Matrix Schubert varieties and Gaussian conditional independence models

TL;DR

This paper explores matrix Schubert varieties and their applications to algebraic statistics, specifically in modeling Gaussian conditional independence, providing combinatorial algorithms and stratification characterizations.

Contribution

It establishes connections between matrix Schubert varieties and Gaussian conditional independence models, offering a primary decomposition algorithm and stratification descriptions.

Findings

Conditional independence models are intersections of matrix Schubert varieties.

Provided a combinatorial primary decomposition algorithm.

Characterized vanishing ideals of Gaussian graphical models.

Abstract

Matrix Schubert varieties are certain varieties in the affine space of square matrices which are determined by specifying rank conditions on submatrices. We study these varieties for generic matrices, symmetric matrices, and upper triangular matrices in view of two applications to algebraic statistics: we observe that special conditional independence models for Gaussian random variables are intersections of matrix Schubert varieties in the symmetric case. Consequently, we obtain a combinatorial primary decomposition algorithm for some conditional independence ideals. We also characterize the vanishing ideals of Gaussian graphical models for generalized Markov chains. In the course of this investigation, we are led to consider three related stratifications, which come from the Schubert stratification of a flag variety. We provide some combinatorial results, including describing the stratifications using the language of rank arrays and enumerating the strata in each case.