A local–global mixed kernel with reproducing property

• We find the basic solution of a generalized differential operator, and prove that this basic solution is a new specific reproducing kernel.
• We prove that the local H-reproducing kernel satisfies the condition of Mercer kernel.
• We prove that the typical polynomial kernel with global property possesses reproducing property.
• We define a novel method named local–global mixed kernel with reproducing property.
• We evaluate the performance of our mixed kernel on standard UCI datasets.
• We demonstrate the effectiveness of the proposed mixed kernel.

A wide variety of kernel-based methods have been developed with great successes in many fields, but very little research has focused on the reproducing kernel function in Reproducing Kernel Hilbert Space (RKHS). In this paper, we propose a novel method which we call a local–global mixed kernel with reproducing property (LGMKRP) to successfully perform a range of classification tasks in the RKHS rather than the more conventionally used Hilbert space. The LGMKRP proposed in this paper consists of two major components. First, we find the basic solution of a generalized differential operator by the delta function, and prove that this basic solution is a new specific reproducing kernel called a local H-reproducing kernel (LHRK) in RKHS. This reproducing kernel has good local properties, including odd order vanishing moment, and fast dilation attenuation. Second, in the RKHS, we prove that the LHRK satisfies the condition of Mercer׳s theorem, and prove that it is a typical polynomial kernel with global property, which also possesses the reproducing property. Furthermore, the novel specific mixed kernel (i.e., LGMKRP) proposed in this paper is based on these two different properties. Experimental results demonstrate that the LGMKRP possesses the approximation and regularization performance of a reproducing kernel, and can enhance the generalization ability of kernel methods.