基于原型超平面的多类最接近支持向量机

基于广义特征值的最接近支持向量机（proximal support vector machine via generalized eigenvalues，GEPSVM）摒弃了传统意义下支持向量机典型平面的平行约束，代之以通过优化使每类原型平面尽可能接近本类样本，同时尽可能远离它类样本的准则来解析获得原型平面；从而避免了SVM的二次规划，其分类性能达到甚至超过了SVM．但GEPSVM仍存在如下不足：①仅对两分类问题而提出，无法直接求解多分类问题；②存在正则化因子的选择问题；③求解原型平面的广义特征值问题中所涉及的矩阵一般仅为半正定，容易导致奇异性问题．通过定义新的准则，构建了一个能直接求解多个原型超平面的多分类方法，称之为基于原型超平面的多类最接近支持向量机，较之GEPSVM，该方法优势在于：①无正则化因子选择的困扰；②可同时求解多个超平面，对两分类问题，分类性能达到甚至优于GEPSVM；③超平面的选择问题转化为简单特征值而非广义特征值求解问题；④原型平面的选择只依赖于本类样本，故不必考虑多分类情形时的数据不平衡问题．

Proximal Support Vector Machine Based on Prototypal Multiclassfication Hyperplanes

Proximal support vector machine via generalized eigenvalues (GEPSVM) casts away the parallelism condition on the canonical planes of the traditional support vector machines (SVM) and analytically seeks two hyperplanes such that each plane is close to the samples of its class and meanwhile far away from the samples of the other classes. Compared with the SVM,GEPSVM does not need quadratic programming and can gain comparable classification performance to SVM. Despite these advantages,GEPSVM is a binary classifier and can not separate multi-class datasets directly. Moreover,it is hard to theoretically set the regularization parameter in it and the generalized eigen-equation problem may be ill-conditioned. In this paper,a novel method,proximal SVM based on prototypal multi-classification hyperplanes (MHPSVM) is proposed,which can directly obtain multi-prototypal hyperplanes for multiple-class classification. Finally,experimental results on both artificial and benchmark datasets show that the classification performance of MHPSVM can be significantly higher than that of GEPSVM,especially in multi-class classification.