Weighted Spectral Filters for Kernel Interpolation on Spheres: Estimates of Prediction Accuracy for Noisy Data

TL;DR

This paper introduces a weighted spectral filter method to improve the stability and accuracy of kernel interpolation on spheres, especially for noisy data, by reducing the kernel matrix's condition number.

Contribution

The paper presents a novel weighted spectral filter approach that enhances kernel interpolation stability on spheres, particularly under noisy conditions, using spherical quadrature rules and spectral filtering.

Findings

Successfully stabilizes kernel interpolation with noisy data

Achieves optimal approximation rates without sacrificing accuracy

Validated through simulations and real-world geophysical and climate data

Abstract

Spherical radial-basis-based kernel interpolation abounds in image sciences including geophysical image reconstruction, climate trends description and image rendering due to its excellent spatial localization property and perfect approximation performance. However, in dealing with noisy data, kernel interpolation frequently behaves not so well due to the large condition number of the kernel matrix and instability of the interpolation process. In this paper, we introduce a weighted spectral filter approach to reduce the condition number of the kernel matrix and then stabilize kernel interpolation. The main building blocks of the proposed method are the well developed spherical positive quadrature rules and high-pass spectral filters. Using a recently developed integral operator approach for spherical data analysis, we theoretically demonstrate that the proposed weighted spectral filter approach succeeds in breaking through the bottleneck of kernel interpolation, especially in fitting noisy data. We provide optimal approximation rates of the new method to show that our approach does not compromise the predicting accuracy. Furthermore, we conduct both toy simulations and two real-world data experiments with synthetically added noise in geophysical image reconstruction and climate image processing to verify our theoretical assertions and show the feasibility of the weighted spectral filter approach.