Characteristic Imsets for Cyclic Linear Causal Models and the Chickering Ideal

TL;DR

This paper extends the understanding of covariance equivalence in linear causal models to cyclic graphs by analyzing characteristic imsets and their associated toric ideals, providing a new algebraic perspective.

Contribution

It introduces an algebraic framework for characterizing covariance equivalence in cyclic linear causal models using characteristic imsets and toric ideals.

Findings

Directed graphs with identical characteristic imsets are covariance equivalent.

The algebraic structure of the toric ideal helps identify covariance equivalence classes.

Imsets reduce the search space in causal discovery algorithms.

Abstract

Two directed graphs are called covariance equivalent if they induce the same set of covariance matrices, up to a Lebesgue measure zero set, on the random variables of their associated linear structural equation models. For acyclic graphs, covariance equivalence is characterized both structurally, via essential graphs and characteristic imsets, and transformationally, through sequences of covered edge flips. However, when cycles are allowed, only a transformational characterization of covariance equivalence has been discovered. We consider a linear map whose fibers correspond to the sets of graphs with identical characteristic imset vectors, and study the toric ideal associated to its integer matrix. Using properties of this ideal we show that directed graphs with the same characteristic imset vectors are covariance equivalent. In applications, imsets form a smaller search space for solving causal discovery via greedy search.