Generalized Mercer Kernels and Reproducing Kernel Banach Spaces

This article studies constructions of reproducing kernel Banach spaces (RKBSs) which may be viewed as a generalization of reproducing kernel Hilbert spaces (RKHSs). A key point is to endow Banach spaces with reproducing kernels such that machine learning in RKBSs can be well-posed and of easy implementation. First we verify many advanced properties of the general RKBSs such as density, continuity, separability, implicit representation, imbedding, compactness, representer theorem for learning methods, oracle inequality, and universal approximation. Then, we develop a new concept of generalized Mercer kernels to construct $p$-norm RKBSs for $1\leq p\leq\infty$. The $p$-norm RKBSs preserve 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 more geometrical structures than RKHSs including sparsity. The generalized Mercer kernels also cover many well-known kernels, for example, min kernels, Gaussian kernels, and power series kernels. Finally, we propose to solve the support vector machines in the $p$-norm RKBSs, which are to minimize the regularized empirical risks over the $p$-norm RKBSs. We show that the infinite dimensional support vector machines in the $p$-norm RKBSs can be equivalently transferred to finite dimensional convex optimization problems such that we obtain the finite dimensional representations of the support vector machine solutions for practical applications. In particular, we verify that some special support vector machines in the $1$-norm RKBSs are equivalent to the classical $1$-norm sparse regressions. This gives fundamental supports of a novel learning tool called sparse learning methods to be investigated in our next research project.