[论文解读] A Centroid Auto-Fused Hierarchical Fuzzy c-Means Clustering

Yunxia Lin, Songcan Chen|arXiv (Cornell University)|Apr 27, 2020

Advanced Clustering Algorithms Research参考文献 33被引用 4

一句话总结

该论文提出了一种新型聚类方法——质心自动融合分层模糊c-均值（CAF-HFCM），通过在优化过程中自适应融合，自动确定最优聚类数，无需依赖外部有效性指数。通过在初始质心上施加ℓ₂-范数惩罚，并利用ADMM进行交替优化，CAF-HFCM实现了快速收敛、对初始化不敏感以及对噪声的鲁棒性，在合成数据集和真实数据集上均优于FCM及当前最先进方法。

ABSTRACT

Like k-means and Gaussian Mixture Model (GMM), fuzzy c-means (FCM) with soft partition has also become a popular clustering algorithm and still is extensively studied. However, these algorithms and their variants still suffer from some difficulties such as determination of the optimal number of clusters which is a key factor for clustering quality. A common approach for overcoming this difficulty is to use the trial-and-validation strategy, i.e., traversing every integer from large number like $\sqrt{n}$ to 2 until finding the optimal number corresponding to the peak value of some cluster validity index. But it is scarcely possible to naturally construct an adaptively agglomerative hierarchical cluster structure as using the trial-and-validation strategy. Even possible, existing different validity indices also lead to different number of clusters. To effectively mitigate the problems while motivated by convex clustering, in this paper we present a Centroid Auto-Fused Hierarchical Fuzzy c-means method (CAF-HFCM) whose optimization procedure can automatically agglomerate to form a cluster hierarchy, more importantly, yielding an optimal number of clusters without resorting to any validity index. Although a recently-proposed robust-learning fuzzy c-means (RL-FCM) can also automatically obtain the best number of clusters without the help of any validity index, so-involved 3 hyper-parameters need to adjust expensively, conversely, our CAF-HFCM involves just 1 hyper-parameter which makes the corresponding adjustment is relatively easier and more operational. Further, as an additional benefit from our optimization objective, the CAF-HFCM effectively reduces the sensitivity to the initialization of clustering performance. Moreover, our proposed CAF-HFCM method is able to be straightforwardly extended to various variants of FCM.

研究动机与目标

• 为解决模糊c-均值聚类中确定最优聚类数的长期挑战，避免依赖试错性验证指数。
• 开发一种从优化过程中自然衍生出的分层聚类方法，避免手动或启发式合并步骤。
• 降低对初始化的敏感性，这是基于FCM算法的已知弱点。
• 提高对噪声数据点的鲁棒性，因为这些点通常会降低标准FCM的聚类性能。
• 设计一种超参数调优最少的方法，以增强实际可用性。

提出的方法

• 该方法以大量数据点作为初始质心，通常设定为√n，以促进分层结构的形成。
• 在目标函数中对初始质心之间的距离施加ℓ₂-范数惩罚，以在优化过程中促进相似聚类的自动融合。
• 通过交替优化策略求解：隶属度矩阵通过解析方法更新，而质心矩阵通过ADMM方法更新。
• 随着惩罚参数γ的增加，算法逐步融合质心，形成自然的聚类层次结构。
• 该过程在无需外部验证指数的情况下自动稳定在最优聚类数量上。
• 该框架可扩展至FCM的各类变体，包括核化FCM（KFCM）。

实验结果

研究问题

• RQ1 能否提出一种模糊c-均值聚类方法，自动确定最优聚类数，而无需依赖外部有效性指数？
• RQ2 FCM的优化过程是否能通过质心融合自然生成分层聚类结构？
• RQ3 与标准FCM相比，所提方法是否降低了对初始化的敏感性？
• RQ4 与最优聚类数下基线FCM相比，该方法在噪声数据集上的表现如何？
• RQ5 该方法能否高效扩展至其他FCM变体，如核化FCM？

主要发现

• CAF-HFCM在噪声均匀数据上的兰德指数（RI）为0.9963 ± 0，显著优于相同聚类数下FCM的0.9887 ± 0.0091。
• CAF-HFCM的调整兰德指数（ARI）为0.9736 ± 0，而FCM为0.9217 ± 0.0630，表明聚类稳定性与准确性更优。
• 标准化互信息（NMI）值分别为CAF-HFCM的0.9804 ± 0和FCM的0.9608 ± 0.0306，证实了更好的划分一致性。
• CAF-HFCM在多次运行中均表现出一致的最优聚类结果，RI、ARI和NMI的方差为零，表明对初始化的敏感性极低。
• 即使初始质心数c=19，CAF-HFCM在高斯混合数据集上也仅用5次迭代即完成收敛，表现出快速收敛性。
• 可视化对比（图15和图16）表明，CAF-HFCM在噪声数据下始终能获得最优划分，而使用最优c=13的FCM仍产生次优结果。