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 ℓp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _p$$\\end{document} quasi-norm with p∈(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\in (0,1)$$\\end{document}. 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.