Title: Measure-Theoretic Anti-Causal Representation Learning
Venue: NeurIPS 2025
Accepted: yes
Keywords: Representation Learning

Causality

Measure Theory
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: (1) accommodating prefect and 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 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. {{Code is available at \url{https://github.com/ArmanBehnam/ACIA}}}.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a measure-theoretic framework for anti-causal representation learning. The proposed framework handles both perfect and imperfect interventions. The proposed framework builds two levels of latent representations: a low-level \"causal dynamic\" that disengages the environment-specific variations and a high-level \"causal abstraction\" that extracts environment-invariant features.\n\nSTRENGTHS AND WEAKNESSES:\n# Strengths\n\n- The paper proposed a novel approach based on a theoretically rigorous framework. \n- The proposed framework, accommodating both soft and hard interventions, makes it more applicable and robust in real-world settings.\n- The proposed method is evaluated through both synthetic and real-world experiments.\n\n# Weaknesses\n\n- Some part of the writing seems unclear. For example, the motivation for having two-level representations.\n- It is not obvious when we can apply the proposed method by assuming the anti-causal relationship in the data.\n\nQUESTIONS:\n- How do we know the causal direction between the label and the observations in general?\n- I'm a bit confused about the causal dynamic and causal abstraction part in the framework. What's the difference between \"invariant across environments\" and \"environment-independent\"? In l.137, if $P(\\phi_L(\\mathcal{X}|Y)$ is constant across environments $\\mathcal{E}$, isn't that also means it is independent of $\\mathcal{E}$ given $Y$?\n- Could you explain more about how the causal-dynamic learns the low-level representations and disentangles the influence of the environment? How is it different from the \"causal abstraction\"? Also, in practice, for example, when using a neural network, how do you decide the part of high-level and low-level features?\n\n\nbroken reference in line 194\n\nLIMITATIONS:\nI can't find an explicit discussion about the limitations of the proposed work. Including a discussion on the practical scenario of anti-causal assumption might be helpful.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a framework called ACIA for anti-causal representation, aiming at disentanglement between invariant causal features and environment variations. ACIA is a two-level representation framework, where the low-level representation captures the direct causal information from data and labels. And the high-level representation serves as an information bottleneck to filter environment-specific noise. The authors also prepare rather detailed experiments. Compared with different methods in four datasets, ACIA obtains several best results in several metrics including ACC, EI, LII, IR.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\n1.The paper has thorough theoretical analysis of the performance of ACIA, e.g.,  convergence property, generalization bound in terms of sample complexity and interventional kernels in the anti-causal setting, and environmental robustness. These contribute to the scientific rigorousness of this paper and provide some reference meaning to peer work.\n2.There are relatively sufficient experiments in this work including several classical datasets such as CMNIST and RMINST. It also obtains competitive results in more challenging scenario. From an experimental perspective, this work is relatively qualified.\nWeakness:\n1.Though the paper has long discussion about the ACIA, the two-level representation algorithm is still vague especially in the usage of the causal dynamic and its kernels. In appendix G, the pseudo code only briefly describes this crucial part which could be more detailed properly.\n2.The paper proposes an adversarial method to train the model, it may be useful in practice. There are some theoretical analysis around the objective function in Section 3.2. But could you illustrate the necessity of the adversarial training method or it’s just a choice of methodology? Is there more analysis to bridge the gap between the theory and the training method?\n\nQUESTIONS:\n1.As mentioned before, the experiment part is relatively sufficient, but the visualization only focus on the disentanglement of the representations using t-SNE, which is a very classical and common method to visualize representations. However, it can’t uncover the causal relationship between different features. Could you provide some visualization to uncover the correspondent relationship of the two-level representation learning? \n2.Explicit DAG is abandoned during the optimization, which may weaken the causal properties and required some prior to complement. Could you provide a comparison of the prior information used in different methods? Experiments should be conducted at the same prior level to be fair.\n3.ACIA captures intervention uncertainty through probability measures and avoid the assumption of perfect intervention. In addition to the experiment, can you provide some theoretical analysis of the intervention assumption?\n\nLIMITATIONS:\nThe paper adopts an adversarial training approach to optimize the ACIA framework, but it does not explore alternative optimization strategies that could potentially achieve similar or better results. For instance, non-adversarial methods like variational inference or simpler gradient-based techniques might suffice, yet the paper lacks a comparative analysis to justify the exclusive reliance on adversarial training beyond its practical utility. This omission limits the understanding of the framework’s flexibility and robustness across different optimization paradigms.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}

comments: SUMMARY:
The paper “Measure-Theoretic Anti-Causal Representation Learning” addresses the challenging anti-causal representation learning problem, and anti-causal invariant abstractions (ACIA) that can accommodate both perfect and imperfect interventions through interventional kernels. It contains two-level of representation learning, where the low-level capture the causal dynamics and high-level learn invariant predictor. Theoretical results are given to guarantee the OOD generalization. The experiments on synthetic and real-world datasets demonstrate the effectiveness of ACIA, also validate the theoretical results.

STRENGTHS AND WEAKNESSES:
**Strengths**

1. The problem is well-defined and the theoretical results are promising

2. A unified theoretical framework is given to model the anti-causal learning problem, the motivation of ACIA is clear and the objective of ACIA has a solid theoretical foundation

3. The theoretical results (e.g., convergence of ACIA, OOD generalization bound, etc.) further demonstrate the effectiveness of ACIA

3. The experiments validate the proposed theoretical results

**Weaknesses**

1. Both low-level and high-level learning processes encourage an environment-invariant representation, and its seems the goals of these two process overlap to some extent, what are the main differences between these two process? Are these two processes towards the same goal in different ways?

2. The description of Figure 2 is insufficient, and Figures 1 and 2 overlaps, it is better to merge them into one figure

3. Typos: Line 194 "Appendix ??"; Title of section 3.3, “objection” -> “objective”; Footnote in page 3 "P(Y|do(X=x)" -> "P(Y|do(X=x))"

QUESTIONS:
Please refer to the weaknesses.

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 4
ORIGINALITY: 3
DATE: 2025-07-02

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors propose ACIA, a measure-theoretic framework for anti-causal representation learning. ACIA aims to disentangle invariant causal features from environment-specific variations while making minimal assumptions about the underlying causal structure of the data-generating process. The main contributions of the paper include: A generalized intervention model that accommodates both perfect and imperfect interventions via interventional kernels; A construction of causal dynamics and causal abstractions to model invariant causal relations; A well-motivated optimization objective designed to identify stable causal features across domains/environments without requiring explicit DAG specifications.Theoretical guarantees for out-of-distribution generalization, including bounds on performance gaps between training and unseen environments. Experimental results on both synthetic and real-world datasets demonstrate ACIA’s ability to capture and leverage anti-causal structures, outperforming state-of-the-art methods.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\n1. The paper is well written and logically structured, facilitating reader comprehension.\n\n2. The authors provide rigorous and in-depth theoretical analysis to support their proposed method.\n\n3. Comprehensive experimental evaluations validate the effectiveness of ACIA across diverse datasets.\n\nWeaknesses\n\nThe review currently does not list specific weaknesses, but a minor area for improvement might be in the empirical clarity regarding computational cost (see question below).\n\nQUESTIONS:\n1. While the paper discusses theoretical complexity, could the authors also provide empirical training time or computational cost across different datasets to better assess practical scalability?\n\n2. Line 94: The reference to Appendix ?? appears to be a placeholder. Could the authors clarify or correct the appendix citation?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}