A Fractional Fox H-Function Kernel for Support Vector Machines: Robust Classification via Weighted Transmutation Operators

Support Vector Machines (SVMs) rely heavily on the choice of the kernel function to map data into high-dimensional feature spaces. While the Gaussian Radial Basis Function (RBF) is the industry standard, its exponential decay makes it highly susceptible to structural noise and outliers, often leading to severe overfitting in complex datasets. In this paper, we propose a novel class of non-stationary kernels derived from the fundamental solution of the generalized time-space fractional diffusion-wave equation. By leveraging a structure-preserving transmutation method over Weighted Sobolev Spaces, we introduce the Amnesia-Weighted Fox Kernel, an exact analytical Mercer kernel governed by the Fox H-function. Unlike standard kernels, our formulation incorporates an aging weight function (the "Amnesia Effect") to penalize distant outliers and a fractional asymptotic power-law decay to allow for robust, heavy-tailed feature mapping (analogous to Lévy flights). Numerical experiments on both synthetic datasets and real-world high-dimensional radar data (Ionosphere) demonstrate that the proposed Amnesia-Weighted Fox Kernel consistently outperforms the standard Gaussian RBF baseline, reducing the classification error rate by approximately 50\% while maintaining structural robustness against outliers.