Orthonormal RBF wavelet and ridgelet-like series and transforms for high-dimensional problems

TL;DR

This paper introduces a systematic framework for RBF-based wavelet and ridgelet transforms, enabling high-dimensional data analysis through novel transforms derived from differential operators and extending existing RBFs.

Contribution

It develops a comprehensive strategy for RBF wavelet analysis, including new transforms based on differential operators and extensions to known RBFs for high-dimensional problems.

Findings

Harmonic Bessel RBF transforms for high-dimensional data

Feasibility of ridgelet-like RBF transforms from convection-diffusion kernels

Extension of methodology to classical RBFs like MQ and Gaussian

Abstract

This paper developed a systematic strategy establishing RBF on the wavelet analysis, which includes continuous and discrete RBF orthonormal wavelet transforms respectively in terms of singular fundamental solutions and nonsingular general solutions of differential operators. In particular, the harmonic Bessel RBF transforms were presented for high-dimensional data processing. It was also found that the kernel functions of convection-diffusion operator are feasible to construct some stable ridgelet-like RBF transforms. We presented time-space RBF transforms based on non-singular solution and fundamental solution of time-dependent differential operators. The present methodology was further extended to analysis of some known RBFs such as the MQ, Gaussian and pre-wavelet kernel RBFs.