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.
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.
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.
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”,
Additional references will be given later.