阿几何方法的支持向量机（SVM）分类.pdfVIP

IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 17, NO. 3, MAY 2006 671 A Geometric Approach to Support Vector Machine (SVM) Classification Michael E. Mavroforakis and Sergios Theodoridis, Senior Member, IEEE Abstract— The geometric framework for the support vector ma- with a different risk (weight). 4) It applies a fast, simple, and chine (SVM) classification problem provides an intuitive ground easily conceivable algorithm to solve the SVM task. 5) It opens for the understanding and the application of geometric optimiza- the road for applying other geometric algorithms, finding the tion algorithms, leading to practical solutions of real world classifi- cation problems. In this work, the notion of “reduced convex hull” closest pair of points between convex sets in Hilbert spaces, for is employed and supported by a set of new theoretical results. These the nonseparable SVM problem. results allow existing geometric algorithms to be directly and prac- Although some authors have presented the theoretical tically applied to solve not only separable, but also nonseparable background of the geometric properties of SVMs, exposed classification problems both accurately and efficiently. As a prac- thoroughly in [10], the main stream of solving methods comes tical application of the new theoretical results, a known geometric algorithm has been employed and transformed accordingly to solve from the algebraic field (mainly decomposition). One of the best nonseparable problems successfully. representative algebraic algorithms with respect to speed and Index Terms—Classification, kernel methods, pattern reco