Sub-One Quasi-Norm-Based k-Means Clustering Algorithm and Analyses

Abstract

Recognizing the pivotal role of choosing an appropriate distance metric in designing the clustering algorithm, our focus is on innovating the k-means method by redefining the distance metric in its distortion. In this study, we introduce a novel k-means clustering algorithm utilizing a distance metric derived from the \(\ell _p\) quasi-norm with \(p\in (0,1)\). Through an illustrative example, we showcase the advantageous properties of the proposed distance metric compared to commonly used alternatives for revealing natural groupings in data. Subsequently, we present a novel k-means type heuristic by integrating this sub-one quasi-norm-based distance, offer a step-by-step iterative relocation scheme, and prove the convergence to the Kuhn-Tucker point. Finally, we empirically validate the effectiveness of our clustering method through experiments on synthetic and real-life datasets, both in their original form and with additional noise introduced. We also investigate the performance of the proposed method as a subroutine in a deep learning clustering algorithm. Our results demonstrate the efficacy of the proposed k-means algorithm in capturing distinctive patterns exhibited by certain data types.