# Random Features for Operator-Valued Kernels: Bridging Kernel Methods and Neural Operators

arXiv: https://arxiv.org/abs/2603.00971 · HTML: https://mlsanity.com/paper/2603.00971/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 48 references that are in the map.
- First posted on arXiv 1 Mar 2026.

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

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Towards moderate overparameterization: global convergence guarantees for training shallow neural networks](https://mlsanity.com/paper/1902.04674/) (2019)
- [Generalization Properties of Learning with Random Features](https://mlsanity.com/paper/1602.04474/) (2016)
- [On the Power and Limitations of Random Features for Understanding Neural Networks](https://mlsanity.com/paper/1904.00687/) (2019)
- [Optimal Rates For Regularization Of Statistical Inverse Learning Problems](https://mlsanity.com/paper/1604.04054/) (2016)
- [Optimal Rates for Spectral Algorithms with Least-Squares Regression over Hilbert Spaces](https://mlsanity.com/paper/1801.06720/) (2018)
- [Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations](https://mlsanity.com/paper/2112.08125/) (2021)
- [Optimal Rates for Averaged Stochastic Gradient Descent under Neural Tangent Kernel Regime](https://mlsanity.com/paper/2006.12297/) (2020)
- [A type of generalization error induced by initialization in deep neural networks](https://mlsanity.com/paper/1905.07777/) (2019)
- [The Future is Log-Gaussian: ResNets and Their Infinite-Depth-and-Width Limit at Initialization](https://mlsanity.com/paper/2106.04013/) (2021)
- [Out-of-distributional risk bounds for neural operators with applications to the Helmholtz equation](https://mlsanity.com/paper/2301.11509/) (2023)
- [Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problem](https://mlsanity.com/paper/2211.08875/) (2022)
- [On the Optimality of Misspecified Spectral Algorithms](https://mlsanity.com/paper/2303.14942/) (2023)
- [Toward a Better Understanding of Fourier Neural Operators from a Spectral Perspective](https://mlsanity.com/paper/2404.07200/) (2024)
- [Error Bounds for Learning with Vector-Valued Random Features](https://mlsanity.com/paper/2305.17170/) (2023)
- [Random Fourier Features for Operator-Valued Kernels](https://mlsanity.com/paper/1605.02536/) (2016)
- [Optimal Convergence for Distributed Learning with Stochastic Gradient Methods and Spectral Algorithms](https://mlsanity.com/paper/1801.07226/) (2018)
- [An operator learning perspective on parameter-to-observable maps](https://mlsanity.com/paper/2402.06031/) (2024)
- [Learning to Learn Kernels with Variational Random Features](https://mlsanity.com/paper/2006.06707/) (2020)
- [Bounding the Rademacher Complexity of Fourier neural operators](https://mlsanity.com/paper/2209.05150/) (2022)
- [Operator-Valued Bochner Theorem, Fourier Feature Maps for Operator-Valued Kernels, and Vector-Valued Learning](https://mlsanity.com/paper/1608.05639/) (2016)
- [Bounding the Width of Neural Networks via Coupled Initialization -- A Worst Case Analysis](https://mlsanity.com/paper/2206.12802/) (2022)
- [Implicit Regularization of Accelerated Methods in Hilbert Spaces](https://mlsanity.com/paper/1905.13000/) (2019)
- [How many Neurons do we need? A refined Analysis for Shallow Networks trained with Gradient Descent](https://mlsanity.com/paper/2309.08044/) (2023)
- [Optimal Convergence Rates for Neural Operators](https://mlsanity.com/paper/2412.17518/) (2024)
- [Deep Learning in High Dimension: Neural Network Approximation of Analytic Functions in L2(ℝd,γd)](https://mlsanity.com/paper/2111.07080/) (2021)

## Built on by (most cited first, 1 of 1)

- [Random Feature Pruning for Multi-Class Classification](https://mlsanity.com/paper/2610.15101/) (2026)

## Abstract (arXiv)

In this work, we investigate the generalization properties of random feature methods. Our analysis extends prior results for Tikhonov regularization to a broad class of spectral regularization techniques and further generalizes the setting to operator-valued kernels. This unified framework enables a rigorous theoretical analysis of neural operators and neural networks through the lens of the Neural Tangent Kernel (NTK). In particular, it allows us to establish optimal learning rates and provides a good understanding of how many neurons are required to achieve a given accuracy. Furthermore, we establish minimax rates in the well-specified case and also in the misspecified case, where the target is not contained in the reproducing kernel Hilbert space. These results sharpen and complete earlier findings for specific kernel algorithms.