# VSFCM: A Novel Viewpoint-Driven Subspace Fuzzy C-Means Algorithm
## 1 Introduction  
Clustering algorithms have received widespread attention and applications in various fields, such as pattern recognition, biology, engineering systems, image processing, and so forth [1, 2, 3, 4, 5, 6, 7]. For clustering algorithms, the data points in a given data set are divided into several clusters, and the similarity between the data points in the same cluster is greater than that in other clusters; meanwhile, there are strong differences among the clusters, which contribute to the presentation of asymmetric data structure. In the early stage, hard (Boolean) clustering was mainly studied, where every object strictly belonged to a single cluster. For example, The DPC algorithm [8] is an excellent representative of this category. Among numerous fuzzy clustering algorithms, the fuzzy c-means (FCM) algorithm is one of the most commonly used methods [9, 10, 11, 12, 13, 14]. In general, FCM attached equal importance to all features of data, establishing a symmetrical structure, which might often be inconsistent with that of the original data. Therefore, the use of weighted processing has become an important development direction. The weighted FCM (WFCM) algorithm [15] grouped data according to the weighted categories of the separated features, and the algorithm incorporated feature weights into commonly used Euclidean distances for clustering. The feature-weighted fuzzy k-means (FWFKM) algorithm [16] was based on thefuzzy k-prototypes algorithm and a supervised algorithm. However, it still required two objective functions to optimize the data partition and feature weights. The attribute weight algorithm (AWA) [17] was a fuzzy weighted subspace clustering algorithm, which could effectively find the important features of each cluster, that is, discover the asymmetry of data. However, the disadvantage of the AWA algorithm was that it did not work when the standard deviation of certain attributes was zero, as zero might be used as the denominator in the learning rules. Improved versions have been proposed to overcome this weakness, including the fuzzy weighted k-means (FWKM) algorithm [18] and the fuzzy subspace clustering (FSC) algorithm [19; 20]. In the objective function of FWKM, a small constant was added when calculating the distance, which effectively avoided the problem caused by the zero standard deviation of some attributes in AWA. [PERSON] et al., proposed the FSC algorithm, which used a strategy similar to the FWKM algorithm discussed earlier. However, they had a significant difference in the method of parameter setting. The constant parameter introduced in FSC should be set manually, while that in FWKM was set through a predefined formula. However, when clustering is completed in a high-dimensional space, the traditional clustering algorithms will expose obvious drawbacks [21; 22; 23]. For example, for any given pair of data points in the cluster of a high-dimensional space, these points may be far apart. Due to the lack of multi-dimensional space thinking, traditional algorithms may have some deviations in calculating the distance, resulting in unsatisfactory clustering performance. For most traditional clustering algorithms, a key challenge is that in many real-world problems, data points in different clusters are often related to different feature subsets; that is, clusters can exist in different subspaces [24; 25; 26]. [PERSON] and [PERSON] [27] proposed a new approach called simultaneous clustering and attribute discrimination (SCAD). It used continuous feature weighting, providing a richer feature correlation representation than feature selection. Moreover, it also independently learned the associative representation of features of each cluster in an unsupervised way. Later, [PERSON] [28] studied the use of intra-class and inter-class information, and proposed a new clustering method, called enhanced soft subspace clustering (ESSC). However, many irrelevant data would affect the clustering performance of fuzzy clustering algorithms.