Preliminary lectures - 2/14(Fri.), Room 1424, KIAS

Youngho Woo

I will give two lectures on rank of tensors. The goal is to understand "rank" of tensors (ordinary, symmetric or multivectors) from a viewpoint of classical algebraic geometry. After a quick reviewing topics in lect.1, I will explain concepts in lect.2 by showing as many examples as possible, rather than proving theorems or introducing general definitions.

13:20-14:20 Lecture 1 (Reviews from linear algebra, Basics on affine and projective varieties)

14:20-14:30 Break

14:30-15:30 Lecture 2 (Secant varieties and rank of tensors)

15:30-16:00 Tea time (Discussion room)

16:00-17:00 Exercise

Contents :

-Tensors, wedges, and symmetric tensors

-Rank of matrices

-Basic concepts on algebraic varieties ; dimension, irreducibility, tangent space.

-Terracini lemma

-Rank and border rank.

-Defect of secant variety

-Polynomial Waring Problem

-Examples ; Segre, Veronese varieties, and Grassmannians

References :

[J. Harris] Algebraic geometry : A first course & [J.M. Landsberg] Tensors : Geometry and applications.

Abstracts of Main lecture series:

Giorgio Ottaviani

*Tensor rank and tensor decomposition.*

A tensor has rank one if it is decomposable. A tensor has rank r if it can be expressed as the sum of r decomposable summands, with minimal r. Such an expression is called a tensor decomposition. The closure of the set of tensors of rank r consists of tensors of border rank r and corresponds geometrically to the r-secant variety to the Segre variety.

We will study the main basic properties of tensor decomposition and tensor rank, and the basic properties

of k-secant varieties, starting from Terracini Lemma.

*Geometric interpretation of Singular Value Decomposition.*

Singular Value Decomposition occurs in a natural way in the problem of computing the closest matrix of rank r to a given matrix (best rank r approximation). It has applications in compressing big data.

We consider this problem from a geometric point of view, computing tangent and normal spaces, and showing the role of Terracini Lemma in this setting. We consider the best approximation problem in the larger setting of computing all the critical points for the euclidean distance.

*Best rank one approximation of tensors, link with euclidean distance degree.*

This lecture is a natural extension of the previous one, from matrices to general tensors. This motivates the general definition of euclidean distance degree. We study its behaviour under projective duality. The best rank r approximation of a tensor is a problem not yet well understood, and from a computational point of view it is ill-posed.

*Symmetric case and polynomials.*

The symmetric case has a long history starting from XIX century, when some beautiful examples were studied. Its understanding is important for the comprehension of the general case. The tensor decomposition specializes to the so called Waring decomposition, expressing a polynomial as a sum of powers. We discuss the Comon conjecture, that the symmetric rank and the rank of a symmetric tensor coincide.

Cristiano Bocci

*Tensors in Algebraic Statistics.*

The basic idea of Algebraic Statistics is that each statistical model has an associated algebraic variety. This permits to use tools from Algebraic Geometry to study statistical events. We introduce the basic facts in Algebraic Statistics focusing on the statistical models whose associated varieties are r-secant varieties of

Segre products, i.e. parameter spaces of tensors.

*Algebraic invariants in Phylogenetics.*

One of the most important field of Algebraic Statistics is the phylogenetic inference. The main statistical model, in this context, is the hidden Markov model on trees. Here the research of invariants for inference could be very hard also for very simple models. However, thank to some recent results on flattenings, this research can be translated in the analysis of specific space of tensors.

*Identifiability and Bernoulli models.*

A statistical model is identifiable if the inference gives a finite number of results. When the variety X associated to the model M is a secant variety of a Segre product, the identifiability of M is related to the

weakly-defectivity of X. We discuss the basic properties of weakly-defectivity and identifiability

and we analyze, in detail the case of a Segre product of projective lines, which correspond to the so-called Bernoulli model.

*Quasi-independence models on tensors.*

On of the most important class of statical models are the so called contingency tables. When the contingency tables is a tensor of rank 1, it represents an independence model. If we permits to have extra values on the main diagonal we obtain quasi-independence models. We discuss recent results for two different classes of quasi-independence models, both from an algebraic and geometry point of view.