# Validity of Clusters Produced By kernel-$k$-means With Kernel-Trick

## TL;DR

This paper revises foundational theorems related to kernel-$k$-means clustering, ensuring the mathematical validity of the kernel trick by correcting previous proofs about kernel functions and their Euclidean embeddings.

## Contribution

It provides corrected proofs for key theorems in Gower's work, clarifying the conditions under which kernel functions are valid for clustering.

## Key findings

- Corrected proof of the existence of kernel functions from distance matrices.
- Clarified conditions for kernel matrices to be embeddable in Euclidean space.
- Ensured the mathematical soundness of kernel-$k$-means clustering methods.

## Abstract

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.