Applications of Algebraic Geometry to Multivariate Gaussian Models

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

The present project aims to do an in depth analysis on the algebraic and geometric structure of two main types of Gaussian models that are commonly chosen in applications: colored Gaussian graphical (CGG) models and Brownian motion tree (BMT) models. CGG models are for modeling interactions among random variables, taking in consideration possible similar traits, and BMT models are Gaussian models for the evolution of continuous traits in mathematical phylogenetics. The investigator will then use this information to compute the complexity of the maximum likelihood estimate problem for each model, a key issue when analyzing data. Some elements of the project will involve undergraduate students majoring in STEM disciplines, especially those from underrepresented groups with limited educational resources. The investigator will use algebra, geometry, combinatorics, and symbolic computations to better understand statistical models and make advancements on their maximum likelihood estimate (MLE) problem. For Gaussian models this starts by identifying Gaussian distributions with their covariance or concentration matrices and analyzing the polynomials vanishing on these matrices. The maximum likelihood degree (MLD) of a statistical model, which computes the complexity of finding the maximum likelihood estimate (MLE) of a statistical model for given data, relies on tools from algebra and geometry such as optimizing over an algebraic variety, intersection theory and polyhedral geometry. Specific questions that this project aims to answer are: (1) determine features in a phylogenetic tree that affect the maximum likelihood degree of its BMT model and connections to the algebraic degree of the vanishing ideal for the BMT model, (2) classify CGG models with toric structure; that is, with toric vanishing ideal or with vanishing ideal that turns toric after an appropriate linear change of variables, (3) find formulas for the maximum likelihood degree of CGG models and for the maximum likelihood estimate function of CGG models with MLD one. The investigator emphasizes CGG models with the algebraic structure of a toric variety because the bimonial equations of a toric statistical model can be used to fasten computations on the MLD of the model, produce Markov bases, contribute in hypothesis testing algorithms, and the polytope associated to the toric model is useful for studying the existence of MLE. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.