Robust Kernels for Robust Location Estimation

This paper shows that least-square estimation (mean calculation) in a reproducing kernel Hilbert space (RKHS) $\mathcal{F}$ corresponds to different M-estimators in the original space depending on the kernel function associated with $\mathcal{F}$. In particular, we present a proof of the correspondence of mean estimation in an RKHS for the Gaussian kernel with robust estimation in the original space performed with the Welsch M-estimator. This result is generalized to other types of M-estimators. This generalization facilitates the definition of new robust kernels associated to Huber, Tukey, Cauchy and Andrews M-estimators. The new kernels are empirically evaluated in different clustering tasks where state-of-the-art robust clustering methods are compared to kernel-based clustering using robust kernels. The results show that some robust kernels perform on a par with the best state-of-the-art robust clustering methods.