A Framework for Directed Acyclic Hypergraph Learning

This work addresses the problem of learning directed acyclic graphs (DAGs) from nodal observations generated by a linear structural equation model. DAG learning is a central task in signal processing, machine learning, and causal inference, but it remains challenging because acyclicity is a global combinatorial property. Continuous acyclicity constraints have led to important algorithmic advances by replacing the discrete DAG constraint with smooth equality constraints. However, existing formulations still involve difficult non-convex optimization landscapes and may suffer from degenerate first-order optimality conditions. Here, we restrict attention to DAGs with non-negative edge weights and exploit this additional structure to obtain a simpler characterization of acyclicity. Building on this characterization, we formulate a regularized non-negative DAG learning problem and develop an algorithm based on the method of multipliers. We further analyze the benign optimization landscape induced by non-negativity. In the population regime, we show that the true DAG is the unique global minimizer of the proposed augmented-Lagrangian formulation; moreover, the landscape contains no spurious interior stationary points, and the true DAG is the only acyclic KKT point. Numerical experiments on synthetic and real-world data show that the proposed method improves over state-of-the-art continuous DAG-learning alternatives.

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.

Many real-world processes can be represented as compositions of functions along a directed acyclic graph (DAG). In causal modelling, these correspond to the underlying mechanisms; in engineering, to multiple fidelity levels; and in gene-regulatory networks, to transcription factors. These functions are partially observed across the DAG, with noisy and heterogeneously sampled measurements, posing significant challenges for reconstruction, uncertainty propagation, and inference. To tackle these challenges, we place priors over functions and naturally arrive at Deep Gaussian Processes over DAGs. We theoretically study their prior-collapse behaviour, and the effect of graph topology and intermediate observations on the preservation of information. We obtain almost-sure lower bounds on the asymptotic frequency of depths at which the distinction between inputs is preserved, identify broad kernel classes for which these hold, and prove an observation by \cite{dunlop2018} on the role of input connections. We offer a structured variational approximation that retains graph dependencies, preserves compositional uncertainty, and captures the explaining-away behaviour of colliders. Finally, we empirically validate our theoretical results and our methodology, and model a latent-collider DAG, a protein signalling network, and a multi-fidelity heavy-ion collision emulation task, attaining state-of-the-art performance while recovering low-fidelity contributions and yielding interpretability of the simulator hierarchy.