Public defence in Mathematics and Statistics, M.Sc. (Tech) / M.Sc. (Econ) Olga Kuznetsova

Title of the doctoral thesis: Interactions of Algebra, Statistics and Optimization

Doctoral student: Olga Kuznetsova
Opponent: Professor Serkan Hoşten, San Francisco State University, USA
Custos: Assistant Professor Kaie Kubjas, Aalto University School of Science, Department of Mathematics and Systems Analysis

The structure and complexity of optimization problems in statistics and algebra 

Consider the question: “How can we predict the duration of hospital stays in different departments based on the observations from the past year?”. Similar problems arise in medical care scheduling, traffic routing, and airline pricing. Optimization theory provides the tools to answer such questions. Its main idea is to use a criterion to identify the best solution from a set of alternatives. 

We study the structure of optimization problems, the conditions for the existence of an optimal solution, and the complexity of computing an optimum. We focus on Gaussian and log-concave maximum likelihood estimation (MLE), which is an optimization method in statistics, and polynomially-constrained optimization. 

Gaussian random variables are popular in statistics and machine learning due to their theoretical properties and the ability to represent many processes in biology and economics. We use graphs to describe the relationships between the random variables, for example, the similarity of the different departments in a hospital. We develop a package GraphicalModelsMLE for the computer algebra system Macaulay2 to study Gaussian graphical models and suggest directions for further research. 

Random variables with log-concave densities are a generalization of Gaussian random variables and have attracted significant attention in recent years. We show that the optimal solution is often transcendental. As a result, modern computers can only solve such problems imprecisely. We also study the properties of log-concave MLE under the additional constraints imposed by undirected graphs. 

The goal of polynomially-constrained optimization is to find an optimal solution among the alternatives that satisfy a system of polynomial equations. Examples of such problems in statistics are the MLE of discrete random variables and the MLE of the means of Gaussian random variables. Estimating the average duration of a hospital stay is an example of the latter. We require that the critical points, i.e., the candidate solutions to the optimization problem, satisfy an explicit formula. The precise form of this formula depends on the data. We show that there is a finite number of critical points whenever the data is sufficiently general. This quantity is a valuable indicator of the complexity of an optimization problem because one often needs to compute all critical points to identify the optimal solution.

Thesis available for public display 10 days prior to the defence at: 
https://aaltodoc.aalto.fi/doc_public/eonly/riiputus/

Contact details:

Doctoral theses in the School of Science: https://aaltodoc.aalto.fi/handle/123456789/52