Networks of Causal Abstractions: A Sheaf-theoretic Framework

arXiv:2509.25236v2 Announce Type: replace
Abstract: Causal artificial intelligence aims to improve explainability, robustness, and trustworthiness by leveraging causal models. Recent work has shown that sheaf-theoretic approaches offer a principled framework for representing and aligning causal knowledge across collections of subjective and imperfect causal models connected by relational structures. In this work, we introduce the causal abstraction network (CAN), a general sheaf-theoretic framework for representing, learning, and reasoning across collections of mixture causal models (MCMs). CAN formalizes causal abstraction relations among subjective MCMs operating at different levels of granularity, while remaining agnostic to explicit causal graphs, functional mechanisms, interventional data, or jointly sampled observations. At the theoretical level, we provide a categorical formulation of MCMs and characterize key properties of CANs, including consistency, smoothness, and the existence of global sections, which are related to spectral properties of an associated combinatorial Laplacian. At the methodological level, we address the problem of learning consistent CANs from data by exploiting the compositionality of causal abstractions and necessary conditions for their existence. The learning task decomposes into local problems on the network edges, for which we propose efficient solutions in Gaussian and Gaussian mixture settings. We validate the proposed learning methods on synthetic data and illustrate the practical relevance of the CAN framework through a financial application, demonstrating both recovery and counterfactual reasoning capabilities.