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A generalization of Haussler's convolution kernel: mapping kernel
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Haussler's convolution kernel provides an effiective 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 effiectiveness 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 effiective 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.