polyDAG: Polynomial Acyclicity Constraints for Efficient Continuous Causal Discovery in Visual Semantic Graphs

2026-06-05

AI Summary

This work addresses the computational inefficiency of existing continuous causal discovery methods, which rely on acyclicity constraints such as matrix exponentials and thus struggle to scale to large-scale visual semantic graph modeling. The authors propose a novel acyclicity constraint based on a finite polynomial trace, which is the first to be strictly equivalent to directed acyclic graphs (DAGs). By integrating geometric series optimization to avoid explicit cyclic summation, the method achieves substantial gains in computational efficiency. Empirical results demonstrate superior structural recovery performance on both synthetic and CelebA image data, reducing the average Structural Hamming Distance (SHD) from 318.4 to 285.4 and achieving an F1 score of 0.756. Furthermore, inference speed improves by 33.4% on 100-node graphs.

Abstract

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.

Problem

• causal discovery
• acyclicity constraint
• visual semantic graphs
• continuous optimization
• computational efficiency

Innovation

• polynomial acyclicity
• continuous causal discovery
• visual semantic graphs
• geometric-series implementation
• directed acyclic graph