# Scalable Operator Learning via Nystr"om Approximation With Denoising Applications

arXiv: https://arxiv.org/abs/2606.26652 · HTML: https://mlsanity.com/paper/2606.26652/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 38 references that are in the map.
- First posted on arXiv 25 Jun 2026.

## Builds on (most cited first, 8 of 8)

- [Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising](https://mlsanity.com/paper/1608.03981/) (2016)
- [Distributed learning with regularized least squares](https://mlsanity.com/paper/1608.03339/) (2016)
- [Towards Optimal Sobolev Norm Rates for the Vector-Valued Regularized Least-Squares Algorithm](https://mlsanity.com/paper/2312.07186/) (2023)
- [Optimal Rates for Vector-Valued Spectral Regularization Learning Algorithms](https://mlsanity.com/paper/2405.14778/) (2024)
- [Optimal rates for the regularized learning algorithms under general source condition](https://mlsanity.com/paper/1611.01900/) (2016)
- [Analysis of regularized Nystr"om subsampling for regression functions of low smoothness](https://mlsanity.com/paper/1806.00826/) (2018)
- [Computational Efficiency under Covariate Shift in Kernel Ridge Regression](https://mlsanity.com/paper/2505.14083/) (2025)
- [Towards regularized learning from functional data with covariate shift](https://mlsanity.com/paper/2601.21019/) (2026)

## Abstract (arXiv)

In this paper, we study Nystr"om subsampling for vector-valued regression in vector-valued reproducing kernel Hilbert spaces. Standard kernel methods often suffer from prohibitive computational costs due to the construction and inversion of large kernel matrices, which limits their scalability to large datasets. To overcome this bottleneck, we propose an efficient operator learning algorithm based on Nystr"om subsampling that accommodates functional outputs. Under general source conditions characterized by index functions-extending beyond the classical H"older-type and operator-monotone frameworks-we establish minimax-optimal convergence rates for the proposed estimator. As an application of the proposed framework, we consider function denoising problems. Unlike classical denoising methods, which are typically tailored to specific signal representations or noise models, our approach formulates denoising within a general operator learning framework. Numerical experiments on signal denoising, real-time audio denoising, image denoising, inverse Radon transform reconstruction, and energy-efficiency prediction confirm that the proposed method achieves performance comparable to full kernel methods while substantially reducing computational cost.