Algebraic Properties of Gaussian HTC-identifiable Graphs

TL;DR

This paper investigates algebraic properties of HTC-identifiable graphs in linear structural equation models, showing how regression coefficients can be recovered from covariance matrices and identifying minimal polynomial generators for certain graph classes.

Contribution

It establishes conditions for HTC-identifiability, links regression recovery to linear algebra, and characterizes the algebraic ideal with minimal generators for specific graph subsets.

Findings

All mixedgraphs are HTC-identifiable iff regression coefficients are recoverable from covariance matrices.

A set of polynomials generating the ideal of model constraints is identified.

Minimal generators of the ideal are characterized for a subset of HTC-identifiable graphs.

Abstract

In this paper, we explore some algebraic properties of linear structural equation modelsthat can be represented by an HTC-identifiable graph. In particular, we prove that all mixedgraphs are HTC-identifiable if and only if all the regression coefficients can be recovered fromthe covariance matrix using straightforward linear algebra operations. We also find a set ofpolynomials that generates the ideal that encompasses all the equality constraints of the modelon the cone of positive definite matrices. We further prove that this set of polynomials are theminimal generators of said ideal for a subset of HTC-identifiable graphs.