Toric Multivariate Gaussian Models from Symmetries in a Tree

TL;DR

This paper explores the algebraic structure of Gaussian models derived from tree symmetries, identifying conditions under which their inverse varieties are toric, and providing explicit parametrizations for these models.

Contribution

It characterizes when the inverse of certain Gaussian models based on trees are toric, linking combinatorial tree structures with algebraic properties of Gaussian models.

Findings

The inverse variety $L_T^{-1}$ is toric under specific graph conditions.

Explicit monomial parametrizations are provided for these toric models.

The paper connects tree symmetries with algebraic properties of Gaussian models.

Abstract

Given a rooted tree $T$ on $n$ non-root leaves with colored and zeroed nodes, we construct a linear space $L_T$ of $n\times n$ symmetric matrices with constraints determined by the combinatorics of the tree. When $L_T$ represents the covariance matrices of a Gaussian model, it provides natural generalizations of Brownian motion tree (BMT) models in phylogenetics. When $L_T$ represents a space of concentration matrices of a Gaussian model, it gives certain colored Gaussian graphical models, which we refer to as BMT derived models. We investigate conditions under which the reciprocal variety $L_T^{-1}$ is toric. Relying on the birational isomorphism of the inverse matrix map, we show that if the BMT derived graph of $T$ is vertex-regular and a block graph, under the derived Laplacian transformation, $L_T^{-1}$ is the vanishing locus of a toric ideal. This ideal is given by the sum of the toric ideal of the Gaussian graphical model on the block graph, the toric ideal of the original BMT model, and binomial linear conditions coming from vertex-regularity. To this end, we provide monomial parametrizations for these toric models realized through paths among leaves in $T$.