## 1 Introduction  
Similar data belongs to a cluster, while different data belongs to different clusters [1; 2; 3]. The fuzzy C-means (FCM) algorithm is a classical pattern recognition method [4], and many FCM-typeclustering algorithms were proposed [5, 6]. However, the between-cluster separation is ignored in these clustering techniques because these algorithms partition data points only by minimizing the distances between data points and cluster centers (_i.e._, the within-cluster compactness). Therefore, [PERSON] _et al._ proposed a fuzzy compactness and separation (FCS) algorithm [7]. The proposed FCS algorithm assigns a crisp boundary for each cluster so that hard memberships and fuzzy memberships can coexist in the clustering results.  
For high dimensional dataset clustering, features of data are assigned weights which illustrate the importance degree of features. A major problem of un-weighted clustering algorithms lies in treating all features equally in the clustering process. Therefore, many contributions attempt to weight features with various methods and to optimize the FCM-type algorithms [8, 9, 10, 11, 12, 13]. [PERSON] and [PERSON] [8] proposed the simultaneous clustering and attribute discrimination algorithm, in which clustering and feature weighting can be performed simultaneously in an unsupervised manner; [PERSON] _et al._[9] discussed that the weight assignment can be given by learning according to the gradient descent technique; [PERSON] _et al._ proposed an EWkmeans [10] which minimizes the within-cluster compactness and maximizes the negative weight entropy to stimulate more features contributing to the identification of a cluster; [PERSON] _et al._[11] presented a new fuzzy C-means algorithm with variable weighting (WFCM) for high dimensional data analysis; [PERSON] _et al._[12] put forward a feature weighting fuzzy clustering algorithm integrating rough sets and shadowed sets (WSRFCM); [PERSON] _et al._[13] introduced the between-cluster separation into the EWkmeans and proposed the enhanced soft subspace clustering (ESSC) algorithm. The WFCM and WSRFCM employ only the within-cluster compactness while updating the membership matrix and feature weights. ESSC uses a parameter \(\eta\) to balance the within-cluster compactness and between-cluster separation. However, negative values may be produced in the membership matrix if the balancing parameter is too large. Therefore, to avoid the negative membership value, \(\eta\) could be set zero. In this case, ESSC would degrade to the EWkmeans.  
In the real world, some data points belong to a cluster strictly (_i.e._, hard clustering) and others belong to a cluster ambiguously (_i.e._, fuzzy clustering). For maximizing the between-cluster separation and minimizing the within-cluster compactness, we proposed a new feature weighting fuzzy compactness and separation (WFCS) algorithm with fusion of hard clustering and fuzzy clustering. The rest of this paper is organized as follows. Section 2 introduces both the FCS and the WFCS algorithms, addresses the flaw of FCS and discusses the adjustment of membership and feature weighting of WFCS. The proposed algorithm is evaluated in section 3. Finally, this paper is concluded and the future work is discussed in Section 4.  
Table 1 illustrates the main symbols that appear in the following formulas.