On a Kernel k-Means Algorithm

This is the extended version of a paper presented at CISP-BMEI 2023. After a general introduction kernels are described by showing how they arise from considerations concerning elementary geometrical properties. They appear as generalizations of the scalarproduct that in turn is the algebraic version of length and angle. By introducing the Reproducing Kernel Hilbert Space it is shown how operations in a high dimensional feature space can be performed without explicitly using an embedding function (the "kernel trick"). The general section of the paper lists some kernels and sophisticated kernel clustering algorithms. Thus the continuing popularity of the k-means algorithm is probably due to its simplicity. This explains why an elegant version of a k-means iterative algorithm originally established by Duda is treated. This was extended to a kernel algorithm by the author. However, its performance still heavily depended on the initialization. In this paper previous results on the original k-means algorithm are transferred to the kernel version thus removing these setbacks. Moreover the algorithm is slightly modified to allow for an easy quantification of the improvements to the target function after initializaztion.