Traditional centroid-based clustering algorithms, such as hard K-means (HKM, or Lloyd's algorithm) and fuzzy K-means (FKM, or Bezdek's algorithm), display degraded performance when true underlying groups of data have varying sizes (i.e., imbalanced data). This paper introduces equilibrium K-means (EKM), a novel fuzzy clustering algorithm that has the robustness to imbalanced data by preventing centroids from crowding together in the center of large clusters. EKM is simple, alternating between two steps; fast, with the same time and space complexity as FKM; and scalable to large datasets. We evaluate the performance of EKM on two synthetic and ten real datasets, comparing it to other centroid-based algorithms, including HKM, FKM, maximum-entropy fuzzy clustering (MEFC), two FKM variations designed for imbalanced data, and the Gaussian mixture model. The results show that EKM performs competitively on balanced data and significantly outperforms other algorithms on imbalanced data. Deep clustering experiments on the MNIST dataset demonstrate the significance of making representation have an EKM-friendly structure when dealing with imbalanced data; In comparison to deep clustering with HKM, deep clustering with EKM obtains a more discriminative representation and a 35% improvement in clustering accuracy. Additionally, we reformulate HKM, FKM, MEFC, and EKM in a general form of gradient descent, where fuzziness is introduced differently and more simply than in Bezdek's work, and demonstrate how the general form facilitates a uniform study of KM algorithms.

翻译：传统的基于质心的聚类算法，如硬K均值（HKM，即Lloyd算法）和模糊K均值（FKM，即Bezdek算法），在面对真实数据组规模差异较大（即不平衡数据）时表现下降。本文提出均衡K均值（EKM），一种新型模糊聚类算法，通过防止质心在大簇中心聚集来增强对不平衡数据的鲁棒性。EKM算法简洁，仅需在两步之间交替迭代；高效，其时间和空间复杂度与FKM相同；且可扩展至大规模数据集。我们在两个合成数据集和十个真实数据集上评估EKM的性能，并将其与HKM、FKM、最大熵模糊聚类（MEFC）、两种针对不平衡数据设计的FKM变体以及高斯混合模型进行对比。结果表明，EKM在平衡数据上具有竞争力，而在不平衡数据上显著优于其他算法。基于MNIST数据集的深度聚类实验表明，在处理不平衡数据时，使表征具有EKM友好结构具有重要意义；与基于HKM的深度聚类相比，基于EKM的深度聚类获得了更具判别性的表征，且聚类准确率提升了35%。此外，我们将HKM、FKM、MEFC和EKM重新表述为梯度下降的统一形式，其中模糊性的引入方式比Bezdek的工作更简洁且不同，并展示了该统一形式如何促进对K均值算法的系统性研究。