# Causal Optimal Transport of Abstractions

arXiv: https://arxiv.org/abs/2312.08107 · HTML: https://mlsanity.com/paper/2312.08107/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 40 references that are in the map.
- First posted on arXiv 13 Dec 2023.

## Builds on (most cited first, 17 of 17)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Causal inference using invariant prediction: identification and confidence intervals](https://mlsanity.com/paper/1501.01332/) (2015)
- [Causal Abstractions of Neural Networks](https://mlsanity.com/paper/2106.02997/) (2021)
- [Large-Scale Optimal Transport and Mapping Estimation](https://mlsanity.com/paper/1711.02283/) (2017)
- [Causal Consistency of Structural Equation Models](https://mlsanity.com/paper/1707.00819/) (2017)
- [Abstracting Causal Models](https://mlsanity.com/paper/1812.03789/) (2018)
- [Approximate Causal Abstraction](https://mlsanity.com/paper/1906.11583/) (2019)
- [Multi-Scale Representation Learning on Proteins](https://mlsanity.com/paper/2204.02337/) (2022)
- [Multi-marginal optimal transport: theory and applications](https://mlsanity.com/paper/1406.0026/) (2014)
- [Compositional Abstraction Error and a Category of Causal Models](https://mlsanity.com/paper/2103.15758/) (2021)
- [The Monge Gap: A Regularizer to Learn All Transport Maps](https://mlsanity.com/paper/2302.04953/) (2023)
- [Causal Modeling of Dynamical Systems](https://mlsanity.com/paper/1803.08784/) (2018)
- [Causal Abstraction with Soft Interventions](https://mlsanity.com/paper/2211.12270/) (2022)
- [Jointly Learning Consistent Causal Abstractions Over Multiple Interventional Distributions](https://mlsanity.com/paper/2301.05893/) (2023)
- [Optimal transport for causal discovery](https://mlsanity.com/paper/2201.09366/) (2022)
- [Optimization-based Causal Estimation from Heterogenous Environments](https://mlsanity.com/paper/2109.11990/) (2021)

## Built on by (most cited first, 13 of 13)

- [Learning Causal Abstractions of Linear Structural Causal Models](https://mlsanity.com/paper/2406.00394/) (2024)
- [Causal Abstraction Learning based on the Semantic Embedding Principle](https://mlsanity.com/paper/2502.00407/) (2025)
- [Causally Abstracted Multi-armed Bandits](https://mlsanity.com/paper/2404.17493/) (2024)
- [Aligning Graphical and Functional Causal Abstractions](https://mlsanity.com/paper/2412.17080/) (2024)
- [The Causal Information Bottleneck and Optimal Causal Variable Abstractions](https://mlsanity.com/paper/2410.00535/) (2024)
- [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)
- [Causal Representation Learning for Generalisable Recommendation](https://mlsanity.com/paper/2605.27043/) (2026)
- [Unsupervised Causal Abstractions Discovery](https://mlsanity.com/paper/2606.19594/) (2026)
- [Multi-Level Causal Embeddings](https://mlsanity.com/paper/2602.22287/) (2026)
- [PLOT: Progressive Localization via Optimal Transport in Neural Causal Abstraction](https://mlsanity.com/paper/2605.06979/) (2026)
- [What Can a Leaderboard Certify? Compositional Controllability for Fair Evaluation and Training of Biomedical Literature-Review Agents](https://mlsanity.com/paper/2609.32318/) (2026)

## Abstract (arXiv)

Causal abstraction (CA) theory establishes formal criteria for relating multiple structural causal models (SCMs) at different levels of granularity by defining maps between them. These maps have significant relevance for real-world challenges such as synthesizing causal evidence from multiple experimental environments, learning causally consistent representations at different resolutions, and linking interventions across multiple SCMs. In this work, we propose COTA, the first method to learn abstraction maps from observational and interventional data without assuming complete knowledge of the underlying SCMs. In particular, we introduce a multi-marginal Optimal Transport (OT) formulation that enforces do-calculus causal constraints, together with a cost function that relies on interventional information. We extensively evaluate COTA on synthetic and real world problems, and showcase its advantages over non-causal, independent and aggregated COTA formulations. Finally, we demonstrate the efficiency of our method as a data augmentation tool by comparing it against the state-of-the-art CA learning framework, which assumes fully specified SCMs, on a real-world downstream task.