On the Existence of Kernel Function for Kernel-Trick of k-Means

This paper corrects the proof of the Theorem 2 from the Gower’s paper [3, page 5]. The correction is needed in order to establish the existence of the kernel function used commonly in the kernel trick e.g. for k-means clustering algorithm, on the grounds of distance matrix. The scope of correction is explained in section 2. 1 The background problem Kernel based k-means clustering algorithm (clustering objects 1,..,m into k clusters 1,. . . ,k) consists in switching to a multidimensional feature space F and searching therein for prototypes μj minimizing the error m ∑ i=1 min 1≤j≤k ‖Φ(i)− μj ‖ (1) where Φ: {1, . . . , m} → F is a (usually non-linear) mapping of the space of objects into the feature space. In analogy to the classical k-means algorithm, the prototype vectors are updated according to the equation