballico, A. Bernardi, P. Santarsiero Mediterranean J. of Mathematics 2021
Entri loci and ranks E. Ballico, E. Ventura Manuscripta Mathematica 2021
Geometric conditions for strict submultiplicativity of rank and border rank E. Ballico, A. Bernardi, F. Gesmundo, A. Oneto, E. Ventura Annali di Matematica Pura ed Applicata 2021
Skew-symmetric tensor decomposition E. Arrondo, A. Bernardi, P. M. Marques, B. Mourrain Communications in Contemporary Mathematics 2021
Waring, tangential and cactus decompositions A. Bernardi, D. Taufer Journal des Mathematiques Pures et Appliquees 2021
On the Terracini Locus of Projective Varieties E. Ballico, L. Chiantini Milan Journal of Mathematics 2021
On Strassen's rank additivity for small three-way tensors J. Buczyński, E. Postinghel, F. Rupniewski SIAM Journal on Matrix Analysis and Applications 2021
Curves in low dimensional projective spaces with the lowest ranks E. Ballico Cubo 2020
On minimal decompositions of low rank symmetric tensors B. Mourrain, A. Oneto Linear Algebra and Its Applications 2020
On the numerical range of matrices defined over a finite field E. Ballico Finite Fields and their Applications 2020
Linearly dependent subsets of Segre varieties E. Ballico Journal of Geometry 2020
Tangential varieties of Segre–Veronese surfaces are never defective M. V. Catalisano, A. Oneto Revista Matematica Complutense 2020
Labels of real projective varieties E. Ballico, E. Ventura Bolletino dell'Unione Matematica Italiana 2020
Large families of homogeneous polynomials with non-unique additive decompositions E. Ballico Beitrage zur Algebra und Geometrie 2020
On the Hilbert function of general fat points in P1 × P1 E. Carlini, M. V. Catalisano, A. Oneto Michigan Mathematical Journal 2020
Dependent subsets of embedded projective varieties E. Ballico Bulletin of the Korean Mathematical Society 2020
The monic rank A. Bik, J. Draisma, A. Oneto, E. Ventura Mathematics of Computation 2020
Birational geometry of defective varieties, II E. Ballico, C. Fontanari Communications in Algebra 2020

Industrial AI Challenge

by HIT - Hub Innovazione Trentino Fondazione

HIT_logo


Industrial AI Challenge is an innovation contest that allows students from the University of Trento with different backgrounds to form teams and work in touch with manufacturing companies between september and december 2021 to find ways to exploit existing datasets about industrial production processes and machinery with artificial intelligence techniques, including machine learning. Goal of the contest will be to deliver advanced statistics, predictive models, and guidelines for companies to better collect and exploit data to support business decisions (e.g. predictive maintenance, optimization of logistics). Students will be supported by academic and business mentors (AI startups from Trentino).



Presentation of the challenge by Nicola Doppio (HIT).

Masterclass: Tensor decompositions and their applications

by Nick Vannieuwenhoven (Assistant Professor, KU Leuven, Belgium)

Trento, 8-17 November 2022.
Multidimensional datasets, in which data can vary in more than two directions, became popular over the past two decades as computational and storage resources increased along with algorithmic innovations for the processing of such data. Multidimensional data poses several challenges, ranging from their interpretation and the extraction of meaningful insights from them, their processing and visualisation, and their storage and archiving. In this Masterclass, we study tensor decompositions, which are algorithmic techniques designed to tackle the foregoing challenges. Tensor decompositions extend the idea of matrix decompositions (like singular value decomposition, principal component analysis, and nonnegative matrix factorization) as instruments for the analysis of data that varies in only two directions to more directions.

All of the main tensor decompositions will be covered, namely:
- tensor rank decomposition or canonical polyadic decomposition,
- Tucker decomposition,
- tensor train decomposition, and
- hierarchical Tucker decomposition.
For each of these decompositions, we will investigate their definition, their main theoretical properties, the main algorithms for their computation, and a worked-out example of how they can be employed to analyse or process multidimensional datasets. nick

Detailed contents.
1. Introduction
  - Recap of matrix decompositions
  - Basic tensor notation
  - The tensor product and rank-1 tensors
  - Tensor product basis
  - Tensor product subspace

2. Tucker decomposition
  - Definition
  - HOSVD algorithm
  - Truncation strategies: parallel, sequential
  - Approximation quality
  - Application: hyperspectral image compression

