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Learning Consistent Causal Abstraction Networks

Gabriele D'Acunto, Paolo Di Lorenzo, Sergio Barbarossa

TL;DR

The paper addresses learning consistent causal abstraction networks (CANs) that connect multiple Gaussian SCMs through SEP-compliant CLCA maps lying on the Stiefel manifold $\mathrm{St}({\ell},{h})$. It formulates edge-wise KL divergence minimization, exploits CA compositionality to prune the search, and introduces a spectral method with an augmented Lagrangian/ADMM to obtain closed-form updates for $\mathbf{V},\mathbf{Y},\mathbf{T}$, applicable to both PD and PSD covariances with $O(\ell^3)$ per update. Empirical results on synthetic data show competitive CA learning and successful CAN recovery, including PSD-input scenarios that previous methods could not handle. These contributions provide a practical pathway to learn globally consistent, collaborative causal knowledge across hierarchical AI systems.

Abstract

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.

Learning Consistent Causal Abstraction Networks

TL;DR

The paper addresses learning consistent causal abstraction networks (CANs) that connect multiple Gaussian SCMs through SEP-compliant CLCA maps lying on the Stiefel manifold . It formulates edge-wise KL divergence minimization, exploits CA compositionality to prune the search, and introduces a spectral method with an augmented Lagrangian/ADMM to obtain closed-form updates for , applicable to both PD and PSD covariances with per update. Empirical results on synthetic data show competitive CA learning and successful CAN recovery, including PSD-input scenarios that previous methods could not handle. These contributions provide a practical pathway to learn globally consistent, collaborative causal knowledge across hierarchical AI systems.

Abstract

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.
Paper Structure (5 sections, 3 theorems, 14 equations, 2 figures)

This paper contains 5 sections, 3 theorems, 14 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A Primer on Causal Abstraction Networks
  3. Learning causal abstraction networks
  4. Empirical assessment
  5. Conclusions and future works

Key Result

Theorem 2.1

Let $\chi^{\ell}\xspace \sim N(\boldsymbol{0}\xspace_\ell, \boldsymbol{\Sigma}^{\ell}\xspace)$, $\chi^{h}\xspace \sim N(\boldsymbol{0}\xspace_h, \boldsymbol{\Sigma}^{h}\xspace)$, where $\boldsymbol{\Sigma}^{\ell}\xspace \in \mathcal{S}_{++}\xspace^{\ell}$ and $\boldsymbol{\Sigma}^{h}\xspace \in \mat

Figures (2)

  • Figure 1: Synthetic results for the solution of the local problem across all settings $(\ell, h)$ from d2025causal.
  • Figure 2: False positive (left) and true positive (right) rates for the proposed search procedure on the three CAN structures.

Theorems & Definitions (5)

  • Theorem 2.1: From d2025causal
  • Definition 2.2: Consistent Causal Abstraction Network d2025cantl
  • Corollary 2.3
  • Lemma 3.1
  • proof