Zhiyuan Dong, Carlos Mundo-Levano, Wei Qian +2cs.LG
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.
Modern image-analysis pipelines often convert images into structured semantic variables, such as facial attributes, object concepts, and scene descriptors. Learning directed dependencies among these variables can produce interpretable visual semantic graphs, but continuous directed acyclic graph learning is limited by the cost of enforcing acyclicity. We present polyDAG, a polynomial acyclicity framework for efficient continuous causal discovery in visual semantic graphs. polyDAG replaces the matrix-exponential acyclicity constraint with a finite polynomial trace constraint and proves that the new constraint is zero exactly for acyclic graphs. We further derive a geometric-series implementation that avoids the explicit summation loop while preserving the same acyclicity condition. Experiments on synthetic Erdos-Renyi graphs and CelebA facial visual attributes show that polyDAG improves efficiency and structure recovery. Averaged over the revised synthetic protocol with d in {100, 200, 500}, polyDAG reduces mean structural Hamming distance from 318.4 to 285.4 and improves mean F1 score from 0.725 to 0.756. At 100 nodes, the geometric variant runs in 3.44 seconds compared with 5.16 seconds for the exponential baseline, corresponding to a 33.4 percent speedup. Code and data are publicly available at https://github.com/wenhaoz-fengcai/polyDAG.