Survey on Twin Support Vector Regression

DING Shi-fei; ZHANG Zi-chen; GUO Li-li; ZHANG Jian; XU Xiao

doi:10.12263/DZXB.20220703

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Survey on Twin Support Vector Regression

SURVEYS AND REVIEWS | 更新时间：2025-12-08

- Survey on Twin Support Vector Regression
- ACTA ELECTRONICA SINICA Vol. 51, Issue 4, Pages: 1117-1134(2023)
- 作者机构：
  
  1.中国矿业大学计算机科学与技术学院,江苏徐州 221116
  2.矿山数字化教育部工程研究中心(中国矿业大学),江苏徐州 221116
- 作者简介：
- 基金信息：
  
  National Natural Science Foundation of China(61976216;62276265)
- DOI：10.12263/DZXB.20220703
  CLC： TP311;
- Received：20 June 2022，
  
  Revised：2022-09-23，
  
  Published：25 April 2023
- 稿件说明：
移动端阅览
丁世飞,张子晨,郭丽丽等.孪生支持向量回归机研究进展[J].电子学报,2023,51(04):1117-1134.

DING Shi-fei,ZHANG Zi-chen,GUO Li-li,et al.Survey on Twin Support Vector Regression[J].ACTA ELECTRONICA SINICA,2023,51(04):1117-1134.
丁世飞,张子晨,郭丽丽等.孪生支持向量回归机研究进展[J].电子学报,2023,51(04):1117-1134. DOI： 10.12263/DZXB.20220703.

DING Shi-fei,ZHANG Zi-chen,GUO Li-li,et al.Survey on Twin Support Vector Regression[J].ACTA ELECTRONICA SINICA,2023,51(04):1117-1134. DOI： 10.12263/DZXB.20220703.

摘要

孪生支持向量回归机（Twin Support Vector Regression，TSVR or TWSVR）是一种基于统计学习理论的回归算法，它以结构风险最小化原理为理论基础，通过适当地选择函数子集及该子集中的判别函数，使学习机的实际风险达到最小，保证了在有限训练样本上得到的小误差分类器对独立测试集的测试误差仍然较小.孪生支持向量回归机通过将线性不可分样本映射到高维特征空间，使得映射后的样本在该高维特征空间内线性可分，保证了其具有较好的泛化性能.孪生支持向量回归机的算法思想基于孪生支持向量机（Twin Support Vector Machine，TWSVM），几何意义是使所有样本点尽可能地处于两条回归超平面的上（下）不敏感边界之间，最终的回归结果由两个超平面的回归值取平均得到.孪生支持向量回归机需求解两个规模较小的二次规划问题（Quadratic Programming Problems，QPPs）便可得到两条具有较小拟合误差的回归超平面，训练时间和拟合精度都高于传统的支持向量回归机（Support Vector Regression，SVR），且其QPPs的对偶问题存在全局最优解，避免了容易陷入局部最优的问题，故孪生支持向量回归机已成为机器学习的热门领域之一.但孪生支持向量回归机作为机器学习领域的一个较新的理论，其数学模型与算法思想都尚不成熟，在泛化性能、求解速度、矩阵稀疏性、参数选取、对偶问题等方面仍存在进一步改进的空间.本文首先给出了两种孪生支持向量回归机的数学模型与几何意义，然后将孪生支持向量回归机的几个常见的改进策略归纳如下.

（1）加权孪生支持向量回归机

由于孪生支持向量回归机中每个训练样本受到的惩罚是相同的，但每个样本对超平面的影响不同，尤其是噪声和离群值会使算法性能降低，并且在不同位置的训练样本应给予不同的处罚更为合理，因此考虑在孪生支持向量回归机的每个QPP中引入一个加权系数，给予不同位置的训练样本不同程度的惩罚.

（2）拉格朗日孪生支持向量回归机

由于孪生支持向量回归机的对偶问题中半正定矩阵的逆矩阵可能不存在，若存在，则对偶问题不是严格凸函数，可能存在多个解，因此考虑使用松弛变量的2范数代替原有的1范数，使对偶问题更简单，易于求解.

（3）最小二乘孪生支持向量回归机

由于孪生支持向量回归机的求解需要在对偶空间进行，得到的解为近似解，考虑通过最小二乘法将原问题的不等式约束转化为等式约束，使得原问题可以在原空间内求解，在很大程度上降低计算时间，提高泛化性能，且不损失精度.

（4）

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-孪生支持向量回归机

通过引入一组参数与

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自动调节

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与

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的值以控制训练样本的特定部分对两条回归超平面所能造成的最大误差，从而自适应给定数据的结构，提高孪生支持向量回归机的拟合精度.

（5）

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-孪生支持向量回归机

在孪生支持向量回归机的原问题中引入正则化项以达到结构风险最小化的目的，使对偶问题转化为稳定的正定二次规划问题，并通过SOR求解对偶问题，加快训练速度.

（6）孪生参数不敏感支持向量回归机

克服参数的选取对孪生支持向量回归机超平面构造的影响，使算法非常适合于存在异方差噪声数据的数据集，训练速度和泛化性能也有提升.

