Asymptotic properties of k-means and its bias correction for high-dimension, low-sample-size data

Although k-means is recognized as an effective method for analyzing high-dimension, low-sample-size (HDLSS) data, its asymptotic properties in such contexts remain underexplored. This study aims to deepen the understanding of k-means clustering under practical high-dimensional settings and investigates the asymptotic properties of k-means algorithms in case of HDLSS conditions. Based on the identified asymptotic properties of conventional k-means, a bias-corrected version of k-means was proposed and assessed for performance improvement. Additionally, in line with the current situation that kernel k-means lacks adequate theoretical exploration in high-dimensional environments, a theoretical analysis of kernel k-means was conducted using a Gaussian kernel function for high-dimensional data, facilitating the theoretical comparison between kernel and standard k-means. Finally, numerical simulations were conducted to demonstrate the effectiveness of both the methods, i.e., the proposed bias-corrected k-means and conventional k-means methods, including kernel k-means, in high-dimensional settings. Practitioners can leverage the bias-corrected k-means for more accurate clustering in high-dimensional contexts, improving data segmentation in fields ranging from bioinformatics to marketing analysis.