A Unified Framework for Structure-Aware Clustering and Heterogeneous Causal Graph Learning
Authors: Honglin Du, Muxuan Liang, Xiang Zhong
Organizations: Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL 32611, USA · Department of Biostatistics, MD Anderson Cancer Center, Houston, TX 77030, USA
Abstract
In complex multivariate systems, interactions among variables are defined by dependency structures, often encoded as directed acyclic graphs (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.
Recent algorithmic advances have made directed acyclic graph (DAG) structure learning scalable for causal discovery. Yet, the currently available techniques assume a completely homogeneous population, precluding their application to clustered data where cluster-specific variations (e.g., patient-specific effects) are common. We address this issue by introducing a new approach that estimates a global structure while accounting for local cluster-level effects. The key idea is to extend the fixed- and random-effects framework of classical mixed models to the structure learning setting. Towards this end, we present a differentiable graph coupling mechanism that guarantees the union of the fixed- and random-effects graphs remains acyclic. Computationally, we provide a provably convergent first-order method and leverage efficient batched updates across clusters. Statistically, we establish identifiability of the model and show that our approach recovers the true structure asymptotically. In experiments on real and synthetic data, our proposal detects dependencies missed by alternative estimators, underscoring its value for structure learning in clustered settings.
Ryan Thompson, Matt P. Wand, Veerabhadran Baladandayuthapani
Continuous optimization methods for learning Directed Acyclic Graphs (DAGs) operate on weighted adjacency matrices and are therefore limited to pairwise causal relationships. We propose a framework for learning Directed Acyclic Hypergraphs (DAHGs) from observational data, capturing joint parental influences that pairwise models cannot represent. Our approach rests on three components: (i) a generalized linear structural equation model (SEM) with multiplicative interaction terms whose non-zero weights correspond one-to-one with directed hyperedges; (ii) a weighted adjacency tensor representation whose acyclicity is characterized via nilpotency under the tensor t-product; and (iii) a differentiable acyclicity constraint derived through the Fourier decomposition of the t-product, which reduces tensor nilpotency to slice-wise matrix nilpotency and enables least-squares learning via the augmented Lagrangian method.
Diagnostic errors and mislabeling are common in biomedicine, which compromise the reliability of predictive models and data-driven outcomes. Stratifying unlabeled biomedical data based on complex relationships between features eliminates the need for data labels and overcomes the limitations of supervised learning. Traditional clustering methods assume restrictive data distributions, making them suboptimal for capturing complex dependencies in high-dimensional biomedical data. This paper introduces a novel cluster-friendly data presentation framework that integrates the non-Gaussian and non-linear feature dependence of copula models with an ensemble of causal structure discovery (CSD) methods based on Directed Acyclic Graphs (DAGs). While copulas model flexible multivariate distributions by relaxing assumptions related to multivariate normality, linear dependence, and symmetric relationships, an ensemble of DAG-based CSD methods identifies stable causal relationships between features. When clustered using K-means, the new data representation obtained by the proposed copula-adapted DAG (CopDAG) ranks first among the 12 methods in normalized clustering accuracy and adjusted Rand index across 16 biomedical datasets. Our CopDAG method predicts ground-truth class labels directly from feature relationships without data annotations and supervised learning, while also providing cluster visualizations and explainable causal structures of the biomedical data features.
Heranga K. Rathnasekara, Norou Diawara, Manar D. Samad