Hybrids of K-means and linkage algorithms

Over the past two decades, there has been a growing interest in delving deep into clustering algorithms, primarily driven by the user’s dilemma. This surge in interest can be traced back to Kleinberg’s Impossibility Theorem for Clustering, [1], which examined the ability of clustering algorithms to meet a series of specific properties aimed at evaluating their robustness and versatility. The overall goal of these studies has been that of classifying clustering algorithms based on their capacity to handle perturbations in their internal parameters effectively and their ability to generate any conceivable partition of a given dataset. However, these studies typically evaluate the algorithms independently, without assessing whether a second algorithm can replicate the results of a given first algorithm under identical conditions (e.g., using the same dissimilarity matrix).This paper aims to compare the behavior of clustering algorithms by addressing the question: What adjustments are necessary to the internal parameters of a clustering algorithm A for this algorithm to be able to replicate the result obtained by a clustering algorithm B without altering the distance matrix? The paper provides necessary and/or sufficient conditions, with no assumptions made on the data generating process, for K-means to reproduce the partition returned by another clustering algorithm while preserving the original distance matrix. The same inquiry is also conducted for linkage-based algorithms. Additionally, the paper introduces a novel class of clustering algorithms called hybrids of K-means and linkage algorithms. These hybrids incorporate a function into K-means to autonomously compute the number of clusters that a linkage-based algorithm would produce for the given data set and distance matrix.