# A Rigorous Theory of Conditional Mean Embeddings

arXiv: https://arxiv.org/abs/1912.00671 · HTML: https://mlsanity.com/paper/1912.00671/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Dec 2019.

## Built on by (most cited first, 30 of 31)

- [Convergence Rates for Learning Linear Operators from Noisy Data](https://mlsanity.com/paper/2108.12515/) (2021)
- [Optimal Rates for Regularized Conditional Mean Embedding Learning](https://mlsanity.com/paper/2208.01711/) (2022)
- [Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problem](https://mlsanity.com/paper/2211.08875/) (2022)
- [Conditional Bures Metric for Domain Adaptation](https://mlsanity.com/paper/2108.00302/) (2021)
- [Efficient Conditionally Invariant Representation Learning](https://mlsanity.com/paper/2212.08645/) (2022)
- [Integrating Prior Knowledge in Contrastive Learning with Kernel](https://mlsanity.com/paper/2206.01646/) (2022)
- [Consistent Long-Term Forecasting of Ergodic Dynamical Systems](https://mlsanity.com/paper/2312.13426/) (2023)
- [Data-Driven Distributionally Robust Safety Verification Using Barrier Certificates and Conditional Mean Embeddings](https://mlsanity.com/paper/2403.10497/) (2024)
- [Nonparametric Instrumental Regression via Kernel Methods is Minimax Optimal](https://mlsanity.com/paper/2411.19653/) (2024)
- [Sobolev Norm Learning Rates for Conditional Mean Embeddings](https://mlsanity.com/paper/2105.07446/) (2021)
- [On the Hardness of Conditional Independence Testing In Practice](https://mlsanity.com/paper/2512.14000/) (2025)
- [Koopman-Equivariant Gaussian Processes](https://mlsanity.com/paper/2502.06645/) (2025)
- [On the Consistency of Kernel Methods with Dependent Observations](https://mlsanity.com/paper/2406.06101/) (2024)
- [Generalized Label Shift Correction via Minimum Uncertainty Principle: Theory and Algorithm](https://mlsanity.com/paper/2202.13043/) (2022)
- [Optimality and Adaptivity of Deep Neural Features for Instrumental Variable Regression](https://mlsanity.com/paper/2501.04898/) (2025)
- [Doubly-Robust Estimation of Counterfactual Policy Mean Embeddings](https://mlsanity.com/paper/2506.02793/) (2025)
- [Adaptive joint distribution learning](https://mlsanity.com/paper/2110.04829/) (2021)
- [Robust Bayesian Inference for Measurement Error Misspecification: The Berkson and Classical Cases](https://mlsanity.com/paper/2306.01468/) (2023)
- [When Invariant Representation Learning Meets Label Shift: Insufficiency and Theoretical Insights](https://mlsanity.com/paper/2406.16608/) (2024)
- [Kernel Density Machines](https://mlsanity.com/paper/2504.21419/) (2025)
- [Recursive Estimation of Conditional Kernel Mean Embeddings](https://mlsanity.com/paper/2302.05955/) (2023)
- [Density Ratio-Free Doubly Robust Proxy Causal Learning](https://mlsanity.com/paper/2505.19807/) (2025)
- [Verifiable Regularity Criterion for Conditional Expectation Operators and Conditional Mean Embeddings with Applications to Nonparametric Regression, Bayesian Inverse Problems, and Koopman Operators](https://mlsanity.com/paper/2608.06155/) (2026)
- [Returning The Favour: When Regression Benefits From Probabilistic Causal Knowledge](https://mlsanity.com/paper/2301.11214/) (2023)
- [Conditional mean embeddings and optimal feature selection via positive definite kernels](https://mlsanity.com/paper/2305.08100/) (2023)
- [Deterministic Fokker-Planck Transport -- With Applications to Sampling, Variational Inference, Kernel Mean Embeddings & Sequential Monte Carlo](https://mlsanity.com/paper/2410.18993/) (2024)
- [Kernel Mean Embedding Topology: Weak and Strong Forms for Stochastic Kernels and Implications for Model Learning](https://mlsanity.com/paper/2502.13486/) (2025)
- [Kernel-Based Learning of Safety Barriers](https://mlsanity.com/paper/2601.12002/) (2026)
- [Kernel Integrated R2: A Measure of Dependence](https://mlsanity.com/paper/2602.22985/) (2026)
- [Detecting Changes in Causal Dependence with Kernels and Copulas](https://mlsanity.com/paper/2605.05809/) (2026)

## 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.