  We prove minimax optimal learning rates for kernel ridge regression,
resp.~support vector machines based on a data dependent partition of the input
space, where the dependence of the dimension of the input space is replaced by
the fractal dimension of the support of the data generating distribution. We
further show that these optimal rates can be achieved by a training validation
procedure without any prior knowledge on this intrinsic dimension of the data.
Finally, we conduct extensive experiments which demonstrate that our considered
learning methods are actually able to generalize from a dataset that is
non-trivially embedded in a much higher dimensional space just as well as from
the original dataset.
