# Online Infinite-Dimensional Regression: Learning Linear Operators

arXiv: https://arxiv.org/abs/2309.06548 · HTML: https://mlsanity.com/paper/2309.06548/

## 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 (most cited first, 12 of 12)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [VC Classes are Adversarially Robustly Learnable, but Only Improperly](https://mlsanity.com/paper/1902.04217/) (2019)
- [Relative Loss Bounds for On-line Density Estimation with the Exponential Family of Distributions](https://mlsanity.com/paper/1301.6677/) (2013)
- [Convergence Rates for Learning Linear Operators from Noisy Data](https://mlsanity.com/paper/2108.12515/) (2021)
- [Operator learning with PCA-Net: upper and lower complexity bounds](https://mlsanity.com/paper/2303.16317/) (2023)
- [Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problem](https://mlsanity.com/paper/2211.08875/) (2022)
- [Multiclass Online Learning and Uniform Convergence](https://mlsanity.com/paper/2303.17716/) (2023)
- [Generalized Deep Image to Image Regression](https://mlsanity.com/paper/1612.03268/) (2016)
- [Learning Schatten--von Neumann Operators](https://mlsanity.com/paper/1901.10076/) (2019)

## Built on by (most cited first, 3 of 3)

- [Controlling Statistical, Discretization, and Truncation Errors in Learning Fourier Linear Operators](https://mlsanity.com/paper/2408.09004/) (2024)
- [Towards Sharp Minimax Risk Bounds for Operator Learning](https://mlsanity.com/paper/2512.17805/) (2025)
- [Optimal Multiscale Learning of Linear Operators](https://mlsanity.com/paper/2606.16913/) (2026)

## 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.