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Statistical machine learning plays an important role in modern statistics andcomputer science. One main goal of statistical machine learning is to provideuniversally consistent algorithms, i.e., the estimator converges in probabilityor in some stronger sense to the Bayes risk or to the Bayes decision function.Kernel methods based on minimizing the regularized risk over a reproducingkernel Hilbert space (RKHS) belong to these statistical machine learningmethods. It is in general unknown which kernel yields optimal results for aparticular data set or for the unknown probability measure. Hence variouskernel learning methods were proposed to choose the kernel and therefore alsoits RKHS in a data adaptive manner. Nevertheless, many practitioners often usethe classical Gaussian RBF kernel or certain Sobolev kernels with good success.The goal of this short note is to offer one possible theoretical explanationfor this empirical fact.

A short note on extension theorems and their connection to universal consistency in machine learning