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.