Papers › Toric Multivariate Gaussian Models from Symmetries in a Tree
Toric Multivariate Gaussian Models from Symmetries in a Tree
Emma Cardwell, Aida Maraj, Alvaro Ribot
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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. 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.
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