Decomposing Determinantal Varieties from Statistics via Matroid Theory

TL;DR

This paper explores the structure of determinantal varieties linked to statistical models with hidden variables, using matroid theory to analyze their decompositions, dimensions, and algebraic properties.

Contribution

It introduces a systematic, matroid-based framework for decomposing determinantal varieties and computing their degrees, advancing beyond existing computational methods.

Findings

Provides a combinatorial method for degree computation

Characterizes irreducible components via matroid flats

Extends analysis of determinantal varieties in statistics

Abstract

We study determinantal varieties from conditional independence models with hidden variables, focusing on their irreducible decompositions, dimensions, degrees, and Gr\"obner bases. Each variety encodes a collection of matroids, whose flats capture algebraic dependencies among variables. Using this approach, we provide a systematic description of the components, their dimensions, and defining equations, and introduce a combinatorial framework for computing the degree of the determinantal variety. Our approach highlights the central role of matroidal structures in the study of determinantal varieties and extends beyond the reach of current computational techniques.