Gregory E. Fasshauer, Fred J. Hickernell, and Qi Ye
Abstract: In this paper we extend support vector machines from reproducing kernel Hilbert spaces into reproducing kernel Banach spaces whose reproducing kernels can be defined on nonsymmetric domains. Using the orthogonality of semi-inner products, we can obtain the empirical representations of support vector machine solutions. In addition, we can set up the reproduction property onto a generalized native space by Fourier transform techniques in order that it becomes a reproducing kernel Banach space and its reproducing kernel is given by the related positive definite function. The reproducing kernel Banach spaces of some reproducing kernels can be even imbedded into Sobolev spaces. We show some special examples of reproducing kernel Banach spaces induced by Matérn functions (Sobolev splines) such that their support vector machine solutions are well computable same as the classical cases but their explicit formulas are totally different from the solutions on reproducing kernel Hilbert spaces. It is possible to produce a new numerical tool for support vector machines.
Journal: Appl. Comput. Harmon. Anal. 38 (2015), no. 1, 115–139
DOI: 10.1016/j.acha.2014.03.007