Learning Consistent Causal Abstraction Networks

Causal artificial intelligence aims to enhance explainability, trustworthiness, and robustness in AI by leveraging structural causal models (SCMs). In this pursuit, recent advances formalize network sheaves and cosheaves of causal knowledge. Pushing in the same direction, we tackle the learning of consistent causal abstraction network (CAN), a sheaf-theoretic framework where (i) SCMs are Gaussian, (ii) restriction maps are transposes of constructive linear causal abstractions (CAs) adhering to the semantic embedding principle, and (iii) edge stalks correspond–up to permutation–to the node stalks of more detailed SCMs. Our problem formulation separates into edge-specific local Riemannian problems and avoids nonconvex objectives. We propose an efficient search procedure, solving the local problems with SPECTRAL, our iterative method with closed-form updates and suitable for positive definite and semidefinite covariance matrices. Experiments on synthetic data show competitive performance in the CA learning task, and successful recovery of diverse CAN structures.

💡 Research Summary

The paper introduces a novel framework called a Consistent Causal Abstraction Network (CAN) for learning linear causal abstractions (CAs) between Gaussian structural causal models (SCMs). The authors start from the observation that modern causal AI seeks explainability, trustworthiness, and robustness, and that recent work has formalized causal knowledge as network sheaves and cosheaves. Building on this, they define CAN as a sheaf‑theoretic structure where each node corresponds to a Gaussian SCM and each edge encodes a linear embedding (the “restriction map”) together with its transpose, which serves as a constructive linear causal abstraction (CLCA). The CLCA must satisfy the Semantic Embedding Principle (SEP), which mathematically forces the embedding matrix V to lie on a Stiefel manifold (VᵀV = I).

A key theoretical ingredient is Theorem 2.1 (borrowed from prior work) that provides a necessary eigenvalue condition linking the covariance matrices Σℓ and Σh of two SCMs: the eigenvalues of Σℓ must interlace those of Σh in a specific way. This condition enables a fast pre‑filtering of candidate edges: only pairs that satisfy the eigenvalue interlacing can possibly admit a CLCA. The authors exploit this to construct a binary matrix P that records admissible edges, filling it diagonal‑by‑diagonal while repeatedly applying transitive closure. This dramatically reduces the number of local learning problems from O(N²) to a much smaller set.

The learning objective (P1) is to minimize the sum of Kullback–Leibler divergences D_KL(Vᵀ_hℓ ∘ χ_h || χ_ℓ) over all admissible edges. In the Gaussian case the KL divergence reduces to a non‑convex function of V involving matrix inverses and log‑determinants, and the Stiefel orthogonality constraint adds further difficulty. Existing methods (