# The linear conditional expectation in Hilbert space

arXiv: https://arxiv.org/abs/2008.12070 · HTML: https://mlsanity.com/paper/2008.12070/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Aug 2020.

## Built on by (most cited first, 8 of 8)

- [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)
- [Uncertainty Quantification and Experimental Design for Large-Scale Linear Inverse Problems under Gaussian Process Priors](https://mlsanity.com/paper/2109.03457/) (2021)
- [Optimal Transportation by Orthogonal Coupling Dynamics](https://mlsanity.com/paper/2410.08060/) (2024)
- [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)
- [Graph Representation Learning via Aggregation Enhancement](https://mlsanity.com/paper/2201.12843/) (2022)
- [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)

## Abstract (arXiv)

The linear conditional expectation (LCE) provides a best linear (or rather, affine) estimate of the conditional expectation and hence plays an important role in approximate Bayesian inference, especially the Bayes linear approach. This article establishes the analytical properties of the LCE in an infinite-dimensional Hilbert space context. In addition, working in the space of affine Hilbert--Schmidt operators, we establish a regularisation procedure for this LCE. As an important application, we obtain a simple alternative derivation and intuitive justification of the conditional mean embedding formula, a concept widely used in machine learning to perform the conditioning of random variables by embedding them into reproducing kernel Hilbert spaces.