Relationship Between Support Vector Set and Kernel Functions in SVM

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. Supported by the National Key Basic Research Program (Grant No.G1998030509) and the National Natural Science Foundation of China (Grant No.60135010). ZHANG ling was born in Fujian, China. He graduated from Nanjing University, Nanjing, China, in 1961. He is now a professor of the Computer Science Department, and the director of Artificial Intelligence Institute, Anhui University, Hefei, Anhui, China. His main research interests include applied mathematics, artificial intelligence and neural networks. He has published more than 100 papers and 4 books in these fields. ZHANG Bo was born in Fujian, China. He graduated from Tsinghua University, Beijing, China, in 1958. He is now a professor of the Computer Science and Technology Department, Tsinghua University, Beijing, China. His main research interests include artificial intelligence, neural networks, robotics and pattern recognition. He has published about 130 papers and 3 books in these fields.