Identifiability of Homoscedastic Linear Structural Equation Models using Algebraic Matroids

TL;DR

This paper investigates the identifiability of homoscedastic linear structural equation models with cyclic graphs using algebraic matroid theory, expanding understanding beyond acyclic models.

Contribution

It introduces algebraic matroid-based conditions for generic identifiability of cyclic SEMs with homoscedastic errors, extending previous acyclic-focused methods.

Findings

Derived sufficient conditions for graph distinguishability.

Identified subclasses of cyclic graphs that are generically identifiable.

Conjectured a stronger graphical criterion for broader identifiability.

Abstract

We consider structural equation models (SEMs), in which every variable is a function of a subset of the other variables and a stochastic error. Each such SEM is naturally associated with a directed graph describing the relationships between variables. When the errors are homoscedastic, recent work has proposed methods for inferring the graph from observational data under the assumption that the graph is acyclic (i.e., the SEM is recursive). In this work, we study the setting of homoscedastic errors but allow the graph to be cyclic (i.e., the SEM to be non-recursive). Using an algebraic approach that compares matroids derived from the parameterizations of the models, we derive sufficient conditions for when two simple directed graphs generate different distributions generically. Based on these conditions, we exhibit subclasses of graphs that allow for directed cycles, yet are generically identifiable. We also conjecture a strengthening of our graphical criterion which can be used to distinguish many more non-complete graphs.