Elucidating the solution structure of the K-means cost function using energy landscape theory

The K-means algorithm, routinely used in many scientific fields, generates clustering solutions that depend on the initial cluster coordinates. The number of solutions may be large, which can make locating the global minimum challenging. Hence, the topography of the cost function surface is crucial to understanding the performance of the algorithm. Here, we employ the energy landscape approach to elucidate the topography of the K-means cost function surface for Fisher's Iris dataset. For any number of clusters, we find that the solution landscapes have a funneled structure that is usually associated with efficient global optimization. An analysis of the barriers between clustering solutions shows that the funneled structures result from remarkably small barriers between almost all clustering solutions. The funneled structure becomes less well-defined as the number of clusters increases, and we analyze kinetic analogs to quantify the increased difficulty in locating the global minimum for these different landscapes. (c) 2022 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Automated summary

The K-means algorithm, commonly used in scientific fields, generates clustering solutions that depend on the initial cluster coordinates. By employing the energy landscape approach, this study reveals that the solution landscapes for K-means cost function exhibit a funnelled structure, which is associated with efficient global optimization. The small barriers between clustering solutions contribute to the observed funnelled structure.