the laboratory on algebra, geometry and applications of tensor
decompositions
of the University of Trento

News

Daniele Taufer (CISPA Helmholtz Saarbrücken, DE) will
give a lecture with title An invitation to computational and symbolic algebra Daniele graduated at U. of Trento in 2020 and his interests vary from cryptography to
tensor decompositions. He will visit us at the end of May 2022.

Luke Oeding (Auburn U., US) will give a PhD course of title "Algebra of tensors and applications" on 2-6 May 2022
at the University of Siena. The course can be attended both online or in presence.
For informations on how to apply follow this
link.

Kaie Kubjas (Aalto U., FI) will be visiting us in the Fall 2022. Kaie is an Assistant Professor at Aalto University (Helsinki, Finland).
She works in the field of applied nonlinear algebra, with particular interest in algebraic statistics and
applications in biology.
Kaie will be teaching a Masterclass from October 18th to October 21st, 2022.

via Sommarive, 14 - 38123 Povo (Trento), IT

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.

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

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.

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

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.

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.

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.

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

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

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

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

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 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.