A Framework for Directed Acyclic Hypergraph Learning

2026-06-19

AI Summary

This work addresses the limitation of existing continuous optimization methods, which can only model pairwise causal relationships and struggle to capture the joint influence of multiple parent nodes. It proposes the first framework for learning directed acyclic hypergraphs (DAHGs) from observational data by extending DAG learning to hypergraph structures. The approach introduces a generalized linear structural equation model with multiplicative interaction terms, establishing a one-to-one correspondence between non-zero weights and directed hyperedges. Acyclicity in the hypergraph is characterized via nilpotency under the tensor t-product, which—combined with Fourier decomposition—is transformed into piecewise matrix nilpotency constraints. This yields a differentiable, efficient end-to-end optimization procedure that overcomes the representational constraints of traditional DAGs and effectively models high-order causal dependencies.

Abstract

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.

Problem

• Directed Acyclic Hypergraph
• causal relationships
• joint parental influences
• observational data
• hypergraph learning

Innovation

• Directed Acyclic Hypergraph
• Structural Equation Model
• Tensor t-product
• Differentiable Acyclicity Constraint
• Multiplicative Interaction