mlsanityData to 7 Oct 2026

Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problem

arXiv 2211.08875 ↗First posted 16 Nov 2022math.ST math.FA math.PRcited by 27

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Abstract (arXiv)

We consider the problem of learning a linear operator θ between two Hilbert spaces from empirical observations, which we interpret as least squares regression in infinite dimensions. We show that this goal can be reformulated as an inverse problem for θ with the feature that its forward operator is generally non-compact (even if θ is assumed to be compact or of p-Schatten class). However, we prove that, in terms of spectral properties and regularisation theory, this inverse problem is equivalent to the known compact inverse problem associated with scalar response regression. Our framework allows for the elegant derivation of dimension-free rates for generic learning algorithms under H"older-type source conditions. The proofs rely on the combination of techniques from kernel regression with recent results on concentration of measure for sub-exponential Hilbertian random variables. The obtained rates hold for a variety of practically-relevant scenarios in functional regression as well as nonlinear regression with operator-valued kernels and match those of classical kernel regression with scalar response.

Key facts. Cited by 27 papers in the map (data through 7 Oct 2026). Builds on 12 of its 55 references that are in the map. First posted on arXiv 16 Nov 2022.

Builds on

12 of 55 references.

References · 12

  1. 2020A Measure-Theoretic Approach to Kernel Conditional Mean Embeddingscited by 86
  2. 2016Optimal Rates For Regularization Of Statistical Inverse Learning Problemscited by 68
  3. 2022Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spacescited by 66
  4. 2015Operator-valued Kernels for Learning from Functional Response Datacited by 59
  5. 2018Optimal Rates for Spectral Algorithms with Least-Squares Regression over Hilbert Spacescited by 59
  6. 2021Convergence Rates for Learning Linear Operators from Noisy Datacited by 55
  7. 2022Optimal Rates for Regularized Conditional Mean Embedding Learningcited by 50
  8. 2014Learning Theory for Distribution Regressioncited by 48
  9. 2019A Rigorous Theory of Conditional Mean Embeddingscited by 31
  10. 2020Nonparametric approximation of conditional expectation operatorscited by 21
  11. 2022Minimax Optimal Kernel Operator Learning via Multilevel Trainingcited by 13
  12. 2020The linear conditional expectation in Hilbert spacecited by 8

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 9; 2024: 7; 2025: 6; 2026: 5.

27 papers.

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  1. 2024Operator Learning: Algorithms and Analysiscited by 84
  2. 2023Error Bounds for Learning with Vector-Valued Random Featurescited by 23
  3. 2023Towards Optimal Sobolev Norm Rates for the Vector-Valued Regularized Least-Squares Algorithmcited by 22
  4. 2024Operator Learning Using Random Features: A Tool for Scientific Computingcited by 21
  5. 2023Machine Learning for Partial Differential Equationscited by 20
  6. 2024Optimal Rates for Vector-Valued Spectral Regularization Learning Algorithmscited by 12
  7. 2024Nonparametric Control Koopman Operatorscited by 7
  8. 2024Nonparametric Instrumental Regression via Kernel Methods is Minimax Optimalcited by 7
  9. 2024Optimal Convergence Rates for Neural Operatorscited by 7
  10. 2023Inverse Problems with Learned Forward Operatorscited by 4
  11. 2023A randomized algorithm to solve reduced rank operator regressioncited by 4
  12. 2023Domain Generalization by Functional Regressioncited by 3
15 more
  1. 2023Online Infinite-Dimensional Regression: Learning Linear Operatorscited by 3
  2. 2025Regularized least squares learning with heavy-tailed noise is minimax optimalcited by 3
  3. 2023On regularized polynomial functional regressioncited by 1
  4. 2023Learned reconstruction methods for inverse problems: sample error estimatescited by 1
  5. 2024Kernel-Based Optimal Control: An Infinitesimal Generator Approachcited by 1
  6. 2025A Kernel-based Stochastic Approximation Framework for Nonlinear Operator Learningcited by 1
  7. 2026Verifiable Regularity Criterion for Conditional Expectation Operators and Conditional Mean Embeddings with Applications to Nonparametric Regression, Bayesian Inverse Problems, and Koopman Operatorscited by 1
  8. 2025Learning convolution operators on compact Abelian groupscited by 0
  9. 2025Kernel Mean Embedding Topology: Weak and Strong Forms for Stochastic Kernels and Implications for Model Learningcited by 0
  10. 2025Operator Learning for Schr"odinger Equation: Unitarity, Error Bounds, and Time Generalizationcited by 0
  11. 2025Operator Models for Continuous-Time Offline Reinforcement Learningcited by 0
  12. 2026Kernel-Based Learning of Safety Barrierscited by 0
  13. 2026Random Features for Operator-Valued Kernels: Bridging Kernel Methods and Neural Operatorscited by 0
  14. 2026Safety Certification is Classificationcited by 0
  15. 2026Optimal Multiscale Learning of Linear Operatorscited by 0