Causal Abstraction Learning based on the Semantic Embedding Principle

Structural causal models (SCMs) allow us to investigate complex systems at multiple levels of resolution.The causal abstraction (CA) framework formalizes the mapping between high- and low-level SCMs. We address CA learning in a challenging and realistic setting, where SCMs are inaccessible, interventional data is unavailable, and sample data is misaligned.A key principle of our framework is semantic embedding, formalized as the high-level distribution lying on a subspace of the low-level one. This principle naturally links linear CA to the geometry of the Stiefel manifold.We present a category-theoretic approach to SCMs that enables the learning of a CA by finding a morphism between the low- and high-level probability measures, adhering to the semantic embedding principle.Consequently, we formulate a general CA learning problem.As an application, we solve the latter problem for linear CA; considering Gaussian measures and the Kullback-Leibler divergence as an objective.Given the nonconvexity of the learning task, we develop three algorithms building upon existing paradigms for Riemannian optimization.We demonstrate that the proposed methods succeed on both synthetic and real-world brain data with different degrees of prior information about the structure of CA.

Consider two scientists using mathematical tools called structural causal models to investigate a city’s dynamics from different vantage points. One works from a bird’s-eye, coarse-grained perspective; the other maps detailed, fine-grained street-level interactions. They aim to link their causal models — without fully sharing either — despite messy, incomplete, or misaligned data.Borrowing tools from pure mathematics (category theory), we formulate a general learning problem for causal abstractions: mapping a fine-grained (e.g., street-level) causal model to a coarse-grained (e.g., bird’s-eye) one while preserving key causal properties. At the core is a novel principle — semantic embedding — which states that coarse-grained causal knowledge resides within the fine-grained one and is preserved when moving from coarse to fine and back again. Interestingly, this principle naturally connects causal abstractions with Riemannian geometry.We exploit this connection to devise three methods for learning a causal abstraction based on semantic embedding. They reliably recover the intended abstractions on both simulated data and real brain recordings, offering promising tools for causal analysis at different levels of granularity in realistic scenarios.