makes a reasonably good choice, if no further domain knowledge about the underlying relation is known. Namely, the theorem says that given a suitable sample of data, the RKHS of the kernel contains functions that are arbitrarily close to any continuous relation in the uniform norm. However, the theorem does not say anything about how likely it is to have, as a training set, such a data sample that can represent the approximating function. Further, the theorem only concerns graded relations that are continuous and therefore crisp relations and graded, discontinuous relations require more detailed considerations.

Other kernel functions might of course outperform the Kronecker product pairwise kernel in applications where domain knowledge can be incorporated in the kernel function. In the following section we discuss reciprocity, symmetry and transitivity as three relational properties that can be represented by means of more specific kernel functions. As a side note, we also introduce the Cartesian pairwise kernel, which is formally defined as follows $$K_{C}^{\Phi}({v},{v}',{\overline{{v}}},{\overline{{v}}}') = K^{\phi}({v}', {\overline{{v}}}') [{v}= {\overline{{v}}}] + K^{\phi}({v},{\overline{{v}}}) [{v}' = {\overline{{v}}}'] \,,$$ with $[.]$ the indicator function, returning one when both elements are identical and zero otherwise. This kernel was recently proposed by [@Kashima2009] as an alternative to 