A Generalization of Haussler's Convolution Kernel — Mapping Kernel and Its Application to Tree Kernels

Abstract

Haussler's convolution kernel provides an effective framework for engineering positive semidefinite kernels, and has a wide range of applications. On the other hand, the mapping kernel that we introduce in this paper is its natural generalization, and will enlarge the range of application significantly. Our main theorem with respect to positive semidefiniteness of the mapping kernel (1) implies Haussler's theorem as a corollary, (2) exhibits an easy-to-check necessary and sufficient condition for mapping kernels to be positive semidefinite, and (3) formalizes the mapping kernel so that significant flexibility is provided in engineering new kernels. As an evidence of the effectiveness of our results, we present a framework to engineer tree kernels. The tree is a data structure widely used in many applications, and tree kernels provide an effective method to analyze tree-type data. Thus, not only is the framework important as an example but also as a practical research tool. The description of the framework accompanies a survey of the tree kernels in the literature, where we see that 18 out of the 19 surveyed tree kernels of different types are instances of the mapping kernel, and examples of novel interesting tree kernels.

Using Classification for Cost Reduction of Applying Mutation Testing

“…In this paper new kernel functions, based on the convolution kernel [37] and on mapping kernel template proposed by Shin et al ( [38]) are introduced for hierarchical control flow graphs.…”

“…The main idea of these kernels is to use the number of substructures of any structured object. This approach was extended by Shin et al [38], who proposed a mapping kernel for tree data.…”

“…To deal with this problem a so called mapping kernel was introduced by Shin et al [38]. It has been successfully used for trees and it allows to limit the calculations to the subset of the cross product.…”

Partitionable Kernels for Mapping Kernels

“…For example, Haussler's convolution kernel ( [2]) is an effective tool to design positive definite kernels. Recently, Shin et al ( [3]) generalized Haussler's convolution kernel, and introduced the notion of mapping kernel. Moreover, they applied the framework of mapping kernel to tree structures, and revealed that 13 of 19 tree kernels known in the literature are defined according to the single template formula of…”

“…Shin and Kuboyama ( [3]) reported that 18 of 19 tree kernels known in the literature can be defined as mapping kernels, and in particular, 13 of 18 are of the form (5) for different M X,Y .…”