标题： 基于树结构中对称性的多元高斯模型的 toric 方法 显示英文标题

标题： Toric Multivariate Gaussian Models from Symmetries in a Tree

摘要： 给定一棵根树$T$，它有$n$个非根叶子节点并且节点带有颜色且被置零，我们构造了一个线性空间$L_T$，它由$n\times n$个满足由树的组合学决定的约束条件的对称矩阵组成。 当$L_T$表示高斯模型的协方差矩阵时，它提供了系统发育学中布朗运动树（BMT）模型的自然推广。 当$L_T$表示高斯模型的浓度矩阵空间时，它给出了某些带色的高斯图模型，我们将其称为派生自 BMT 的模型。 我们研究了在什么条件下互反簇$L_T^{-1}$是环簇。 利用逆矩阵映射的有理同构性，我们证明了如果$T$的 BMT 导出图是顶点正则的且为块图，则在导出的拉普拉斯变换下，$L_T^{-1}$是某个环面理想的消没点集。 该理想由块图上的高斯图模型的环面理想、原始 BMT 模型的环面理想以及来自顶点正则性的二项式线性条件的和组成。 为此，我们在$T$的叶间路径上为这些实现的环面模型提供单变元参数化。 显示英文摘要

摘要： 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 $T$.