# Minimax Optimal Kernel Operator Learning via Multilevel Training

arXiv: https://arxiv.org/abs/2209.14430 · HTML: https://mlsanity.com/paper/2209.14430/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 77 references that are in the map.
- First posted on arXiv 28 Sep 2022.

## Builds on (most cited first, 30 of 31)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Kernel Instrumental Variable Regression](https://mlsanity.com/paper/1906.00232/) (2019)
- [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)
- [Minimax Estimation of Conditional Moment Models](https://mlsanity.com/paper/2006.07201/) (2020)
- [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)
- [Deep Nonparametric Estimation of Operators between Infinite Dimensional Spaces](https://mlsanity.com/paper/2201.00217/) (2022)
- [Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimality](https://mlsanity.com/paper/2110.06897/) (2021)
- [Kernel Methods for Causal Functions: Dose, Heterogeneous, and Incremental Response Curves](https://mlsanity.com/paper/2010.04855/) (2020)
- [Meta-Auto-Decoder for Solving Parametric Partial Differential Equations](https://mlsanity.com/paper/2111.08823/) (2021)
- [Solving for high dimensional committor functions using artificial neural networks](https://mlsanity.com/paper/1802.10275/) (2018)
- [Learning Green's functions associated with time-dependent partial differential equations](https://mlsanity.com/paper/2204.12789/) (2022)
- [Breaking the Curse of Many Agents: Provable Mean Embedding Q-Iteration for Mean-Field Reinforcement Learning](https://mlsanity.com/paper/2006.11917/) (2020)
- [Nonparametric approximation of conditional expectation operators](https://mlsanity.com/paper/2012.12917/) (2020)
- [Meta-learning Pseudo-differential Operators with Deep Neural Networks](https://mlsanity.com/paper/1906.06782/) (2019)
- [Kernel Conditional Density Operators](https://mlsanity.com/paper/1905.11255/) (2019)
- [A Multi-level procedure for enhancing accuracy of machine learning algorithms](https://mlsanity.com/paper/1909.09448/) (2019)
- [Sobolev Acceleration and Statistical Optimality for Learning Elliptic Equations via Gradient Descent](https://mlsanity.com/paper/2205.07331/) (2022)
- [Functional Linear Regression with Mixed Predictors](https://mlsanity.com/paper/2012.00460/) (2020)
- [Sobolev Norm Learning Rates for Conditional Mean Embeddings](https://mlsanity.com/paper/2105.07446/) (2021)

## Built on by (most cited first, 13 of 13)

- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [A Mathematical Guide to Operator Learning](https://mlsanity.com/paper/2312.14688/) (2023)
- [Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problem](https://mlsanity.com/paper/2211.08875/) (2022)
- [Error Bounds for Learning with Vector-Valued Random Features](https://mlsanity.com/paper/2305.17170/) (2023)
- [Operator Learning Using Random Features: A Tool for Scientific Computing](https://mlsanity.com/paper/2408.06526/) (2024)
- [Optimal Rates for Vector-Valued Spectral Regularization Learning Algorithms](https://mlsanity.com/paper/2405.14778/) (2024)
- [Towards Sharp Minimax Risk Bounds for Operator Learning](https://mlsanity.com/paper/2512.17805/) (2025)
- [Multi-Level Monte Carlo Training of Neural Operators](https://mlsanity.com/paper/2505.12940/) (2025)
- [Domain Generalization by Functional Regression](https://mlsanity.com/paper/2302.04724/) (2023)
- [Spectrally Transformed Kernel Regression](https://mlsanity.com/paper/2402.00645/) (2024)
- [Base Models for Parabolic Partial Differential Equations](https://mlsanity.com/paper/2407.12234/) (2024)
- [Optimal Multiscale Learning of Linear Operators](https://mlsanity.com/paper/2606.16913/) (2026)
- [Benign overfitting in Fixed Dimension via Physics-Informed Learning with Smooth Inductive Bias](https://mlsanity.com/paper/2406.09194/) (2024)

## Abstract (arXiv)

Learning mappings between infinite-dimensional function spaces has achieved empirical success in many disciplines of machine learning, including generative modeling, functional data analysis, causal inference, and multi-agent reinforcement learning. In this paper, we study the statistical limit of learning a Hilbert-Schmidt operator between two infinite-dimensional Sobolev reproducing kernel Hilbert spaces. We establish the information-theoretic lower bound in terms of the Sobolev Hilbert-Schmidt norm and show that a regularization that learns the spectral components below the bias contour and ignores the ones that are above the variance contour can achieve the optimal learning rate. At the same time, the spectral components between the bias and variance contours give us flexibility in designing computationally feasible machine learning algorithms. Based on this observation, we develop a multilevel kernel operator learning algorithm that is optimal when learning linear operators between infinite-dimensional function spaces.