  Centroid-based clustering algorithms, such as hard K-means (HKM) and fuzzy
K-means (FKM), have suffered from learning bias towards large clusters. Their
centroids tend to be crowded in large clusters, compromising performance when
the true underlying data groups vary in size (i.e., imbalanced data). To
address this, we propose a new clustering objective function based on the
Boltzmann operator, which introduces a novel centroid repulsion mechanism,
where data points surrounding the centroids repel other centroids. Larger
clusters repel more, effectively mitigating the issue of large cluster learning
bias. The proposed new algorithm, called equilibrium K-means (EKM), is simple,
alternating between two steps; resource-saving, with the same time and space
complexity as FKM; and scalable to large datasets via batch learning. We
substantially evaluate the performance of EKM on synthetic and real-world
datasets. The results show that EKM performs competitively on balanced data and
significantly outperforms benchmark algorithms on imbalanced data. Deep
clustering experiments demonstrate that EKM is a better alternative to HKM and
FKM on imbalanced data as more discriminative representation can be obtained.
Additionally, we reformulate HKM, FKM, and EKM in a general form of gradient
descent and demonstrate how this general form facilitates a uniform study of
K-means algorithms.
