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Relationship Between Support Vector Set and Kernel Functions in SVM
作者姓名:张铃  张钹
作者单位:[1]ArtificialIntelligenceInstitute,AnhuiUniversity,Anhui230039,P.R.China [2]DepartmentofComputerScience,TsinghuaUniversity,Beijing100084,P.R.China
基金项目:National Key Basic Research Program,国家自然科学基金 
摘    要:Based on a constructive learning approach,covering algorithms,we investigate the relationship between support vector sets and kernel functions in support vector machines (SVM).An interesting result is obtained.That is,in the linearly non-separable case,any sample of a given sample set K can become a support vector under a certain kernel function.The result shows that when the sample set K is linearly non-separable,although the chosen kernel function satisfies Mercer‘s condition its corresponding support vector set is not necessarily the subset of K that plays a crucial role in classifying K.For a given sample set,what is the subset that plays the crucial role in classification?In order to explore the problem,a new concept,boundary or boundary points,is defined and its properties are discussed.Given a sample set K,we show that the decision functions for classifying the boundary points of K are the same as that for classifying the K itself.And the boundary points of K only depend on K and the structure of the space at which k is located and independent of the chosen approach for finding the boundary.Therefore,the boundary point set may become the subset of K that plays a crucial role in classification.These results are of importance to understand the principle of the support vector machine(SVM) and to develop new learning algorithms.

关 键 词:机器学习  支持向量机  学习理论

Relationship between support vector set and kernel functions in SVM
Ling Zhang,Bo Zhang.Relationship Between Support Vector Set and Kernel Functions in SVM[J].Journal of Computer Science and Technology,2002,17(5):0-0.
Authors:Ling Zhang  Bo Zhang
Affiliation:(1) Artificial Intelligence Institute, Anhui University, 230039 Anhui, P.R. China;(2) Department of Computer Science, Tsinghua University, 100084 Beijing, P.R. China;(3) State Key Lab of Intelligent Technology and System, Tsinghua University, 100084 Beijing, P.R. China
Abstract:Based on a constructive learning approach, covering algorithms, we investigate the relationship between support vector sets and kernel functions in support vector machines (SVM). An interesting result is obtained. That is, in the linearly non-separable case, any sample of a given sample setK can become a support vector under a certain kernel function. The result shows that when the sample setK is linearly non-separable, although the chosen kernel function satisfies Mercer’s condition its corresponding support vector set is not necessarily the subset ofK that plays a crucial role in classifyingK. For a given sample set, what is the subset that plays the crucial role in classification? In order to explore the problem, a new concept, boundary or boundary points, is defined and its properties are discussed. Given a sample setK, we show that the decision functions for classifying the boundary points ofK are the same as that for classifying theK itself. And the boundary points ofK only depend onK and the structure of the space at whichK is located and independent of the chosen approach for finding the boundary. Therefore, the boundary point set may become the subset ofK that plays a crucial role in classification. These results are of importance to understand the principle of the support vector machine (SVM) and to develop new learning algorithms.
Keywords:support vector machine (SVM)  support vector  kernel function  constructive learning theory  cover  boundary
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