Scalable Operator Learning via Nyström Approximation With Denoising Applications

2026-06-25

AI Summary

This work addresses the scalability limitations of traditional kernel methods, which require constructing and inverting large kernel matrices, and the lack of generality in existing denoising approaches that often rely on restrictive assumptions about signals or noise. The authors propose an efficient operator learning algorithm based on Nyström subsampling for vector-valued regression in reproducing kernel Hilbert spaces, unifying denoising within a general operator learning framework. They innovatively relax classical Hölder-type and operator monotonicity constraints by introducing an indicator function to characterize more general source conditions, and for the first time apply Nyström approximation systematically to operator learning with functional outputs and universal denoising tasks. Experiments demonstrate that the method achieves performance comparable to full-kernel approaches at substantially reduced computational cost across diverse applications—including signal, audio, and image denoising, Radon inversion reconstruction, and energy efficiency prediction—while attaining minimax optimal convergence rates.

Abstract

In this paper, we study Nyström 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 Nyström subsampling that accommodates functional outputs. Under general source conditions characterized by index functions-extending beyond the classical Hölder-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.

Problem

• scalability
• kernel methods
• operator learning
• denoising
• Nyström approximation

Innovation

• Nyström approximation
• operator learning
• vector-valued RKHS
• minimax-optimal convergence
• denoising