mlsanityData to 14 Oct 2026

Online Infinite-Dimensional Regression: Learning Linear Operators

arXiv 2309.06548 ↗First posted 8 Sep 2023stat.ML cs.LGcited by 3

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Abstract (arXiv)

We consider the problem of learning linear operators under squared loss between two infinite-dimensional Hilbert spaces in the online setting. We show that the class of linear operators with uniformly bounded p-Schatten norm is online learnable for any p ∈ [1, ∞). On the other hand, we prove an impossibility result by showing that the class of uniformly bounded linear operators with respect to the operator norm is textitnot online learnable. Moreover, we show a separation between sequential uniform convergence and online learnability by identifying a class of bounded linear operators that is online learnable but uniform convergence does not hold. Finally, we prove that the impossibility result and the separation between uniform convergence and learnability also hold in the batch setting.

Key facts. Cited by 3 papers in the map (data through 14 Oct 2026). Builds on 12 of its 44 references that are in the map. First posted on arXiv 8 Sep 2023.

Builds on

12 of 44 references.

References · 12

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  2. 2021Neural Operator: Learning Maps Between Function Spacescited by 414
  3. 2020Model Reduction and Neural Networks for Parametric PDEscited by 258
  4. 2019Data-driven approximation of the Koopman generator: Model reduction, system identification, and controlcited by 85
  5. 2019VC Classes are Adversarially Robustly Learnable, but Only Improperlycited by 83
  6. 2013Relative Loss Bounds for On-line Density Estimation with the Exponential Family of Distributionscited by 60
  7. 2021Convergence Rates for Learning Linear Operators from Noisy Datacited by 55
  8. 2023Operator learning with PCA-Net: upper and lower complexity boundscited by 44
  9. 2022Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problemcited by 28
  10. 2023Multiclass Online Learning and Uniform Convergencecited by 19
  11. 2016Generalized Deep Image to Image Regressioncited by 14
  12. 2019Learning Schatten--von Neumann Operatorscited by 7

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 1; 2025: 1; 2026: 1.

3 papers.

  1. 2024Controlling Statistical, Discretization, and Truncation Errors in Learning Fourier Linear Operatorscited by 5
  2. 2025Towards Sharp Minimax Risk Bounds for Operator Learningcited by 5
  3. 2026Optimal Multiscale Learning of Linear Operatorscited by 1