The geometric relationship between Core Vector Machine and Support Vector Machine

Core vector machine (CVM) is an efficient kernel method for large data classification. It has prominent advantages in dealing with large data sets in high-dimensional space. This paper presents a novel geometric framework between CVM and the traditional support vector machine (SVM). We proved theoretically that: (1) In one-class classification, non-training examples on the surface of the exact minimum enclosing ball (MEB) in CVM belong to the optimal separating hyperplane in SVM; (2) In one-class classification, training examples on the surface of the exact MEB in CVM correspond to the support vectors in SVM; (3) In two-class classification, non-training examples on the surface of the exact MEB in CVM belong to the bounding hyperplanes in SVM; (4) In two-class classification, training examples on the surface of the exact MEB in CVM correspond to the support vectors in SVM. Geometric interpretations for points on the (1 + epsiv)-approximate MEB in CVM are presented as well. It is believed that the obtained geometric relationship will be helpful in analyzing CVM and inspiring new classification algorithms.