本文同时对以上算法的数学模型、改进算法及应用进行了系统地分析与总结，给出了以上算法在9个UCI基准数据集上的回归性能与计算时间，并在模型结构层面逐一分析每个算法的表现与耗时的根本原因.对于其他不便于归类的孪生支持向量回归机改进算法及应用，本文也对其作逐一总结.整体来看，最小二乘孪生支持向量回归机在性能和计算时间方面表现最佳，拉格朗日孪生支持向量回归机、

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-孪生支持向量回归机的性能并列次优且计算时间接近，加权孪生支持向量回归机、ε-孪生支持向量回归机和孪生参数不敏感支持向量回归机的性能不理想，但计算时间接近.本文旨在使读者对孪生支持向量回归机的不同改进算法之间的异同点与优缺点产生更深刻的理解与认识，从而将更多优秀的改进策略应用于孪生支持

向量回归机，最终为进一步提高孪生支持向量回归机的性能以及扩展孪生支持向量回归机的应用范围提供较为清晰的思路.

Abstract

Twin support vector regression is a regression algorithm based on statistical learning theory. It employs the theoretical principle of structural risk minimization; by appropriately selecting a subset of functions and obtaining the discriminant functions in that subset

it minimizes the actual risk of the learning machine

ensuring that the small test error of a classifier obtained on a limited training sample remains small on an independent test set. Twin support vector regression ensures better generalization performance because it maps linearly inseparable samples to a high-dimensional feature space

making the mapped samples linearly separable within that high-dimensional feature space. The algorithm of twin support vector regression is based on the twin support vector machine

and the geometric meaning of twin support vector regression is to make all sample points as far as possible between the upper (lower) insensitive boundaries of the two regression hyperplanes. The final regression result is then obtained by averaging the regression values of the two hyperplanes. In twin support vector regression

it is only necessary to solve two small-scale quadratic programming problems (QPPs) to obtain two regression hyperplanes with small fitting errors. Hence

its training time and fitting accuracy are higher than those of traditional support vector regression. Moreover

the dual problems of two QPPs have a globally optimal solution

and this makes it harder for the algorithm to become trapped in local optima. Hence

twin support vector regression has become a popular field of machine learning. However

there is still room for further improvement in the generalization performance

solution speed

matrix sparsity

parameter selection

and dual problem of twin support vector regression. In this paper

the mathematical models and geometric meanings of two types of twin support vector regression are presented; then

the following common improvement strategies for twin support vector regression are summarized:

(1) Weighted twin support vector regression

Because each training sample in twin support vector regression receives the same penalty but has different effects on the hyperplane

noise and outliers can degrade the performance of twin support vector regression. It is more reasonable for training samples at different locations to be given different penalties; thus

the introduction of a weighted factor in each QPP of twin support vector regression has been proposed to give training samples at different locations different degrees of penalties.

(2) Lagrange twin support vector regression

The inverse matrix of the semi-positive definite matrix in the dual problem of twin support vector regression may not exist. Even if it does

the dual problem is not a strictly convex function and may have multiple solutions. Thus the use of the 2-norm of the slack variables to replace the original 1-norm has been considered to make the dual problem simpler and easier to solve.

(3) Least squares twin support vector regression

Twin support vector regression needs to be solved in dual space

and the obtained solution is approximate. Hence

a method was proposed that transforms the inequality constraint of the original problem into an equation constraint using the least squares method so that the original problem can be solved in the original space. This substantially reduces the computation time and improves the generalization performance without loss of accuracy.

(4)

-twin support vector regression

This approach introduces a set of parameters

and

to control the maximum error that a particular part of the training sample can cause to the two regression hyperplanes. This adapts the hyperplanes to the structure of the given data and improves the fitting accuracy of the twin support vector r

egression.

(5) ε-twin support vector regression

A regularization term is introduced into the original formulation of twin support vector regression to minimize the structural risk so that the dual problem is transformed into a stable positive definite quadratic programming problem. The dual problem is solved by successive over relaxation (SOR) method to speed up the training.

(6) Twin parameter insensitive support vector regression

This method overcomes the effect of parameter selection on twin support vector regression

making the algorithm well suited for data sets that contain heteroscedastic noisy data as well as enhancing training speed and generalization performance.

This paper also systematically analyzes and summarizes the mathematical models

improved algorithms

and applications of the above algorithms; analyzes the regression performance and calculation time of the above algorithms on nine UCI benchmark datasets through experiments; and analyzes the root cause of each algorithm’s performance and time consumption at the level of model structure. This paper also summarizes other improved twin support vector regression algorithms and applications of twin support vector regression that are not easy to categorize. Overall

the least squares twin support vector regression performs the best in terms of performance and computation time. Lagrangian twin support vector regression and

-twin support vector regression both obtain the next best performance and have similar computation times. Weighted twin support vector regression

ε-twin support vector regression

and twin parameter insensitive support vector regression do not perform well

but have similar computation times. The purpose of this paper is to provide readers with a deeper understanding and perception of the similarities

differences

advantages

and disadvantages of different improved algorithms based on twin support vector regression

so that they may apply better improvement strategies to twin support

vector regression. Moreover

this paper aims to eventually provide a clear idea for further improving the performance of twin support vector regression and expanding its range of applications.

关键词

Keywords

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