A Unified Framework for Structure-Aware Clustering and Heterogeneous Causal Graph Learning

TL;DR

This paper introduces a novel framework that jointly learns subject clusters and their specific causal dependency structures using DAGs, addressing heterogeneity in multivariate systems.

Contribution

It proposes DAG-DC-ADMM, a unified method combining SEM, acyclicity constraints, and clustering via gTLP, with convergence guarantees for certain graph structures.

Findings

High true positive rate in recovering cluster-specific causal structures

Low false discovery rate in dependency structure estimation

Effective clustering of subjects based on structural similarity

Abstract

In complex multivariate systems, interactions among variables are defined by dependency structures, often encoded as directed acyclic graphs ($\text{DAGs}$). However, dependency structures can vary across subjects, and ignoring this structural heterogeneity introduces bias and obscures subpopulation-specific dependencies. To address this, we propose Directed Acyclic Graph-based Dependency Clustering via Alternating Direction Method of Multipliers (DAG-DC-ADMM), a unified framework built upon Structural Equation Modeling (SEM) that jointly learns cluster assignments and cluster-specific dependency structures. We encode acyclicity via a smooth constraint and integrate a groupwise truncated Lasso fusion penalty (gTLP) to cluster subjects based on their structural similarity. This yields a nonconvex optimization problem that incorporates sparsity, acyclicity, and structural consensus constraints. We address the nonconvexity by using the augmented Lagrangian method and solve it with an adapted version of the Alternating Direction Method of Multipliers (ADMM) for difference-of-convex programs. For certain graph structures, such as upper triangular adjacency matrices, our algorithm is guaranteed to converge to a Karush-Kuhn-Tucker (KKT) point. Experiments demonstrate that our method recovers cluster-specific causal dependency structures with a high true positive rate and a low false discovery rate. This capability enables the robust discovery of heterogeneous dependencies across subjects where the subpopulation label is unknown.