3. Tensor trains and hierarchical Tucker decomposition
  - Definition
  - Nested tensor subspaces
  - SVD-based leaves-to-root and root-to-leaves algorithms
  - Approximation quality
  - Application: accelerating scientific computations

4-5. Tensor rank decomposition
  - Definition
  - Theoretical properties: subspace constraint, ill-posedness, conditioning, identifiability
  - Jennrich's pencil-based algorithm including numerical properties
  - Optimization algorithms: steepest descent, Gauss-Newton, Riemannian Gauss-Newton
  - Extra constraints (smoothness, nonnegativity), regularization
  - Application: full data analysis example for solar power prediction


LECTURE 1: INTRODUCTION

LECTURE 2: TUCKER DECOMPOSITION

LECTURE 3: TENSOR TRAINS DECOMPOSITION

LECTURE 4: TENSOR RANK DECOMPOSITION

LECTURE 5: APPROXIMATION BY A TENSOR RANK DECOMPOSITION

Masterclass: Algebraic statistics and related topics

by Kaie Kubjas (Assistant Professor, AAlto University, Finland)

Trento, 17-21 October 2022.

In this course, we will cover selected topics from algebraic statistics including:
  - conditional independence,
  - likelihood inference,
  - graphical models,
  - nonnegative matrix factorizations.
Time permitting further topics will be covered.

At the end of this course, the student can:
  - list topics in algebraic statistics;
  - recognize problems in statistics that are answerable by algebraic methods;
  - assess which algebraic methods are suitable for solving a problem;
  - apply basic algebraic tools to solve a problem. kaie

EDUCATION

Courses

  • Tensor Decomposition for Big Data Analysis

    by A. Bernardi. Master level course in Data Science, Mathematics and Statistics for Life and Social Sciences, Mathematics for Life and Data Sciences.
    An itroduction to big data science from tensor decompositons perspective.

  • Geometry and Topology for Data Analysis

    by A. Oneto. Master level course in Data Science, Mathematics and Statistics for Life and Social Sciences, Mathematics for Life and Data Sciences.
    A first course in algebraic topology, numerical algebraic geometry, with a view towards applications in data analysis.

Activities

TensorDec Lab Seminars

Within the activities of the laboratory, we organize seminars that are aimed not only to our colleagues researchers but also to mater and PhD students. Here the page dedicated to past and future seminars.

Applied Algebraic Geometry Seminar

Joint cycle of Seminars between Bologna, Ferrara, Firenze, Siena, Trento, Torino on Applied Algebraic Geometry in particular on arguments related to tensor decompositions. The seminars are held between Bologna e Firenze, every three weeks.
Organisers: E. Angelini, A. Bernardi, L. Chiantini, M. Mella, G. Ottaviani, E. Turatti.

Information, Algebra and Geometry Workgroup

Q@TN_logo
Since 2016, we co-organize a cycle of interdisciplinary meetings at the University of Trento in which we try to lay the foundations for a common language between Geometry, Algebra and Physics starting from Quantum Information. Form June 2018 we are part of the Q@TN initiative.
Organizers: A. Bernardi (DM, UniTn), I. Carusotto (CNR), F. Pederiva (DF, UniTn), F. Hauke (DF, UniTn), A. Oneto (DM, UniTn)

Online PhD seminar (2020-2021)

Our graduate students collected all Italian graduate students on topics related to tensor decompositions in a weekly cycle of seminars where graduate students have the possibility to share their latest discoveries or the problems they are currently studying in front of the senior members of the Italian community.
Organizers: C. Delazzari, V. Galgano, P. Santarsiero, R. Staffolani

Industrial AI Challenge (2021), for students

by HIT - Hub Innovazione Trentino Fondazione

HIT_logo

Industrial AI Challenge is an innovation contest that allows students from the University of Trento with different backgrounds to form teams and work in touch with manufacturing companies between September and December 2021 to find ways to exploit existing datasets about industrial production processes and machinery with artificial intelligence techniques, including machine learning. Goal of the contest will be to deliver advanced statistics, predictive models, and guidelines for companies to better collect and exploit data to support business decisions (e.g. predictive maintenance, optimization of logistics). Students will be supported by academic and business mentors (AI startups from Trentino).


