Measure-Theoretic Anti-Causal Representation Learning
Arman Behnam, Binghui Wang
Causal representation learning for the anti-causal setting—labels cause features rather than the reverse—presents unique challenges that require specialized approaches. This paper introduces a novel measure-theoretic framework called anti-causal invariant abstractions (ACIA) that effectively handles both perfect and imperfect interventions. ACIA learns two-level representations where low-level representations capture causal dynamics from input data, while high-level abstractions learn environment-invariant features. ACIA addresses key limitations of existing approaches by: (1) accommodating imperfect interventions through interventional kernels, (2) eliminating dependency on explicit causal structures, (3) handling high-dimensional data effectively, and (4) providing theoretical guarantees for out-of-distribution generalization. Experiments on synthetic  and real-world medical datasets demonstrate that our ACIA consistently outperforms state-of-the-art methods across accuracy and invariance metrics. Our theoretical results establish tight bounds on performance gaps between training and unseen environments, confirming the efficacy of our approach for robust anti-causal learning.