mlsanityData to 7 Oct 2026

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

arXiv 2603.00971 ↗First posted 1 Mar 2026stat.ML cs.LG math.STcited by 0

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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.

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References · 30

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2018Neural Tangent Kernel: Convergence and Generalization in Neural Networkscited by 2,597
  3. 2019Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descentcited by 732
  4. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  5. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 537
  6. 2019Towards moderate overparameterization: global convergence guarantees for training shallow neural networkscited by 176
  7. 2016Generalization Properties of Learning with Random Featurescited by 163
  8. 2019On the Power and Limitations of Random Features for Understanding Neural Networkscited by 103
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  10. 2018Optimal Rates for Spectral Algorithms with Least-Squares Regression over Hilbert Spacescited by 59
  11. 2021Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equationscited by 44
  12. 2020Optimal Rates for Averaged Stochastic Gradient Descent under Neural Tangent Kernel Regimecited by 35
18 more
  1. 2019A type of generalization error induced by initialization in deep neural networkscited by 33
  2. 2021The Future is Log-Gaussian: ResNets and Their Infinite-Depth-and-Width Limit at Initializationcited by 33
  3. 2023Out-of-distributional risk bounds for neural operators with applications to the Helmholtz equationcited by 32
  4. 2023On the Optimality of Misspecified Spectral Algorithmscited by 28
  5. 2022Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problemcited by 27
  6. 2024Toward a Better Understanding of Fourier Neural Operators from a Spectral Perspectivecited by 24
  7. 2023Error Bounds for Learning with Vector-Valued Random Featurescited by 23
  8. 2016Random Fourier Features for Operator-Valued Kernelscited by 22
  9. 2018Optimal Convergence for Distributed Learning with Stochastic Gradient Methods and Spectral Algorithmscited by 22
  10. 2024An operator learning perspective on parameter-to-observable mapscited by 22
  11. 2020Learning to Learn Kernels with Variational Random Featurescited by 18
  12. 2022Bounding the Rademacher Complexity of Fourier neural operatorscited by 16
  13. 2016Operator-Valued Bochner Theorem, Fourier Feature Maps for Operator-Valued Kernels, and Vector-Valued Learningcited by 15
  14. 2022Bounding the Width of Neural Networks via Coupled Initialization -- A Worst Case Analysiscited by 15
  15. 2019Implicit Regularization of Accelerated Methods in Hilbert Spacescited by 14
  16. 2023How many Neurons do we need? A refined Analysis for Shallow Networks trained with Gradient Descentcited by 10
  17. 2024Optimal Convergence Rates for Neural Operatorscited by 7
  18. 2021Deep Learning in High Dimension: Neural Network Approximation of Analytic Functions in L2(ℝd,γd)cited by 3

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