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Robust kernel-free quadratic surface twin support vector machine with capped $L_1$-norm distance metric

Qi Si, Zhi Xia Yang

TL;DR

The paper tackles the sensitivity of twin SVMs to outliers and the burden of kernel choice by introducing a kernel-free quadratic surface TSVM that employs a capped $L_1$-norm distance. It combines this non-smooth loss with an $L_2$ regularization term and solved via an efficient re-weighted iterative algorithm, with convergence and time complexity analyzed. Empirical results across synthetic, benchmark, and image datasets demonstrate improved robustness and competitive accuracy, while the kernel-free design avoids kernel parameter tuning. This approach offers a practical, scalable alternative for robust pattern classification without kernel selection, with strong potential for extensions to multi-view, multi-class, and clustering tasks.

Abstract

Twin support vector machine (TSVM) is a very classical and practical classifier for pattern classification. However, the traditional TSVM has two limitations. Firstly, it uses the L_2-norm distance metric that leads to its sensitivity to outliers. Second, it needs to select the appropriate kernel function and the kernel parameters for nonlinear classification. To effectively avoid these two problems, this paper proposes a robust capped L_1-norm kernel-free quadratic surface twin support vector machine (CL_1QTSVM). The strengths of our model are briefly summarized as follows. 1) The robustness of our model is further improved by employing the capped L_1 norm distance metric. 2) Our model is a kernel-free method that avoids the time-consuming process of selecting appropriate kernel functions and kernel parameters. 3) The introduction of L_2-norm regularization term to improve the generalization ability of the model. 4) To efficiently solve the proposed model, an iterative algorithm is developed. 5) The convergence, time complexity and existence of locally optimal solutions of the developed algorithms are further discussed. Numerical experiments on numerous types of datasets validate the classification performance and robustness of the proposed model.

Robust kernel-free quadratic surface twin support vector machine with capped $L_1$-norm distance metric

TL;DR

The paper tackles the sensitivity of twin SVMs to outliers and the burden of kernel choice by introducing a kernel-free quadratic surface TSVM that employs a capped -norm distance. It combines this non-smooth loss with an regularization term and solved via an efficient re-weighted iterative algorithm, with convergence and time complexity analyzed. Empirical results across synthetic, benchmark, and image datasets demonstrate improved robustness and competitive accuracy, while the kernel-free design avoids kernel parameter tuning. This approach offers a practical, scalable alternative for robust pattern classification without kernel selection, with strong potential for extensions to multi-view, multi-class, and clustering tasks.

Abstract

Twin support vector machine (TSVM) is a very classical and practical classifier for pattern classification. However, the traditional TSVM has two limitations. Firstly, it uses the L_2-norm distance metric that leads to its sensitivity to outliers. Second, it needs to select the appropriate kernel function and the kernel parameters for nonlinear classification. To effectively avoid these two problems, this paper proposes a robust capped L_1-norm kernel-free quadratic surface twin support vector machine (CL_1QTSVM). The strengths of our model are briefly summarized as follows. 1) The robustness of our model is further improved by employing the capped L_1 norm distance metric. 2) Our model is a kernel-free method that avoids the time-consuming process of selecting appropriate kernel functions and kernel parameters. 3) The introduction of L_2-norm regularization term to improve the generalization ability of the model. 4) To efficiently solve the proposed model, an iterative algorithm is developed. 5) The convergence, time complexity and existence of locally optimal solutions of the developed algorithms are further discussed. Numerical experiments on numerous types of datasets validate the classification performance and robustness of the proposed model.
Paper Structure (23 sections, 4 theorems, 65 equations, 9 figures, 8 tables, 1 algorithm)

This paper contains 23 sections, 4 theorems, 65 equations, 9 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Notation
  4. Related models
  5. LSQTSVM
  6. FRTBSVM
  7. Capped $L_1$-norm
  8. The proposed method
  9. C$L_1$QTSVM
  10. Convergence analysis
  11. Time complexity
  12. Comparison methods
  13. Numerical experiments
  14. Experiment setups
  15. Experiments results
  16. ...and 8 more sections

Key Result

Lemma 1

For any non-negative variables $x,y\in\mathbb{R}_+$, the following inequality holds:

Figures (9)

  • Figure 1: The geometric interpretations of different norm metrics
  • Figure 2: The distribution of data for Example 1 with different label noise ratios.
  • Figure 3: The distribution of data for Example 2 with different label noise ratios.
  • Figure 4: The distribution of data for Example 3 with different label noise ratios.
  • Figure 5: Visualization results for four image datasets.
  • ...and 4 more figures

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof