Support Vector Machines in Reproducing Kernel Hilbert Spaces Versus Banach Spaces

Abstract

AbstractIn this article, we compare the support vector classifiers in Hilbert spaces versus those in Banach spaces. Recently, we developed a new concept of reproducing kernel Banach spaces (RKBSs). These spaces are a natural generalization of reproducing kernel Hilbert spaces (RKHSs) by extending the reproduction property from inner products to dual bilinear products. Based on the techniques of Fourier transforms, we can construct RKBSs by many well-known positive definite functions, e.g., Matérn functions and Gaussian functions. In addition, we can obtain finite-dimensional solutions of support vector machines defined in infinite-dimensional RKBSs. Finally, the numerical examples provided in this paper show that the solution of support vector machines in a RKBS can be computed and easily coded just as the classical algorithms given in RKHSs.KeywordsSupport vector machineReproducing kernel Banach spacePositive definite functionMatérn functionSobolev splineGaussian function