Validity of Clusters Produced By kernel-$k$-means With Kernel-Trick
This addresses foundational mathematical issues in kernel methods for clustering, but it is incremental as it corrects existing proofs rather than introducing new techniques.
The paper corrects two theorems from prior work by Gower, establishing the existence of the kernel function for kernel-k-means based on distance matrices and enabling transformation of kernel matrices for Euclidean embedding.
This paper corrects the proof of the Theorem 2 from the Gower's paper \cite[page 5]{Gower:1982} as well as corrects the Theorem 7 from Gower's paper \cite{Gower:1986}. The first 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 correction encompasses the missing if-part proof and dropping unnecessary conditions. The second correction deals with transformation of the kernel matrix into a one embeddable in Euclidean space.