# Summer School on Algebra, Statistics and Combinatorics

Summer School on Algebra, Statistics and Combinatorics was organised by Aalto Science Institute.

When: June 27-July 2, 2016

Where: Aalto University, Department of Mathematics and Systems Analysis, room M1 (on Wednesday room E), Otakaari 1, 02150 Espoo, Finland

Schedule

### Mini courses

• Kaie Kubjas (Aalto University): Geometry of the nonnegative and positive semidefinite rank
• Patrik Norén (North Carolina State University): Algebraic graph limits
• Louis Theran (St Andrews): Combinatorics and algebra of low-rank matrix completion
• Piotr Zwiernik (Universitat Pompeu Fabra): Semialgebraic statistics and latent tree models

## Abstracts

### The geometry of the nonnegative and positive semidefinite rank, by Kaie Kubjas (Aalto University)

Matrix rank is often defined as the minimal number of linearly independent rows or columns, but it could be also defined as the smallest natural number r such that a matrix M can be written as M=AB where the matrix A has r columns and the matrix B has r rows. Nonnegative and positive semidefinite rank are generalizations of this definition.

One of the motivations for nonnegative and positive semidefinite rank comes from optimization -- in the nonnegative rank case from linear programming and in the positive semidefinite case from semidefinite programming. Nonnegative rank is also closely related to mixture models in statistics.

In this mini course, we will define nonnegative and positive semidefinite rank, introduce some of their applications, study their geometric characterizations using polytopes and explore how real algebraic geometry is related to different notions of rank.

### Algebraic graph limits, by Patrik Norén (North Carolina State University)

The theory of algebraic graph limits combine graph theory, real algebraic geometry and statistics.

The theory of graph limits or graphons is a powerful tool in for example extremal graph theory. The graph limits serves two purposes in parallel, they both encode many standard and non-standard random graph models, and they are the natural objects to compactify the space of graphs.

We study a certain subclass of graph limits that in a natural way correspond to polynomials. These algebraic graph limits have many desirable properties and in particular they can be used to study very large networks.

### Combinatorics and algebra of low-rank matrix completion, by Louis Theran (Aalto University)

Low-rank matrix completion is the task to impute missing entries from a partial matrix, under the assumption that the true matrix has low rank.  This kind of model has applications in a number of learning problems, including recommendation systems, transductive learning, and collaborative filtering, to name a few.  In this course, we will study how the structure of low-rank matrix completion is related to that of determinantal varieties and ideals, and how much information about specific instances can be extracted from knowing only the rank and positions of the observed entries.

### Semialgebraic statistics and latent tree models, by Piotr Zwiernik (Pompeu Fabra University)

This short lecture course is divided into three parts. In part 1, I will present a basic geometric perspective on statistical models and show how algebraic geometry can be useful in statistics. In part 2, I will define graphical models and graphical models with hidden variables focusing on latent tree models.  Graphical models with hidden variables form a rich class of statistical models that leads to many interesting open problems in (real) algebraic geometry. In part 3, I am planning to give a brief introduction to cumulants and their combinatorics. The aim of this section is to show how cumulants can be used in statistics to study graphical models with hidden varables and in algebraic geometry to study secant varieties.

This course will be based on my book:

P. Zwiernik, “Semialgebraic statistics and latent tree models”, Chapman&Hall, 2015.

## Publications

Louis Theran, Anthony Nixon, Elissa Ross, Mahdi Sadjadi, Brigitte Servatius, and Michael Thorpe, Anchored boundary conditions for locally isostatic networks​, Physical Review E, to appear.

Karim A. Adiprasito, Arnau Padrol, and Louis Theran, Universality theorems for inscribed polytopes and Delaunay triangulations, Discrete & Computational Geometry, 2015.

Justin Malestein and Louis Theran, Frameworks with forced symmetry I: reflections and rotations, Discrete & Computational Geometry, 2015.

Franz J. Király, Louis Theran, and Ryota Tomioka, The algebraic combinatorial approach for Low-Rank Matrix Completion, Journal of Machine Learning Research, 2015.

Arnau Padrol and Louis Theran, Delaunay triangulations with disconnected realization spaces, in Proceedings of SoCG, 2014.

James Farre, Helena Kleinschmidt, Jessica Sidman, Audrey Lee-St. John, Stephanie Stark, and Louis Theran, Detecting dependencies in geometric constraint systems, in Proceedings of ADG, 2014.

Justin Malestein and Louis Theran, Generic rigidity with forced symmetry and sparse colored graphs, in Proceedings of Fields Workshop on Rigidity and Symmetry, 2014.

## Organisers

Ragnar Freij-Hollanti, Kaie Kubjas, Louis Theran and Emanuele Ventura
Supported by Aalto Science Institute

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