Relationship between support vector set and kernel functions in SVM

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 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.