mlsanityData to 7 Oct 2026

Error Bounds for Learning with Vector-Valued Random Features

arXiv 2305.17170 ↗First posted 26 May 2023stat.ML cs.LG math.STcited by 23

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Abstract (arXiv)

This paper provides a comprehensive error analysis of learning with vector-valued random features (RF). The theory is developed for RF ridge regression in a fully general infinite-dimensional input-output setting, but nonetheless applies to and improves existing finite-dimensional analyses. In contrast to comparable work in the literature, the approach proposed here relies on a direct analysis of the underlying risk functional and completely avoids the explicit RF ridge regression solution formula in terms of random matrices. This removes the need for concentration results in random matrix theory or their generalizations to random operators. The main results established in this paper include strong consistency of vector-valued RF estimators under model misspecification and minimax optimal convergence rates in the well-specified setting. The parameter complexity (number of random features) and sample complexity (number of labeled data) required to achieve such rates are comparable with Monte Carlo intuition and free from logarithmic factors.

Key facts. Cited by 23 papers in the map (data through 7 Oct 2026). Builds on 19 of its 48 references that are in the map. First posted on arXiv 26 May 2023.

Builds on

19 of 48 references.

References · 19

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  3. 2020Model Reduction and Neural Networks for Parametric PDEscited by 257
  4. 2016Generalization Properties of Learning with Random Featurescited by 163
  5. 2019Linearized two-layers neural networks in high dimensioncited by 155
  6. 2015On the Equivalence between Kernel Quadrature Rules and Random Feature Expansionscited by 136
  7. 2021Generalization error of random features and kernel methods: hypercontractivity and kernel matrix concentrationcited by 110
  8. 2022Generic bounds on the approximation error for physics-informed (and) operator learningcited by 65
  9. 2020DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problemscited by 61
  10. 2021Convergence Rates for Learning Linear Operators from Noisy Datacited by 55
  11. 2020Approximation Bounds for Random Neural Networks and Reservoir Systemscited by 52
  12. 2021Learning elliptic partial differential equations with randomized linear algebracited by 29
7 more
  1. 2018But How Does It Work in Theory? Linear SVM with Random Featurescited by 28
  2. 2022Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problemcited by 27
  3. 2016Random Fourier Features for Operator-Valued Kernelscited by 22
  4. 2021Generalization Bounds for Sparse Random Feature Expansionscited by 22
  5. 2016Operator-Valued Bochner Theorem, Fourier Feature Maps for Operator-Valued Kernels, and Vector-Valued Learningcited by 15
  6. 2022Minimax Optimal Kernel Operator Learning via Multilevel Trainingcited by 13
  7. 2022Concentration of Random Feature Matrices in High-Dimensionscited by 8

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 4; 2024: 8; 2025: 6; 2026: 5.

23 papers.

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  1. 2024Operator Learning: Algorithms and Analysiscited by 84
  2. 2024Data Complexity Estimates for Operator Learningcited by 21
  3. 2024Operator Learning Using Random Features: A Tool for Scientific Computingcited by 21
  4. 2025Operator Learning: A Statistical Perspectivecited by 14
  5. 2024Optimal Rates for Vector-Valued Spectral Regularization Learning Algorithmscited by 12
  6. 2023Random Fourier Signature Featurescited by 9
  7. 2024Operator Learning of Lipschitz Operators: An Information-Theoretic Perspectivecited by 9
  8. 2024Hyperparameter Optimization for Randomized Algorithms: A Case Study on Random Featurescited by 7
  9. 2024Solving Partial Differential Equations with Random Feature Modelscited by 6
  10. 2023Inverse Problems with Learned Forward Operatorscited by 4
  11. 2025Random Feature Representation Boostingcited by 4
  12. 2025Cauchy Random Features for Operator Learning in Sobolev Spacecited by 3
11 more
  1. 2023Gaussian Process Regression under Computational and Epistemic Misspecificationcited by 2
  2. 2024Differentially Private Random Feature Modelcited by 2
  3. 2025Random feature approximation for general spectral methodscited by 2
  4. 2026A short tour of operator learning theory: Convergence rates, statistical limits, and open questionscited by 2
  5. 2025Regularized Random Fourier Features and Finite Element Reconstruction for Operator Learning in Sobolev Spacecited by 1
  6. 2026DeepAFL: Deep Analytic Federated Learningcited by 1
  7. 2023Random feature approximation for general spectral methodscited by 0
  8. 2025Joker: Joint Optimization Framework for Lightweight Kernel Machinescited by 0
  9. 2026Random Features for Operator-Valued Kernels: Bridging Kernel Methods and Neural Operatorscited by 0
  10. 2026Geometric Dictionary Learning of Dynamical Systems with Optimal Transportcited by 0
  11. 2026Random Neural Network Expressivity for Non-Linear Partial Differential Equationscited by 0