# VSFCM: A Novel Viewpoint-Driven Subspace Fuzzy C-Means Algorithm
## 2 Related Work  
In this section, we review several clustering algorithms closely related to our work. Assuming that the data set X\(=\{x_{j}\}_{j=1}^{n}\) is a set of \(n\) samples, we divide the data into \(c\) (\(2\leq c\leq n\)) clusters to get a set of cluster centers \(V=\{v_{i}\}_{i=1}^{c}\). Each sample \(x_{j}\) and cluster center \(v_{i}\) are positioned in \(R^{l}\) space where \(l\) is the data dimension. The objective function of the FCM algorithm [33] is as follows:  
\[J_{\text{FCM}}=\sum\limits_{j=1}^{n}\sum\limits_{i=1}^{c}u_{ij}^{m}d_{ij}^{2} \tag{1}\]  
Among them, \(u_{ij}\) is the degree of membership, which ranges from 0 to 1, and needs to satisfy the constraint \(\sum\limits_{i=1}^{c}u_{ij}=1(j=1,2,\cdots,n)\). \({d_{ij}}^{2}=\left\|x_{j}-v_{i}\right\|^{2}\) represents the Euclidean distance between the \(i\)-th cluster center and the \(j\)-th sample, and \(m\in(1,~{}+\infty)\) is a fuzzy coefficient. [PERSON] and [PERSON] [27] established two versions of the SCAD algorithm. SCAD1 attempts to balance the two terms of a composite objective function and introduces a penalty term to determine the optimal attribute-related weight. Moreover, SCAD2 introduces a fuzzy weighted index to minimize the single-item criterion. In subsequent experiments, we choose to compare our algorithm with SCAD1, so only its expression is shown here:  
\[J_{\text{SCAD1}}=\sum\limits_{j=1}^{n}\sum\limits_{i=1}^{c}u_{ij}^{m}\sum \limits_{k=1}^{l}w_{ik}d_{ijk}^{2}+\sum\limits_{i=1}^{c}\theta_{i}\sum\limits_ {k=1}^{l}w_{ik}^{2}, \tag{2}\]  
Thereinto, the distance formula is \(d_{ijk}=\left|x_{jk}-v_{ik}\right|\), \(\theta_{i}\) is a weighted constraint expressed as  
\[\theta_{i}^{(t)}=K\frac{\sum\limits_{j=1}^{n}{(u_{ij}^{t-1})}^{m}\sum\limits_ {k=1}^{l}w_{ik}^{t-1}d_{ijk}^{t-1}}{\sum\limits_{k=1}^{l}{(w_{ik}^{t-1})}^{2}}. \tag{3}\]  
Here \(K\) is a constant, and the superscript \(t-1\) is the previous iteration of the \(t\)-th. [PERSON] and [PERSON] proved that SCAD1 and SCAD2 had similar behavior and yielded similar clustering results. [PERSON] et al. [28] proposed the enhanced fuzzy weighted soft subspace clustering (ESSC) algorithm. The most prominent advantage of this algorithm is its ability to minimize the intra-class distance while maximizing the inter-class distance. The objective function of the algorithm is as follows:  
\[J_{\text{ESSC}}=\sum\limits_{i=1}^{c}\sum\limits_{j=1}^{n}u_{ij}^{m}\sum \limits_{k=1}^{l}w_{ik}\left|\left|x_{jk}-v_{ik}\right|\right|^{2}+\gamma\sum \limits_{i=1}^{c}\sum\limits_{k=1}^{l}w_{ik}\ln(w_{ik})-\eta\sum\limits_{i=1} ^{c}{(\sum\limits_{j=1}^{n}u_{ij}^{m})}\sum\limits_{k=1}^{l}w_{ik}(v_{ik}-v_{ 0k})^{2}. \tag{4}\]  
Here \(\eta\) and \(\gamma\) are constants, \(\eta\in[0,1],~{}\gamma\in[1,+\infty]\). Moreover, \(w_{ik}\) is the feature weight, which satisfies \(\sum\limits_{i=1}^{c}w_{ik}=1(k=1,2,\cdots,l)\). The third term is the weighted inter-class separation term, \(v_{0}\) represents the center point of this data set, and the expression is  
\[v_{0}=\frac{\sum\limits_{j=1}^{n}x_{j}}{n} \tag{5}\]  
Because the ESSC algorithm considers both distances within and between classes and uses the idea of entropy information, it has certain advantages in soft subspace clustering algorithms belonging to the same category. One disadvantage of ESSC is that more parameters need to be set manually, and how to select the appropriate parameters of it is still an open question. [PERSON] and [PERSON] [29] came up with the feature-reduction FCM (FRFCM) algorithm. It calculates a new weight for each feature by adding feature-weighted entropy to the FRFCM objective function. Then, during the iteration, the cluster centers and the fuzzy membership matrix are updated with these new weights. It not only improves the performance compared with the FCM algorithm but also can select important features by weighting and reducing the feature dimension by discarding unimportant features. Therefore, the algorithm can automatically reduce the feature dimension and achieves a good clustering effect.