Relaxed Sparsest-Permutation Formulation for Causal Discovery at Scale

2026-05-06

AI Summary

This work addresses the scalability challenge in large-scale causal structure learning, which is hindered by high computational complexity in high-dimensional settings. The authors propose SCOPE, a novel method that demonstrates for the first time that exact Cholesky decomposition is not essential for causal structure recovery. Instead, SCOPE employs sparse triangular factor search over the precision support graph, integrating an incomplete Cholesky decomposition with mask-based zero filling and a support-level relaxation strategy. This approach preserves the theoretical correctness of Markov equivalence class (MEC) recovery while enhancing robustness to ordering errors. Experiments on both synthetic and real-world datasets show that SCOPE achieves accuracy comparable to slower baselines but with substantially reduced runtime, enabling efficient scaling to problems involving tens of thousands of variables.

Abstract

Despite the growing availability of large datasets, causal structure learning remains computationally prohibitive at scale. We revisit sparsest-permutation learning for linear structural equation models and show that exact Cholesky factorization is unnecessary for structure recovery. This observation motivates a support-level relaxation that searches for sparse triangular factors over a precision-support screening graph. The relaxed formulation can be efficiently evaluated via masked zero-fill incomplete Cholesky factorization, enabling scalable comparison of candidate orderings. At the population level, we establish soundness for Markov equivalence class (MEC) recovery under no-cancellation and sparsest Markov representation assumptions, as well as robustness to ordering misspecification. Motivated by these guarantees, we introduce SCOPE, a sparse-Cholesky pipeline that provides a scalable implementation of the relaxed formulation. Experiments on synthetic and real datasets demonstrate that SCOPE matches the MEC recovery accuracy of substantially slower baselines, while achieving significantly reduced runtime and scaling to 10k variables.

Problem

• causal discovery
• scalability
• sparsest permutation
• structural equation models
• Markov equivalence class

Innovation

• causal discovery
• sparsest permutation
• Cholesky factorization
• Markov equivalence class
• scalable algorithm