# Sobolev Norm Learning Rates for Conditional Mean Embeddings

arXiv: https://arxiv.org/abs/2105.07446 · HTML: https://mlsanity.com/paper/2105.07446/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 25 references that are in the map.
- First posted on arXiv 16 May 2021.

## Builds on (most cited first, 9 of 9)

- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [Sobolev Norm Learning Rates for Regularized Least-Squares Algorithm](https://mlsanity.com/paper/1702.07254/) (2017)
- [A Measure-Theoretic Approach to Kernel Conditional Mean Embeddings](https://mlsanity.com/paper/2002.03689/) (2020)
- [Optimal Rates for Spectral Algorithms with Least-Squares Regression over Hilbert Spaces](https://mlsanity.com/paper/1801.06720/) (2018)
- [Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/1712.01572/) (2017)
- [A Rigorous Theory of Conditional Mean Embeddings](https://mlsanity.com/paper/1912.00671/) (2019)
- [Optimal Convergence for Distributed Learning with Stochastic Gradient Methods and Spectral Algorithms](https://mlsanity.com/paper/1801.07226/) (2018)
- [Nonparametric approximation of conditional expectation operators](https://mlsanity.com/paper/2012.12917/) (2020)
- [Kernel Autocovariance Operators of Stationary Processes: Estimation and Convergence](https://mlsanity.com/paper/2004.00891/) (2020)

## Built on by (most cited first, 7 of 7)

- [Optimal Rates for Regularized Conditional Mean Embedding Learning](https://mlsanity.com/paper/2208.01711/) (2022)
- [Minimax Optimal Kernel Operator Learning via Multilevel Training](https://mlsanity.com/paper/2209.14430/) (2022)
- [Spectral Representation Learning for Conditional Moment Models](https://mlsanity.com/paper/2210.16525/) (2022)
- [Nonparametric Instrumental Variable Regression through Stochastic Approximate Gradients](https://mlsanity.com/paper/2402.05639/) (2024)
- [Optimal Learning Rates for Regularized Least-Squares with a Fourier Capacity Condition](https://mlsanity.com/paper/2204.07856/) (2022)
- [Spectrally Transformed Kernel Regression](https://mlsanity.com/paper/2402.00645/) (2024)
- [Supervised learning with probabilistic morphisms and kernel mean embeddings](https://mlsanity.com/paper/2305.06348/) (2023)

## Abstract (arXiv)

We develop novel learning rates for conditional mean embeddings by applying the theory of interpolation for reproducing kernel Hilbert spaces (RKHS). We derive explicit, adaptive convergence rates for the sample estimator under the misspecifed setting, where the target operator is not Hilbert-Schmidt or bounded with respect to the input/output RKHSs. We demonstrate that in certain parameter regimes, we can achieve uniform convergence rates in the output RKHS. We hope our analyses will allow the much broader application of conditional mean embeddings to more complex ML/RL settings involving infinite dimensional RKHSs and continuous state spaces.