Relationship Between Support Vector Set and Kernel Functions in SVM

Based on a constructive learning approach, covering algorithms, weinvestigate the relationship between support vector sets and kernelfunctions in support vector machines (SVM). An interesting result isobtained. That is, in the linearly non-separable case, any sample of agiven sample set K can become a support vector under a certain kernelfunction. The result shows that when the sample set K is linearlynon-separable, although the chosen kernel function satisfies Mercer'scondition its corresponding support vector set is not necessarily thesubset of K that plays a crucial role in classifying K. For a givensample set, what is the subset that plays the crucial rolein classification? In order to explore the problem, a new concept,boundary or boundary points, is defined and its properties arediscussed. Given a sample set K, we show that the decision functionsfor classifying the boundary points of K are the same as that forclassifying the K itself. And the boundary points of K only dependon K and the structure of the space at which K is located andindependent of the chosen approach for finding the boundary.Therefore, the boundary point set may become the subset of K thatplays a crucial role in classification. These results are of importanceto understand the principle of the support vector machine (SVM) and todevelop new learning algorithms.