# Causal Abstraction Learning based on the Semantic Embedding Principle

arXiv: https://arxiv.org/abs/2502.00407 · HTML: https://mlsanity.com/paper/2502.00407/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 53 references that are in the map.
- First posted on arXiv 1 Feb 2025.

## Builds on (most cited first, 26 of 26)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [Causal Abstractions of Neural Networks](https://mlsanity.com/paper/2106.02997/) (2021)
- [Mechanistic Interpretability for AI Safety -- A Review](https://mlsanity.com/paper/2404.14082/) (2024)
- [Finding Alignments Between Interpretable Causal Variables and Distributed Neural Representations](https://mlsanity.com/paper/2303.02536/) (2023)
- [OSQP: An Operator Splitting Solver for Quadratic Programs](https://mlsanity.com/paper/1711.08013/) (2017)
- [Causal Consistency of Structural Equation Models](https://mlsanity.com/paper/1707.00819/) (2017)
- [Manopt, a Matlab toolbox for optimization on manifolds](https://mlsanity.com/paper/1308.5200/) (2013)
- [Abstracting Causal Models](https://mlsanity.com/paper/1812.03789/) (2018)
- [Goal-Oriented and Semantic Communication in 6G AI-Native Networks: The 6G-GOALS Approach](https://mlsanity.com/paper/2402.07573/) (2024)
- [Causal Inference by String Diagram Surgery](https://mlsanity.com/paper/1811.08338/) (2018)
- [Jointly Learning Consistent Causal Abstractions Over Multiple Interventional Distributions](https://mlsanity.com/paper/2301.05893/) (2023)
- [Neuro-Symbolic Causal Reasoning Meets Signaling Game for Emergent Semantic Communications](https://mlsanity.com/paper/2210.12040/) (2022)
- [Targeted Reduction of Causal Models](https://mlsanity.com/paper/2311.18639/) (2023)
- [A Review of the Role of Causality in Developing Trustworthy AI Systems](https://mlsanity.com/paper/2302.06975/) (2023)
- [Causal Semantic Communication for Digital Twins: A Generalizable Imitation Learning Approach](https://mlsanity.com/paper/2304.12502/) (2023)
- [On the Equivalence of Causal Models: A Category-Theoretic Approach](https://mlsanity.com/paper/2201.06981/) (2022)
- [Causal Optimal Transport of Abstractions](https://mlsanity.com/paper/2312.08107/) (2023)
- [MADMM: a generic algorithm for non-smooth optimization on manifolds](https://mlsanity.com/paper/1505.07676/) (2015)
- [Learning Causal Abstractions of Linear Structural Causal Models](https://mlsanity.com/paper/2406.00394/) (2024)
- [Parallel and Distributed Successive Convex Approximation Methods for Big-Data Optimization](https://mlsanity.com/paper/1805.06963/) (2018)
- [Aligning Graphical and Functional Causal Abstractions](https://mlsanity.com/paper/2412.17080/) (2024)
- [A Riemannian Proximal Newton Method](https://mlsanity.com/paper/2304.04032/) (2023)
- [A Regularized Semi-Smooth Newton Method With Projection Steps for Composite Convex Programs](https://mlsanity.com/paper/1603.07870/) (2016)
- [Causal Responsibility Attribution for Human-AI Collaboration](https://mlsanity.com/paper/2411.03275/) (2024)
- [Extracting the Multiscale Causal Backbone of Brain Dynamics](https://mlsanity.com/paper/2311.00118/) (2023)

## Built on by (most cited first, 9 of 9)

- [Coarsening Causal DAG Models](https://mlsanity.com/paper/2601.10531/) (2026)
- [Structural Causal Bottleneck Models](https://mlsanity.com/paper/2603.08682/) (2026)
- [Distributionally Robust Causal Abstractions](https://mlsanity.com/paper/2510.04842/) (2025)
- [The Relativity of Causal Knowledge](https://mlsanity.com/paper/2503.11718/) (2025)
- [Learning Consistent Causal Abstraction Networks](https://mlsanity.com/paper/2602.02623/) (2026)
- [Unsupervised Causal Abstractions Discovery](https://mlsanity.com/paper/2606.19594/) (2026)
- [Sheaf-Based Federated Representation Learning](https://mlsanity.com/paper/2608.10016/) (2026)
- [Operationalising Relative Causal Knowledge: Backbone Identifiability from Private Reports on a Shared Outcome](https://mlsanity.com/paper/2608.10664/) (2026)
- [Score-Based Learning of Cluster DAGs from Interventions](https://mlsanity.com/paper/2610.11947/) (2026)

## Abstract (arXiv)

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.