Presentation of the challenge by Nicola Doppio (HIT):

TensorDec Lab Seminars


Future seminars


Thursday 19 May 2022, h. 08.30 - 11.30, room A213 (Povo1)
Daniele Taufer, CISPA Helmotz Center, Saarbrücken (DE)
An invitation to computational and symbolic algebra

Computational algebra constitutes one of the most fruitful tools that mathematicians have been developing during the last century. The advances in this field have been leading to astonishing results, playing a crucial role in modern breakthroughs and proofs (and disproofs) of intriguing conjectures. However, their wide adoption and daily usage by researchers all over the world is arguably their most relevant aspect, which turned them into an indispensable instrument for mathematicians of any age. In this lecture, a gentle introduction to this topic is given, and concrete examples and use-cases are presented. Gröbner bases and resultant theory are recalled and employed for addressing concrete instances of problems arising from elliptic curves, which constitute an inexhaustible supply of challenges in algebra, geometry and number theory. This lecture targets bachelor's and master's students that are familiar with the basic algebraic structures (rings, polynomials, ideals). Far from being a comprehensive treatment of this topic, this talk aims at providing concrete reasons for investigating this fruitful subject. The practical examples are addressed with the Magma Computational Algebra System, which may be accessed online (http://magma.maths.usyd.edu.au/calc/). No preliminary knowledge of this software is required, all the notions and material needed will be provided during the lecture. A computer with an internet connection is advisable.



GEOMETRY AND TOPOLOGY FOR DATA ANALYSIS
This series of seminars is part of the homonym master level course. Each seminar (2 hours) will be also available in streaming on ZOOM at: https://unitn.zoom.us/j/88115614484
Contact person: Alessandro Oneto - alessandro.oneto@unitn.it (Write me to get the password of the zoom call)


Wednesday 11 May 2022, h. 17.30 - 19.30, room A213 (Povo1)
Algebraic degrees of phylogenetic varieties
Marina Garrote-López (U. Alaska Fairbanks)

Thursday 12 May 2022, h. 14.30 - 16.30, room A215 (Povo1)
Persistent homology for shape comparison
Ulderico Fugacci (CNR-IMATI Genova)

Thursday 19 May 2022, h. 14.30 - 16.30, room 215 (Povo1)
Topological applications for pattern discovery in precision medicine
Nicole Bussola (OROBIX Life, Bergamo)

Wednesday 25 May 2022, h. 14.30 - 16.30, room 215 (Povo1)
Numerical Algebraic Geometry
Paul Braiding (U. Osnabrück))

Thursday 26 May 2022, h. 15.30 - 17.30, room A205 (Povo1)
Topological data analysis methods for neuroscience
Martina Scolamiero (KTH Stockholm)


Past seminars


Wednesday 20 April 2022
Daniele Castellana, U. Pisa (IT)
A Tensor Framework for Learning in Structured Domains
video


Events

Masterclass: Algebraic Statistics and related topics

Trento, 17-21 October 2022.
Held by K. Kubjas (Aalto U., Finland).



In this course, we will cover selected topics from algebraic statistics including conditional independence likelihood inference, graphical models, nonnegative matrix factorizations. Time permitting further topics will be covered.
At the end of this course, the student can:
  • list topics in algebraic statistics
  • recognize problems in statistics that are answerable by algebraic methods
  • assess which algebraic methods are suitable for solving a problem
  • apply basic algebraic tools to solve a problem.

Masterclass: Tensor decompositions and their applications

Trento, 8-17 November 2021.
Held by N. Vannieuwenhoven (KU Leuven, Belgium).



Multidimensional datasets, in which data can vary in more than two directions, became popular over the past two decades as computational and storage resources increased along with algorithmic innovations for the processing of such data. Multidimensional data poses several challenges, ranging from their interpretation and the extraction of meaningful insights from them, their processing and visualisation, and their storage and archiving. In this Masterclass, we will study tensor decompositions, which are algorithmic techniques designed to tackle the foregoing challenges. Tensor decompositions extend the idea of matrix decompositions (like singular value decomposition, principal component analysis, and nonnegative matrix factorization) as instruments for the analysis of data that varies in only two directions to more directions.

The slides of the course can be found here.

Tensor Networks: Quantum Physics, Geometry and Applications

Levico Terme, 26-28 July 2021

The goal of the workshop is to develop the connections between the mathematics and physics communities working on tensor network. It is aimed at researchers from both areas that are active on the topic, from experienced PhD students over postdocs to permanent faculty. As a key novelty, it will be a hands-on event with a limited number of talks and several sessions of active work. During these sessions, participants will distribute themselves in small groups and will actively work on a specific cutting-edge research problem. Each problem will be proposed by a world expert who will be in charge of directing the activities of the group. Thus, we hope to foster strong scientific collaborations that last well beyond the workshop.