Work

Toric multivariate Gaussian models from symmetries in a tree

Year: 2026

Type: article

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 × 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 and a step toward a more accurate model for phylogenetic networks with symmetries for species hybridization. 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, which we introduce, 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 .

Source: Advances in Applied Mathematics

DOI: 10.1016/j.aam.2025.103024

Authors: Emma Cardwell, Aida Maraj, Álvaro Ribot

Institutions: Harvard University, Brown University, Center for Systems Biology Dresden, Max Planck Institute of Molecular Cell Biology and Genetics, Technische Universität Dresden

Language: English

Topic: Polynomial and algebraic computation

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,380