mlsanityData to 14 Oct 2026

Convergence Rates for Learning Linear Operators from Noisy Data

arXiv 2108.12515 ↗First posted 27 Aug 2021math.ST cs.LG stat.MEcited by 55

Not on the map. Why?

Abstract (arXiv)

This paper studies the learning of linear operators between infinite-dimensional Hilbert spaces. The training data comprises pairs of random input vectors in a Hilbert space and their noisy images under an unknown self-adjoint linear operator. Assuming that the operator is diagonalizable in a known basis, this work solves the equivalent inverse problem of estimating the operator's eigenvalues given the data. Adopting a Bayesian approach, the theoretical analysis establishes posterior contraction rates in the infinite data limit with Gaussian priors that are not directly linked to the forward map of the inverse problem. The main results also include learning-theoretic generalization error guarantees for a wide range of distribution shifts. These convergence rates quantify the effects of data smoothness and true eigenvalue decay or growth, for compact or unbounded operators, respectively, on sample complexity. Numerical evidence supports the theory in diagonal and non-diagonal settings.

Key facts. Cited by 55 papers in the map (data through 14 Oct 2026). Builds on 25 of its 87 references that are in the map. First posted on arXiv 27 Aug 2021.

Builds on

25 of 87 references.

References · 25

  1. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  2. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 734
  3. 2021Neural Operator: Learning Maps Between Function Spacescited by 414
  4. 2020The Random Feature Model for Input-Output Maps between Banach Spacescited by 90
  5. 2019Data-driven approximation of the Koopman generator: Model reduction, system identification, and controlcited by 85
  6. 2020A physics-informed operator regression framework for extracting data-driven continuum modelscited by 60
  7. 2021Can stable and accurate neural networks be computed? -- On the barriers of deep learning and Smale's 18th problemcited by 53
  8. 2017Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spacescited by 48
  9. 2020Derivative-Informed Projected Neural Networks for High-Dimensional Parametric Maps Governed by PDEscited by 40
  10. 2019A Rigorous Theory of Conditional Mean Embeddingscited by 31
  11. 2021Learning elliptic partial differential equations with randomized linear algebracited by 29
  12. 2021Two-layer neural networks with values in a Banach spacecited by 26
13 more
  1. 2021Learning the optimal Tikhonov regularizer for inverse problemscited by 26
  2. 2020Deep Neural Networks Are Effective At Learning High-Dimensional Hilbert-Valued Functions From Limited Datacited by 22
  3. 2020Nonparametric approximation of conditional expectation operatorscited by 21
  4. 2019Solving Electrical Impedance Tomography with Deep Learningcited by 18
  5. 2019Deep learning architectures for nonlinear operator functions and nonlinear inverse problemscited by 17
  6. 2020Deep neural networks for inverse problems with pseudodifferential operators: an application to limited-angle tomographycited by 15
  7. 2019Excess risk bounds in robust empirical risk minimizationcited by 9
  8. 2020Functional Linear Regression with Mixed Predictorscited by 8
  9. 2019Learning Schatten--von Neumann Operatorscited by 7
  10. 2019Convergence analysis of Tikhonov regularization for non-linear statistical inverse learning problemscited by 7
  11. 2017Efficient Nonparametric Bayesian Inference For X-Ray Transformscited by 4
  12. 2018Bayesian inverse problems with unknown operatorscited by 4
  13. 2021Bayesian inference of an uncertain generalized diffusion operatorcited by 3

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2021: 1; 2022: 10; 2023: 12; 2024: 15; 2025: 9; 2026: 8.

55 papers.

