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

Gustavo Dorrego

Abstract

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.

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

Abstract

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.
Paper Structure (15 sections, 1 theorem, 4 equations, 4 figures, 1 table)

This paper contains 15 sections, 1 theorem, 4 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Mathematical Framework
  3. Weighted Spaces and the Transmutation Operator
  4. The Weighted Fractional Laplacian and Fox H-Function
  5. Derivation of the Fox-Dorrego Fractional Kernel
  6. The Asymptotic Fox-Dorrego Kernel
  7. Proof of Mercer's Condition for the Asymptotic Kernel
  8. Numerical Experiments
  9. Experiment 1: Extreme Structural Noise
  10. Experiment 2: Real-World Generalization in High Dimensions
  11. Experiment 3: Complex Topological Boundaries
  12. Hyperparameter Sensitivity Analysis
  13. Computational Complexity and Execution Time
  14. Case Study: Radar Signal Classification (Ionosphere Dataset)
  15. Conclusions

Key Result

Theorem 3.1

For any $s > 0$, $\lambda > 0$, and strictly positive weight function $\omega(x)$, the Asymptotic Fox-Dorrego Kernel $K_{FD}$ is a valid Mercer kernel.

Figures (4)

  • Figure 1: Comparison of SVM decision boundaries under extreme structural noise.
  • Figure 2: Confusion matrices for the 30-dimensional Breast Cancer Wisconsin dataset. Left: The highly optimized Classical RBF SVM achieves $97.66\%$ accuracy in this standard noise-free environment. Right: The proposed Fractional Fox-Dorrego SVM maintains a highly competitive $95.91\%$ accuracy. The marginal difference reflects the inherent "robustness trade-off", where the fractional kernel preserves its active amnesia weights to defend against potential structural anomalies in real-world deployment.
  • Figure 3: Topological robustness on the noisy Two Moons dataset. Left: The RBF kernel exhibits severe local overfitting, fracturing the decision boundary into isolated "bubbles" around noisy data points due to its exponential decay. Right: The proposed Fox-Dorrego kernel ($s=0.5$) leverages fractional wave-diffusion and heavy tails to understand the global S-shaped topology, maintaining a mathematically pure boundary while ignoring local perturbations.
  • Figure 4: Heatmap demonstrating hyperparameter sensitivity. The region of optimal accuracy (lighter colors) reveals a broad and stable operational zone for the Fox-Dorrego kernel, effectively balancing the heavy-tailed diffusion ($s$) and the amnesia effect ($\eta$).

Theorems & Definitions (2)

  • Theorem 3.1
  • proof