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Measure-Theoretic Anti-Causal Representation Learning

Arman Behnam, Binghui Wang

TL;DR

This work introduces ACIA, a measure-theoretic framework for anti-causal representation learning that learns low-level causal dynamics and high-level environment-invariant abstractions through interventional kernels. By formulating learning as a min–max optimization with environment-independence and causal-structure regularizers, ACIA achieves strong out-of-distribution generalization across diverse environments and interventions. The approach is grounded in a product causal space with causal and interventional kernels, and it provides convergence and generalization guarantees. Empirically, ACIA outperforms strong baselines on synthetic anti-causal datasets (CMNIST, RMNIST, Ball Agent) and a real medical dataset (Camelyon17), demonstrating robust invariance to environmental variations and interventions. These results indicate that measure-theoretic anti-causal representations can enhance robustness and transferability in domains where causal directions are inverted and interventions are imperfect.

Abstract

Causal representation learning in the anti-causal setting (labels cause features rather than the reverse) presents unique challenges requiring specialized approaches. We propose Anti-Causal Invariant Abstractions (ACIA), a novel measure-theoretic framework for anti-causal representation learning. ACIA employs a two-level design, low-level representations capture how labels generate observations, while high-level representations learn stable causal patterns across environment-specific variations. ACIA addresses key limitations of existing approaches by accommodating prefect and imperfect interventions through interventional kernels, eliminating dependency on explicit causal structures, handling high-dimensional data effectively, and providing theoretical guarantees for out-of-distribution generalization. Experiments on synthetic and real-world medical datasets demonstrate that ACIA consistently outperforms state-of-the-art methods in both accuracy and invariance metrics. Furthermore, 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.

Measure-Theoretic Anti-Causal Representation Learning

TL;DR

This work introduces ACIA, a measure-theoretic framework for anti-causal representation learning that learns low-level causal dynamics and high-level environment-invariant abstractions through interventional kernels. By formulating learning as a min–max optimization with environment-independence and causal-structure regularizers, ACIA achieves strong out-of-distribution generalization across diverse environments and interventions. The approach is grounded in a product causal space with causal and interventional kernels, and it provides convergence and generalization guarantees. Empirically, ACIA outperforms strong baselines on synthetic anti-causal datasets (CMNIST, RMNIST, Ball Agent) and a real medical dataset (Camelyon17), demonstrating robust invariance to environmental variations and interventions. These results indicate that measure-theoretic anti-causal representations can enhance robustness and transferability in domains where causal directions are inverted and interventions are imperfect.

Abstract

Causal representation learning in the anti-causal setting (labels cause features rather than the reverse) presents unique challenges requiring specialized approaches. We propose Anti-Causal Invariant Abstractions (ACIA), a novel measure-theoretic framework for anti-causal representation learning. ACIA employs a two-level design, low-level representations capture how labels generate observations, while high-level representations learn stable causal patterns across environment-specific variations. ACIA addresses key limitations of existing approaches by accommodating prefect and imperfect interventions through interventional kernels, eliminating dependency on explicit causal structures, handling high-dimensional data effectively, and providing theoretical guarantees for out-of-distribution generalization. Experiments on synthetic and real-world medical datasets demonstrate that ACIA consistently outperforms state-of-the-art methods in both accuracy and invariance metrics. Furthermore, 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.
Paper Structure (66 sections, 22 theorems, 136 equations, 12 figures, 10 tables, 3 algorithms)

This paper contains 66 sections, 22 theorems, 136 equations, 12 figures, 10 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background on Measure Theory and Causality
  3. Measure Theory
  4. Causality
  5. ACIA: Measure-Theoretic Anti-Causal Representation Learning
  6. Problem Formulation
  7. The Theoretical Framework
  8. Objective Function of ACIA
  9. ACIA Algorithm Details
  10. Experiments
  11. Experimental Setup
  12. Experimental Results
  13. Results under Perfect Intervention
  14. Results under Imperfect Intervention
  15. Visualizing Learnt Representations
  16. ...and 51 more sections

Key Result

Theorem 1

For an anti-causal structure with arbitrary feature space $\mathcal{X}$, label space $\mathcal{Y}$, and environments $\mathcal{E}$, the causal kernel satisfies: where $\mu_Y$ is the marginal measure on $\mathcal{Y}$.

Figures (12)

  • Figure 1: Anti-Causal Diagram: causal (orange), spurious (blue), and confounding (dashed blue) dependencies.
  • Figure 2: ACIA: Anti-Causal Invariant Abstraction Framework.
  • Figure 3: t-SNE visualization of ACIA representations on CMNIST. From left to right: (1) Low-level representations show initial digit clustering with color influence; (2) High-level representations show improved digit separation; (3) Environment visualization demonstrate removal of environment-specific information; (4) Parity analysis reveals clear separation between even and odd digits.
  • Figure 4: t-SNE visualization of ACIA representations on RMNIST. From left to right: (1) Low-level representations show digit clustering but with rotation influence; (2) High-level representations with better digit boundaries; (3) Rotation angle visualization shows uniform distribution across the representation space; (4) Digit complexity reveals semantic organization by structural properties.
  • Figure 5: t-SNE visualization of ACIA representations on Ball Agent. From left to right: (1) Low-level representations display position-based organization with environmental mixing, (2) High-level representations show more pronounced position clustering; (3) Intervention visualization categorized by intervention patterns, and (4) Prediction error shows areas of high accuracy (green) versus areas requiring improvement (red).
  • ...and 7 more figures

Theorems & Definitions (49)

  • Definition 1: Environment Measurable Space
  • Definition 2: Data Space
  • Definition 3: Representation
  • Definition 4: Kernel klenke2014probability
  • Definition 5: Intervention kocaoglu2019characterization
  • Definition 6: Causal Independence pearl2009causality
  • Definition 7: Causal Space park2023measure
  • Definition 8: Product Causal Space
  • Definition 9: Sub-$\sigma$-algebra
  • Definition 10: Causal Kernel
  • ...and 39 more