mlsanityData to 7 Oct 2026

A Rigorous Theory of Conditional Mean Embeddings

arXiv 1912.00671 ↗First posted 2 Dec 2019math.ST math.FA stat.MLcited by 31

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Abstract (arXiv)

Conditional mean embeddings (CMEs) have proven themselves to be a powerful tool in many machine learning applications. They allow the efficient conditioning of probability distributions within the corresponding reproducing kernel Hilbert spaces (RKHSs) by providing a linear-algebraic relation for the kernel mean embeddings of the respective joint and conditional probability distributions. Both centred and uncentred covariance operators have been used to define CMEs in the existing literature. In this paper, we develop a mathematically rigorous theory for both variants, discuss the merits and problems of each, and significantly weaken the conditions for applicability of CMEs. In the course of this, we demonstrate a beautiful connection to Gaussian conditioning in Hilbert spaces.

Key facts. Cited by 31 papers in the map (data through 7 Oct 2026). First posted on arXiv 2 Dec 2019.

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Citing papers per year: 2021: 4; 2022: 5; 2023: 5; 2024: 5; 2025: 7; 2026: 5.

31 papers.

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  1. 2021Convergence Rates for Learning Linear Operators from Noisy Datacited by 55
  2. 2022Optimal Rates for Regularized Conditional Mean Embedding Learningcited by 50
  3. 2022Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problemcited by 27
  4. 2021Conditional Bures Metric for Domain Adaptationcited by 25
  5. 2022Efficient Conditionally Invariant Representation Learningcited by 19
  6. 2022Integrating Prior Knowledge in Contrastive Learning with Kernelcited by 11
  7. 2023Consistent Long-Term Forecasting of Ergodic Dynamical Systemscited by 10
  8. 2024Data-Driven Distributionally Robust Safety Verification Using Barrier Certificates and Conditional Mean Embeddingscited by 9
  9. 2021Sobolev Norm Learning Rates for Conditional Mean Embeddingscited by 7
  10. 2024Nonparametric Instrumental Regression via Kernel Methods is Minimax Optimalcited by 7
  11. 2025Koopman-Equivariant Gaussian Processescited by 6
  12. 2025On the Hardness of Conditional Independence Testing In Practicecited by 6
19 more
  1. 2022Generalized Label Shift Correction via Minimum Uncertainty Principle: Theory and Algorithmcited by 3
  2. 2024On the Consistency of Kernel Methods with Dependent Observationscited by 3
  3. 2025Optimality and Adaptivity of Deep Neural Features for Instrumental Variable Regressioncited by 3
  4. 2025Doubly-Robust Estimation of Counterfactual Policy Mean Embeddingscited by 3
  5. 2021Adaptive joint distribution learningcited by 2
  6. 2023Robust Bayesian Inference for Measurement Error Misspecification: The Berkson and Classical Casescited by 2
  7. 2024When Invariant Representation Learning Meets Label Shift: Insufficiency and Theoretical Insightscited by 2
  8. 2025Kernel Density Machinescited by 2
  9. 2023Recursive Estimation of Conditional Kernel Mean Embeddingscited by 1
  10. 2025Density Ratio-Free Doubly Robust Proxy Causal Learningcited by 1
  11. 2026Verifiable Regularity Criterion for Conditional Expectation Operators and Conditional Mean Embeddings with Applications to Nonparametric Regression, Bayesian Inverse Problems, and Koopman Operatorscited by 1
  12. 2023Returning The Favour: When Regression Benefits From Probabilistic Causal Knowledgecited by 0
  13. 2023Conditional mean embeddings and optimal feature selection via positive definite kernelscited by 0
  14. 2024Deterministic Fokker-Planck Transport -- With Applications to Sampling, Variational Inference, Kernel Mean Embeddings & Sequential Monte Carlocited by 0
  15. 2025Kernel Mean Embedding Topology: Weak and Strong Forms for Stochastic Kernels and Implications for Model Learningcited by 0
  16. 2026Kernel-Based Learning of Safety Barrierscited by 0
  17. 2026Kernel Integrated R2: A Measure of Dependencecited by 0
  18. 2026Detecting Changes in Causal Dependence with Kernels and Copulascited by 0
  19. 2026Kernel-Based Safe Exploration in Deep Reinforcement Learningcited by 0