On the generalization of Tanimoto-type kernels to real valued functions

The Tanimoto kernel (Jaccard index) is a well known tool to describe the similarity between sets of binary attributes. It has been extended to the case when the attributes are nonnegative real values. This paper introduces a more general Tanimoto kernel formulation which allows to measure the similarity of arbitrary real-valued functions. This extension is constructed by unifying the representation of the attributes via properly chosen sets. After deriving the general form of the kernel, explicit feature representation is extracted from the kernel function, and a simply way of including general kernels into the Tanimoto kernel is shown. Finally, the kernel is also expressed as a quotient of piecewise linear functions, and a smooth approximation is provided.