Algebraic statistics: Problems in phylogenetics and wasserstein distance optimization

Abstract

Algebraic statistics is an emerging field of applied mathematics at the intersectionof algebraic geometry, commutative algebra, and statistics, driven by the observation that polynomial equations govern many statistical models and inference problems. This dissertation contributes to the theoretical foundations of algebraic statistics by addressing three problems of contemporary interest: the structure of phylogenetic invariants for network-based models, the algebraic complexity of maximum likelihood estimation in Brownian motion tree models, and the computation of Wasserstein distance to statistical models. The first contribution characterizes the phylogenetic invariants of group-based Cavender-Farris-Neyman (CFN) models on level-1 phylogenetic networks. Building on recent work that reduces the study of invariants of arbitrary level-1 networks to that of sunlet networks, we show that the CFN model on a sunlet network admits a parameterization that factors through the space of skew-symmetric matrices. In this formulation, the associated phylogenetic variety is cut out by specific sub-Pfaffians, leading to an explicit description of the generators of their defining ideal. Second, we focus on the maximum likelihood degree (ML-degree) of Brownian motion tree models, which are Gaussian graphical models encoding trait evolution. We show that the ML-degree is invariant under root choice and derive a closed-form expression for the ML-degree in the case of star trees. Finally, we examine the algebraic complexity of computing the Wasserstein distance to statistical models, utilizing recent geometric formulations that lead to the computation of polar degrees of projective varieties. Restricting to small toric models, we compute the polar degrees of rational normal scrolls and some graphical models. Together, these results deepen our understanding of the polynomial structure and algebraic complexity of statistical models central to phylogenetics and optimization, offering new tools and directions for research in algebraic statistics.