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  1. 2022Physics-Informed Machine Learning: A Survey on Problems, Methods and Applicationscited by 92
  2. 2024Operator Learning: Algorithms and Analysiscited by 85
  3. 2023A Mathematical Guide to Operator Learningcited by 76
  4. 2022Deep Nonparametric Estimation of Operators between Infinite Dimensional Spacescited by 45
  5. 2023Operator learning with PCA-Net: upper and lower complexity boundscited by 44
  6. 2023Nonlocality and Nonlinearity Implies Universality in Operator Learningcited by 43
  7. 2023Out-of-distributional risk bounds for neural operators with applications to the Helmholtz equationcited by 32
  8. 2022Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problemcited by 28
  9. 2023Uncertainty quantification for noisy inputs-outputs in physics-informed neural networks and neural operatorscited by 28
  10. 2022Learning Green's functions associated with time-dependent partial differential equationscited by 26
  11. 2023Elliptic PDE learning is provably data-efficientcited by 26
  12. 2023Error Bounds for Learning with Vector-Valued Random Featurescited by 23
38 more
  1. 2023Machine Learning for Partial Differential Equationscited by 21
  2. 2024Operator Learning Using Random Features: A Tool for Scientific Computingcited by 21
  3. 2023Deep Operator Learning Lessens the Curse of Dimensionality for PDEscited by 18
  4. 2025Operator Learning: A Statistical Perspectivecited by 17
  5. 2021Learning High-Dimensional Parametric Maps via Reduced Basis Adaptive Residual Networkscited by 16
  6. 2024MODNO: Multi Operator Learning With Distributed Neural Operatorscited by 14
  7. 2022Minimax Optimal Kernel Operator Learning via Multilevel Trainingcited by 13
  8. 2022Sobolev Acceleration and Statistical Optimality for Learning Elliptic Equations via Gradient Descentcited by 11
  9. 2024Spectral-Refiner: Accurate Fine-Tuning of Spatiotemporal Fourier Neural Operator for Turbulent Flowscited by 11
  10. 2022On the sample complexity of stabilizing linear dynamical systems from datacited by 8
  11. 2022Approximation of Functionals by Neural Network without Curse of Dimensionalitycited by 8
  12. 2024Operator learning without the adjointcited by 8
  13. 2024Nonlinear functional regression by functional deep neural network with kernel embeddingcited by 7
  14. 2023Operator learning for hyperbolic partial differential equationscited by 5
  15. 2024Controlling Statistical, Discretization, and Truncation Errors in Learning Fourier Linear Operatorscited by 5
  16. 2025Towards Sharp Minimax Risk Bounds for Operator Learningcited by 5
  17. 2023Inverse Problems with Learned Forward Operatorscited by 4
  18. 2025Theory-to-Practice Gap for Neural Networks and Neural Operatorscited by 4
  19. 2023Online Infinite-Dimensional Regression: Learning Linear Operatorscited by 3
  20. 2024Learning a Gaussian Mixture for Sparsity Regularization in Inverse Problemscited by 3
  21. 2024On the estimation rate of Bayesian PINN for inverse problemscited by 3
  22. 2024Structure-Preserving Operator Learningcited by 3
  23. 2024Transfer Learning on Multi-Dimensional Data: A Novel Approach to Neural Network-Based Surrogate Modelingcited by 3
  24. 2024Learning sparsity-promoting regularizers for linear inverse problemscited by 3
  25. 2025Learning where to learn: Training data distribution optimization for scientific machine learningcited by 3
  26. 2025Method of Manufactured Learning for Solver-free Training of Neural Operatorscited by 3
  27. 2024Spectrally Transformed Kernel Regressioncited by 2
  28. 2025Deep Symmetric Autoencoders from the Eckart-Young-Schmidt Perspectivecited by 2
  29. 2022A Data-Adaptive Prior for Bayesian Learning of Kernels in Operatorscited by 1
  30. 2025Rates and architectures for learning geometrically non-trivial operatorscited by 1
  31. 2026Supervised and Unsupervised Neural Network Solver for First Order Hyperbolic Nonlinear PDEscited by 1
  32. 2026Optimal Multiscale Learning of Linear Operatorscited by 1
  33. 2022Data-driven discovery of Green's functionscited by 0
  34. 2024Benign overfitting in Fixed Dimension via Physics-Informed Learning with Smooth Inductive Biascited by 0
  35. 2024A DeepONet for inverting the Neumann-to-Dirichlet Operator in Electrical Impedance Tomography: An approximation theoretic perspective and numerical resultscited by 0
  36. 2025Learning convolution operators on compact Abelian groupscited by 0
  37. 2025Operator Learning for Schr"odinger Equation: Unitarity, Error Bounds, and Time Generalizationcited by 0
  38. 2026Interpretable Operator Learning for Inverse Problems via Adaptive Spectral Filtering: Convergence and Discretization Invariancecited by 0

50 most cited of 55 All 55 (JSON)