mlsanityData to 7 Oct 2026

A Measure-Theoretic Approach to Kernel Conditional Mean Embeddings

arXiv 2002.03689 ↗First posted 10 Feb 2020cs.LG stat.MLcited by 86

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Abstract (arXiv)

We present an operator-free, measure-theoretic approach to the conditional mean embedding (CME) as a random variable taking values in a reproducing kernel Hilbert space. While the kernel mean embedding of unconditional distributions has been defined rigorously, the existing operator-based approach of the conditional version depends on stringent assumptions that hinder its analysis. We overcome this limitation via a measure-theoretic treatment of CMEs. We derive a natural regression interpretation to obtain empirical estimates, and provide a thorough theoretical analysis thereof, including universal consistency. As natural by-products, we obtain the conditional analogues of the maximum mean discrepancy and Hilbert-Schmidt independence criterion, and demonstrate their behaviour via simulations.

Key facts. Cited by 86 papers in the map (data through 7 Oct 2026). First posted on arXiv 10 Feb 2020.

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Citing papers per year: 2021: 8; 2022: 13; 2023: 12; 2024: 16; 2025: 26; 2026: 11.

86 papers.

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  1. 2022Optimal Rates for Regularized Conditional Mean Embedding Learningcited by 50
  2. 2021Conditional Distributional Treatment Effect with Kernel Conditional Mean Embeddings and U-Statistic Regressioncited by 40
  3. 2022From Statistical to Causal Learningcited by 40
  4. 2021RKHS-SHAP: Shapley Values for Kernel Methodscited by 30
  5. 2021Higher Order Kernel Mean Embeddings to Capture Filtrations of Stochastic Processescited by 27
  6. 2022Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problemcited by 27
  7. 2023Towards Optimal Sobolev Norm Rates for the Vector-Valued Regularized Least-Squares Algorithmcited by 22
  8. 2022Calibration tests beyond classificationcited by 20
  9. 2022Efficient Conditionally Invariant Representation Learningcited by 19
  10. 2024Signature Kernel Conditional Independence Tests in Causal Discovery for Stochastic Processescited by 18
  11. 2022Minimax Optimal Kernel Operator Learning via Multilevel Trainingcited by 13
  12. 2024Practical Kernel Tests of Conditional Independencecited by 13
38 more
  1. 2021An Asymptotic Test for Conditional Independence using Analytic Kernel Embeddingscited by 12
  2. 2022Spectral Representation Learning for Conditional Moment Modelscited by 12
  3. 2024Optimal Rates for Vector-Valued Spectral Regularization Learning Algorithmscited by 12
  4. 2023MIST: Defending Against Membership Inference Attacks Through Membership-Invariant Subspace Trainingcited by 11
  5. 2022Learning Counterfactually Invariant Predictorscited by 10
  6. 2022Context-Aware Drift Detectioncited by 9
  7. 2024Data-Driven Distributionally Robust Safety Verification Using Barrier Certificates and Conditional Mean Embeddingscited by 9
  8. 2022Evaluating Aleatoric Uncertainty via Conditional Generative Modelscited by 8
  9. 2021Sobolev Norm Learning Rates for Conditional Mean Embeddingscited by 7
  10. 2024Nonparametric Instrumental Regression via Kernel Methods is Minimax Optimalcited by 7
  11. 2023The Conditional Cauchy-Schwarz Divergence with Applications to Time-Series Data and Sequential Decision Makingcited by 6
  12. 2024Doubly Robust Conditional Independence Testing with Generative Neural Networkscited by 6
  13. 2025On the Hardness of Conditional Independence Testing In Practicecited by 6
  14. 2022Distance and Kernel-Based Measures for Global and Local Two-Sample Conditional Distribution Testingcited by 5
  15. 2023Confidence and Uncertainty Assessment for Distributional Random Forestscited by 5
  16. 2023Kernel Single Proxy Control for Deterministic Confoundingcited by 5
  17. 2024A Kernel-Based Conditional Two-Sample Test Using Nearest Neighbors (with Applications to Calibration, Regression Curves, and Simulation-Based Inference)cited by 5
  18. 2024An Overview of Causal Inference using Kernel Embeddingscited by 5
  19. 2021Exact Distribution-Free Hypothesis Tests for the Regression Function of Binary Classification via Conditional Kernel Mean Embeddingscited by 4
  20. 2022Generalized Label Shift Correction via Minimum Uncertainty Principle: Theory and Algorithmcited by 3
  21. 2023A continuous Structural Intervention Distance to compare Causal Graphscited by 3
  22. 2023Double Machine Learning Based Structure Identification from Temporal Datacited by 3
  23. 2024I Bet You Did Not Mean That: Testing Semantic Importance via Bettingcited by 3
  24. 2025Optimality and Adaptivity of Deep Neural Features for Instrumental Variable Regressioncited by 3
  25. 2025Doubly-Robust Estimation of Counterfactual Policy Mean Embeddingscited by 3
  26. 2021Adaptive joint distribution learningcited by 2
  27. 2024When Invariant Representation Learning Meets Label Shift: Insufficiency and Theoretical Insightscited by 2
  28. 2024Unifying Invariant and Variant Features for Graph Out-of-Distribution via Probability of Necessity and Sufficiencycited by 2
  29. 2024Interventional Processes for Causal Uncertainty Quantificationcited by 2
  30. 2025All Models Are Miscalibrated, But Some Less So: Comparing Calibration with Conditional Mean Operatorscited by 2
  31. 2025Kernel Density Machinescited by 2
  32. 2025Score-based Generative Modeling for Conditional Independence Testingcited by 2
  33. 2025Informed Asymmetric Actor-Critic: Leveraging Privileged Signals Beyond Full-State Accesscited by 2
  34. 2025Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operatorscited by 2
  35. 2025Sequence Modeling with Spectral Mean Flowscited by 2
  36. 2025LUCID: Learning-Enabled Uncertainty-Aware Certification of Stochastic Dynamical Systemscited by 2
  37. 2026The Generalised Kernel Covariance Measurecited by 2
  38. 2021Learning to Transfer with von Neumann Conditional Divergencecited by 1

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