Theoretical analyses on a class of nested RKHS´s

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 forming a class of nested reproducing kernel Hilbert spaces with 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. In this paper, we consider a class of kernels forming a class of nested reproducing kernel Hilbert spaces whose metrics are not always invariant and show that a similar result to the invariant case is not obtained by providing a counter example using a class of Gaussian kernels.