Partitionable Kernels for Mapping Kernels

Many of tree kernels in the literature are designed tanking advantage of the mapping kernel framework. The most important advantage of using this framework is that we have a strong theorem to examine positive definiteness of the resulting tree kernels. In the mapping kernel framework, each data object is viewed as a collection of components, and a mapping kernel for a pair of data objects is determined as a sum of kernel values of component pairs over a certain range determined according to the purpose of use of the resulting mapping kernel. For those tree kernels known to belong to the mapping kernel category, the string kernel of the product type is commonly used to compute the kernel values of component pairs. This is because it is known that use of the product-type string kernel together with the mapping kernel framework allows us to have recursive formulas to calculate the resulting tree kernels efficiently. We significantly generalizes this result. In fact, we show that we can use partition able kernels, a new class of string kernels instead of the product-type string kernel to enjoy the same advantage, that is, efficient computation based on recursive formulas. The class of partition able kernels is abundant, and contains the product-type string kernels just as an instance. Also, this result, not limited to tree kernels, can be applied to general mapping kernels after we formalize the decomposition properties of trees as the new notion of pretty decomposability.