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# Mapping Kernels Defined Over Countably Infinite Mapping Systems and their Application

*Proceedings of the Asian Conference on Machine Learning*, PMLR 20:367-382, 2011.

#### Abstract

The mapping kernel is a generalization of Haussler’s convolution kernel, and has a wide range of application including kernels for higher degree structures such as trees. Like Haussler’s convolution kernel, a mapping kernel is a finite sum of values of a primitive kernel. One of the major reasons to use the mapping kernel template in engineering novel kernels is because a strong theorem is known for positive definiteness of the resulting mapping kernels. If the mapping kernel meets the transitivity condition and if the primitive kernel is positive definite, the mapping kernel is also positive definite. In this paper, we generalize this theorem by showing, even when we extend the definition of mapping kernels so that a mapping kernel can be a converging sum of countably infinite primitive kernel values, the transitivity condition is still a criteria to determine positive definiteness of mapping kernels according to the extended definition. Interestingly, this result is also useful to investigate positive definiteness of mapping kernels determined as finite sums, when they do not meet the transitivity condition. For this purpose, we introduce a general method that we call covering technique.