Generalized Mercer Kernels and Reproducing Kernel Banach Spaces

摘要

This article studies constructions of reproducing kernel Banach spaces(RKBSs) which may be viewed as a generalization of reproducing kernel Hilbertspaces (RKHSs). A key point is to endow Banach spaces with reproducing kernelssuch that machine learning in RKBSs can be well-posed and of easyimplementation. First we verify many advanced properties of the general RKBSssuch as density, continuity, separability, implicit representation, imbedding,compactness, representer theorem for learning methods, oracle inequality, anduniversal approximation. Then, we develop a new concept of generalized Mercerkernels to construct $p$-norm RKBSs for $1leq pleqinfty$. The $p$-norm RKBSspreserve the same simple format as the Mercer representation of RKHSs.Moreover, the $p$-norm RKBSs are isometrically equivalent to the standard$p$-norm spaces of countable sequences. Hence, the $p$-norm RKBSs possess moregeometrical structures than RKHSs including sparsity. The generalized Mercerkernels also cover many well-known kernels, for example, min kernels, Gaussiankernels, and power series kernels. Finally, we propose to solve the supportvector machines in the $p$-norm RKBSs, which are to minimize the regularizedempirical risks over the $p$-norm RKBSs. We show that the infinite dimensionalsupport vector machines in the $p$-norm RKBSs can be equivalently transferredto finite dimensional convex optimization problems such that we obtain thefinite dimensional representations of the support vector machine solutions forpractical applications. In particular, we verify that some special supportvector machines in the $1$-norm RKBSs are equivalent to the classical $1$-normsparse regressions. This gives fundamental supports of a novel learning toolcalled sparse learning methods to be investigated in our next research project.