# Learning convolution operators on compact Abelian groups

arXiv: https://arxiv.org/abs/2501.05279 · HTML: https://mlsanity.com/paper/2501.05279/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 47 references that are in the map.
- First posted on arXiv 9 Jan 2025.

## Builds on (most cited first, 10 of 10)

- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Physics-Informed Deep Neural Operator Networks](https://mlsanity.com/paper/2207.05748/) (2022)
- [Learning to Deblur](https://mlsanity.com/paper/1406.7444/) (2014)
- [Operator-valued Kernels for Learning from Functional Response Data](https://mlsanity.com/paper/1510.08231/) (2015)
- [Convergence Rates for Learning Linear Operators from Noisy Data](https://mlsanity.com/paper/2108.12515/) (2021)
- [Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problem](https://mlsanity.com/paper/2211.08875/) (2022)
- [Learning Green's functions associated with time-dependent partial differential equations](https://mlsanity.com/paper/2204.12789/) (2022)
- [On the Sample Complexity of Subspace Learning](https://mlsanity.com/paper/1408.5032/) (2014)
- [Learning Schatten--von Neumann Operators](https://mlsanity.com/paper/1901.10076/) (2019)
- [Bayesian inverse problems with unknown operators](https://mlsanity.com/paper/1801.09894/) (2018)

## Abstract (arXiv)

We consider the problem of learning convolution operators associated to compact Abelian groups. We study a regularization-based approach and provide corresponding learning guarantees under natural regularity conditions on the convolution kernel. More precisely, we assume the convolution kernel is a function in a translation invariant Hilbert space and analyze a natural ridge regression (RR) estimator. Building on existing results for RR, we characterize the accuracy of the estimator in terms of finite sample bounds. Interestingly, regularity assumptions which are classical in the analysis of RR, have a novel and natural interpretation in terms of space/frequency localization. Theoretical results are illustrated by numerical simulations.