Nonparametric regression using needlet kernels for spherical data

TL;DR

This paper develops needlet kernel methods for nonparametric regression on spherical data, demonstrating their rate optimality and robustness across various regularization strategies due to their excellent localization properties.

Contribution

It introduces kernel methods based on needlet kernels for spherical data, proving their rate optimality and showing that the choice of regularization parameter q has minimal impact on generalization.

Findings

Regularization parameter can decrease arbitrarily fast.

Kernel ridge regression may not require regularization for optimal rates.

Different q-regularization estimates have similar generalization capabilities.

Abstract

Needlets have been recognized as state-of-the-art tools to tackle spherical data, due to their excellent localization properties in both spacial and frequency domains. This paper considers developing kernel methods associated with the needlet kernel for nonparametric regression problems whose predictor variables are defined on a sphere. Due to the localization property in the frequency domain, we prove that the regularization parameter of the kernel ridge regression associated with the needlet kernel can decrease arbitrarily fast. A natural consequence is that the regularization term for the kernel ridge regression is not necessary in the sense of rate optimality. Based on the excellent localization property in the spacial domain further, we also prove that all the $l^{q}$ $(01\leq q < \infty)$ kernel regularization estimates associated with the needlet kernel, including the kernel lasso estimate and the kernel bridge estimate, possess almost the same generalization capability for a large range of regularization parameters in the sense of rate optimality. This finding tentatively reveals that, if the needlet kernel is utilized, then the choice of $q$ might not have a strong impact in terms of the generalization capability in some modeling contexts. From this perspective, $q$ can be arbitrarily specified, or specified merely by other no generalization criteria like smoothness, computational complexity, sparsity, etc..