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Causal Learning with the Invariance Principle

arXiv 2605.13589 ↗First posted 13 May 2026stat.ML cs.LGcited by 0

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Abstract (arXiv)

Causal discovery, the problem of inferring the direction of causality, is generally ill-posed. We use the language of structural causal models (SCM) to show that assuming that the causal relations are acyclic and invariant across multiple environments (e.g., the way minimum wage affects employment rate is stable across different geographical regions), textitonly two auxiliary environments are sufficient to infer the causal graph for arbitrary nonlinear mechanisms. Moreover, we demonstrate that this implies identifiability of the SCM functional mechanisms: as a corollary, we show that textittwo auxiliary environments are sufficient to guarantee correct counterfactual inference. We empirically support our theoretical results on synthetic data.

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References · 35

  1. 2015Causal inference using invariant prediction: identification and confidence intervalscited by 536
  2. 2019Variational Autoencoders and Nonlinear ICA: A Unifying Frameworkcited by 503
  3. 2018Nonlinear ICA Using Auxiliary Variables and Generalized Contrastive Learningcited by 282
  4. 2016Unsupervised Feature Extraction by Time-Contrastive Learning and Nonlinear ICAcited by 273
  5. 2013Causal Discovery with Continuous Additive Noise Modelscited by 255
  6. 2020Differentiable Causal Discovery from Interventional Datacited by 218
  7. 2013CAM: Causal additive models, high-dimensional order search and penalized regressioncited by 178
  8. 2019Causal Discovery from Heterogeneous/Nonstationary Data with Independent Changescited by 166
  9. 2021Disentanglement via Mechanism Sparsity Regularization: A New Principle for Nonlinear ICAcited by 160
  10. 2017Invariant Causal Prediction for Nonlinear Modelscited by 140
  11. 2022Score matching enables causal discovery of nonlinear additive noise modelscited by 111
  12. 2023Learning Linear Causal Representations from Interventions under General Nonlinear Mixingcited by 92
23 more
  1. 2019The Incomplete Rosetta Stone Problem: Identifiability Results for Multi-View Nonlinear ICAcited by 86
  2. 2020Hidden Markov Nonlinear ICA: Unsupervised Learning from Nonstationary Time Seriescited by 77
  3. 2018Characterizing and Learning Equivalence Classes of Causal DAGs under Interventionscited by 76
  4. 2022Learning to Induce Causal Structurecited by 68
  5. 2021Disentangling Identifiable Features from Noisy Data with Structured Nonlinear ICAcited by 64
  6. 2017Gradient Estimators for Implicit Modelscited by 58
  7. 2019Causal Discovery with General Non-Linear Relationships Using Non-Linear ICAcited by 58
  8. 2023Identifiability of latent-variable and structural-equation models: from linear to nonlinearcited by 58
  9. 2022On the Identifiability and Estimation of Causal Location-Scale Noise Modelscited by 56
  10. 2022Causal Discovery in Heterogeneous Environments Under the Sparse Mechanism Shift Hypothesiscited by 55
  11. 2023Counterfactual Identifiability of Bijective Causal Modelscited by 43
  12. 2023Causal Discovery with Score Matching on Additive Models with Arbitrary Noisecited by 43
  13. 2020Active Structure Learning of Causal DAGs via Directed Clique Treecited by 38
  14. 2015backShift: Learning causal cyclic graphs from unknown shift interventionscited by 37
  15. 2024Unifying Causal Representation Learning with the Invariance Principlecited by 31
  16. 2023Assumption violations in causal discovery and the robustness of score matchingcited by 28
  17. 2024Score-based Causal Representation Learning: Linear and General Transformationscited by 27
  18. 2023Multi-Domain Causal Representation Learning via Weak Distributional Invariancescited by 23
  19. 2024Score matching through the roof: linear, nonlinear, and latent variables causal discoverycited by 10
  20. 2024Are Bayesian networks typically faithful?cited by 6
  21. 2024Deriving Causal Order from Single-Variable Interventions: Guarantees & Algorithmcited by 5
  22. 2026Causal Representation Learning from General Environments under Nonparametric Mixingcited by 5
  23. 2025Multi-Domain Causal Discovery in Bijective Causal Modelscited by 4

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