Kernel-free quadratic proximal support vector machine with Lp-norm regularization

For binary classification problems, real-world data often exhibit class imbalance, noise, and outliers, and their complex distribution requires the use of kernel functions to separate the data non-linearly. In this paper, we propose a novel nonlinear classifier, called the kernel-free quadratic proximal support vector machine with an arbitrary $L_p$-norm regularization (L_p-QPSVM), where $p>0$. The goal of our L_p-QPSVM is to find two quadratic hypersurfaces to non-linearly classify. By introducing the L_p-norm regularization term in our method, L_p-QPSVM allows for flexible adjustment of $p$, enhancing its robustness and generalization. To strengthen practical applications, a simplified version of L_p-QPSVM is proposed. Additionally, we transform the two optimization problems of L_p-QPSVM into the convex quadratic programming problems, and design an iterative algorithm to solve them. The convergence, interpretability and computational complexity of L_p-QPSVM are provided. Numerical experiments on the artificial and benchmark datasets validate the effectiveness of our L_p-QPSVM, demonstrating its state-of-the-art performance compared with the representative classification methods.