Theoretical analyses for a class of kernels with an invariant metric

One of central topics of kernel machines in the field of machine learning is a model selection, especially a selection of a kernel or its parameters. In our previous work, we discussed a class of kernels whose corresponding reproducing kernel Hilbert spaces have an invariant metric and proved that the kernel corresponding to the smallest reproducing kernel Hilbert space, including an unknown true function, gives the optimal model. However, discussions for properties that make the metrics of reproducing kernel Hilbert spaces invariant are insufficient. In this paper, we show a necessary and sufficient condition that makes the metrics of reproducing kernel Hilbert spaces invariant.