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

TL;DR

This work tackles the instability of kernel interpolation on the sphere under noisy observations, which stems from large condition numbers of the kernel matrix $\\Phi_D$.It introduces Weighted Spectral Filters (WSF) that combine positive spherical quadrature weights with high-pass spectral filters $g_\lambda$, forming the stabilized estimator $f_{D,\lambda}$ and its Ada-WSF variant via weighted cross-validation.Theoretical results establish rate-optimal approximation guarantees for WSF under deterministic spherical sampling, decomposing error into stability, fitting, and approximation, and showing that the optimal rate $|D|^{-\\frac{\\gamma\\alpha}{2\\gamma\\alpha+d}}$ is achievable with appropriate scaling of $\lambda$ and quadrature order.Extensive toy and real-data experiments (geophysical geomagnetic and climate wind-field data) validate improved stability and predictive accuracy over classical kernel interpolation, and provide practical parameter-selection guidance for noisy spherical 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.

Figures (13)

• Figure 1: Different roles of the samples in KI on the sphere
• Figure 1: Relations between training sample size and RMSE (rooted mean square error) of KI/CNKM (condition number of kernel matrix)/$\sigma_{|D|}(\Phi_D)$. The training sample inputs $\{x_i\}_{i=1}^{|D|}$ are generated by Womersley's symmetric spherical $t$-design on $\mathbb S^2$, and the corresponding outputs are generated by the model \ref{['Model1:fixed']}, where $f^*$ is defined by \ref{['Model_toysimulation']} and $\varepsilon_i$ are independent truncated Gaussian noise $\mathcal{N}(0,\delta^2)$. We ran simulations 5 times and recorded the average RMSE using the kernel given as in \ref{['Wendland_function']}.
• Figure 1: WSF to reduce the CNKM for rotated data with $|D|=1200$ and truncated Gaussian noise with standard deviation $\delta=0.5$.
• Figure 2: Relations between training sample size and RMSE of KI/CNKM/$\tau_{\Lambda}$. The training sample inputs $\{x_i\}_{i=1}^{|D|}$ are generated by rotating $t$-designs on $\mathbb S^2$ whose details can be found in \ref{['Sec.Numerical']}, and the corresponding outputs are generated by the same way as in \ref{['fig: drawbackofKI_tdesign']}. We ran simulations 5 times and recorded the average RMSE.
• Figure 2: WSF to reduce the stability error with truncated Gaussian noise with $\delta=0.5$.

Theorems & Definitions (19)

• Lemma 2.1
• Lemma 2.2
• Definition 2.3
• Definition 3.1
• Theorem 3.2
• Corollary 3.3
• Corollary 3.4
• Corollary 3.5
• Theorem 3.6
• Corollary 3.7