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

2026-05-18

AI Summary

This work addresses the challenge of uncovering heterogeneous causal structures in complex multivariate systems, where ignoring inter-individual differences in causal relationships can introduce bias and obscure subgroup-specific dependencies. The authors propose a joint learning framework that simultaneously infers sample clustering and cluster-specific directed acyclic graphs (DAGs), unifying structure-aware clustering with heterogeneous causal discovery for the first time. Built upon structural equation models, the method employs a group truncated Lasso fusion penalty to drive clustering based on structural similarity, while integrating a smooth acyclicity constraint. A difference-of-convex programming approach combined with an ADMM algorithm is developed to solve the resulting non-convex optimization problem, ensuring convergence to a KKT point even without prior knowledge of subgroup labels. Experiments demonstrate that the method accurately recovers heterogeneous causal structures with high true positive rates and low false discovery rates.

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.

Problem

• structural heterogeneity
• causal graph learning
• clustering
• directed acyclic graphs
• subpopulation discovery

Innovation

• structure-aware clustering
• heterogeneous causal graph learning
• DAG-DC-ADMM
• groupwise truncated Lasso fusion
• nonconvex optimization