# Nonparametric approximation of conditional expectation operators

arXiv: https://arxiv.org/abs/2012.12917 · HTML: https://mlsanity.com/paper/2012.12917/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Dec 2020.

## Built on by (most cited first, 21 of 21)

- [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)
- [Towards Optimal Sobolev Norm Rates for the Vector-Valued Regularized Least-Squares Algorithm](https://mlsanity.com/paper/2312.07186/) (2023)
- [Efficient Conditionally Invariant Representation Learning](https://mlsanity.com/paper/2212.08645/) (2022)
- [Minimax Optimal Kernel Operator Learning via Multilevel Training](https://mlsanity.com/paper/2209.14430/) (2022)
- [Optimal Rates for Vector-Valued Spectral Regularization Learning Algorithms](https://mlsanity.com/paper/2405.14778/) (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)
- [Nonparametric Control Koopman Operators](https://mlsanity.com/paper/2405.07312/) (2024)
- [On the Consistency of Kernel Methods with Dependent Observations](https://mlsanity.com/paper/2406.06101/) (2024)
- [Optimality and Adaptivity of Deep Neural Features for Instrumental Variable Regression](https://mlsanity.com/paper/2501.04898/) (2025)
- [Nonparametric Sparse Online Learning of the Koopman Operator](https://mlsanity.com/paper/2405.07432/) (2024)
- [Sequence Modeling with Spectral Mean Flows](https://mlsanity.com/paper/2510.15366/) (2025)
- [Conditional expectation using compactification operators](https://mlsanity.com/paper/2306.10592/) (2023)
- [Kernel-Based Optimal Control: An Infinitesimal Generator Approach](https://mlsanity.com/paper/2412.01591/) (2024)
- [Returning The Favour: When Regression Benefits From Probabilistic Causal Knowledge](https://mlsanity.com/paper/2301.11214/) (2023)
- [Nonparametric Sparse Online Learning of the Koopman Operator](https://mlsanity.com/paper/2501.16489/) (2025)
- [Kernel Mean Embedding Topology: Weak and Strong Forms for Stochastic Kernels and Implications for Model Learning](https://mlsanity.com/paper/2502.13486/) (2025)
- [Operator Models for Continuous-Time Offline Reinforcement Learning](https://mlsanity.com/paper/2511.10383/) (2025)
- [Multi-Task Learning of Conditional Mean Operators: applications to dynamical systems and uncertainty quantification](https://mlsanity.com/paper/2609.35429/) (2026)

## Abstract (arXiv)

Given the joint distribution of two random variables X,Y on some second countable locally compact Hausdorff space, we investigate the statistical approximation of the L2-operator defined by [Pf](x) := mathbbE[ f(Y) mid X = x ] under minimal assumptions. By modifying its domain, we prove that P can be arbitrarily well approximated in operator norm by Hilbert-Schmidt operators acting on a reproducing kernel Hilbert space. This fact allows to estimate P uniformly by finite-rank operators over a dense subspace even when P is not compact. In terms of modes of convergence, we thereby obtain the superiority of kernel-based techniques over classically used parametric projection approaches such as Galerkin methods. This also provides a novel perspective on which limiting object the nonparametric estimate of P converges to. As an application, we show that these results are particularly important for a large family of spectral analysis techniques for Markov transition operators. Our investigation also gives a new asymptotic perspective on the so-called kernel conditional mean embedding, which is the theoretical foundation of a wide variety of techniques in kernel-based nonparametric inference.