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

Standard Gradient Descent and its modern variants assume local, Markovian weight updates, making them highly susceptible to noise and overfitting. This limitation becomes critically severe in extremely imbalanced datasets such as financial fraud detection where dominant class gradients systematically overwrite the subtle signals of the minority class. In this paper, we introduce a novel optimization algorithm grounded in Fractional Calculus. By isolating the core memory engine of the generalized fractional derivative, the Weighted Fractional Weyl Integral, we replace the instantaneous gradient with a dynamically weighted historical sequence. This fractional memory operator acts as a natural regularizer. Empirical evaluations demonstrate that our method prevents overfitting in medical diagnostics and achieves an approximately 40 percent improvement in PR-AUC over classical optimizers in financial fraud detection, establishing a robust bridge between pure fractional topology and applied Machine Learning.

The theoretical elegance of Support Vector Machines (SVMs), grounded in statistical learning theory, establishes them as a reference for robust classification. However, in the era of Big Data, classical SVMs face two major bottlenecks: the prohibitive cubic complexity $\mathcal{O}(N^3)$ of training and the rigidity of static kernels. Traditional methodologies treat kernel selection as a discrete, "brute-force" GridSearch problem, effectively decoupling the learning of data representation from the optimization of the decision boundary. This paper proposes a paradigm shift with a so-called "Revisited SVM", a unified framework that recasts the learning process as a continuous %bi-level joint optimization problem. We derive a novel objective function $J(α, θ)$ that allows for the simultaneous optimization of dual Lagrange multipliers and kernel hyper-parameters. To solve this challenging non-convex problem efficiently, we introduce a hybrid Block Coordinate Ascent (BCA) algorithm, tightly integrated with Nyström-based low-rank approximation, which drastically reduces computational complexity to $\mathcal{O}(Nm)$. Extensive experiments conducted across four heterogeneous domains (Bioinformatics, Finance, Cybersecurity, and Computer Vision) demonstrate that our approach not only matches but often exceeds exhaustive GridSearch accuracy, while delivering training speedups of up to 100x. Notably, on the complex NSL-KDD intrusion detection dataset, our adaptive kernel approach achieves 96.25% accuracy compared to a mere 50% for standard baselines, proving its unique ability to model complex non-linear manifolds in high-dimensional spaces. We validate our results using rigorous statistical tests and provide a theoretical convergence analysis of the proposed algorithm.

Kernel methods are typically formulated under the assumption of exact, noise-free access to the Gram matrix. However, in emerging settings such as quantum machine learning, each kernel entry must be inferred from noisy observations, and its accuracy depends on how a limited measurement budget is allocated. Despite this, existing approaches overwhelmingly rely on uniform allocation, which equalizes estimator variance but ignores the highly non-uniform dependence of kernelized classifiers on the Gram matrix. In this work, we introduce an adaptive measurement-allocation strategy for learning kernelized Support Vector Machines (SVMs) from noisy Bernoulli observations. Our approach combines two complementary principles: (i) geometric sensitivity, capturing how perturbations of individual kernel entries affect the classifier margin, and (ii) active-set instability, quantifying the probability of discrete changes in support-vector membership induced by measurement noise. These signals define a task-aware allocation scheme that concentrates measurements on the most decision-critical regions of the kernel matrix. We provide a theoretical analysis showing that the benefit of adaptive allocation is governed by the heterogeneity of the induced kernel importance structure, leading to distinct regimes in which adaptive or uniform strategies are preferable. Empirical evaluations on synthetic datasets demonstrate that adaptive allocation significantly improves support-vector recovery, margin estimation, and decision-function accuracy under fixed measurement budgets. A dual-coefficient stability criterion further enables early stopping, achieving near-optimal performance while using only a fraction of the measurement cost. Additional experiments on quantum kernels derived from real-world data reveal a regime-dependent behavior aligned with known phenomena such as kernel concentration. Together...