This paper introduces Geometric-k-means (or \({\mathsf{G}}k\) -means for short), a novel approach that significantly enhances the efficiency and energy economy of the widely utilized k-means algorithm, which, despite its inception over five decades ago, remains a cornerstone in machine learning applications. The essence of \({\mathsf{G}}k\) -means lies in its active utilization of geometric principles, specifically scalar projection, to significantly accelerate the algorithm without sacrificing solution quality. This geometric strategy enables a more discerning focus on data points that are most likely to influence cluster updates, which we call as high expressive data (HE). In contrast, low expressive data (LE), does not impact clustering outcome, is effectively bypassed, leading to considerable reductions in computational overhead. Experiments spanning synthetic, real-world and high-dimensional datasets, demonstrate \({\mathsf{G}}k\) -means is significantly better than traditional and state of the art (SOTA) \(k\) -means variants in runtime and distance computations (DC). Moreover, \({\mathsf{G}}k\) -means exhibits better resource efficiency, as evidenced by its reduced energy footprint, placing it as more sustainable alternative. The software code and data for our algorithm is available at https://github.com/parichit/Geometric-k-means.
