  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.
