# VSFCM: A Novel Viewpoint-Driven Subspace Fuzzy C-Means Algorithm
## 2 Related Work
The V-FCM [31] algorithm optimizes the FCM algorithm by introducing the viewpoints. The viewpoints are represented by typical characteristic data of artificial selection, such as average value, maximum value, and minimum value. The objective function of the algorithm is as follows:  
\[J_{\text{V-FCM}}=\sum_{j=1}^{n}\sum_{i=1}^{c}\sum_{k=1}^{l}u_{ik}^{m}\parallel x _{jk}-g_{ik}\parallel^{2}. \tag{6}\]  
The V-FCM algorithm still has problems that it is not suitable for processing high-dimensional data, is sensitive to the initialization of cluster centers, and the selection of viewpoint types will also affect the clustering results. The DPC algorithm [8] assumes that the cluster center is surrounded by neighbors with lower local density, and they are at a relatively large distance from any points with a higher local density. For all data points \(x_{j}\), it is necessary to calculate their local density and the minimum distance \(\delta_{j}\) between itself and other points with higher local density \(\rho_{k}\). The formulas are (\(j\in\{1,2,\cdots,n\}\)):  
\[\rho_{j}=\sum f(d_{jk}-r), \tag{7}\]  
\[f(x)= \begin{cases}1,&x=d_{jk}-r<0\\ 0,&\text{other}\end{cases} , \tag{8}\]  
\[\delta_{j}=\min\Bigl{\{}d_{jk}\Bigl{|}\rho_{k}>\rho_{j},\ \ k\in\{1,2,\cdots,n\} \Bigr{\}}. \tag{9}\]  
Here \(d_{jk}\) is the distance between two data points and \(r\) is the density radius. The local density reflects the number of data points within the radius \(r\). For the point with the largest local density value, its \(\delta_{j}=\max\Bigl{\{}d_{jk}\Bigl{|}k\
eq j,\ \ k\in\{1,2,\cdots,n\} \Bigr{\}}(j\in\{1,2,\cdots,n\})\). In the DPC algorithm, according to the \(\rho-\delta\) distribution map, the data point with high density and a larger distance from the data point with a higher density are regarded as the cluster center. The reason is that the true cluster center has a large value of \(\rho_{j}\) and \(\delta_{j}\), while the \(\delta\) of the noise point is small. Recently, [PERSON] et al. [32] proposed the density viewpoint-induced possibilistic fuzzy c-means (DVPFCM) algorithm. It provides a new initial method called the hypersphere density-based cluster center initialization (HDCCI) method, which is served for the viewpoint selection. On the basis of it, and combined with the advantages of FCM and PFCM, the DVPFCM algorithm is proposed. The specific objective function of DVPFCM is expressed as follows:  
\[J_{\text{DVPFCM}}=\sum_{j=1}^{n}\sum_{i=1}^{c}\Bigl{(}au_{ij}^{m}+bt_{ij}^{p} \Bigr{)}\left\lVert x_{j}-h_{i}\right\rVert^{2}+\frac{\sigma^{2}}{m^{2}c}{ \sum_{i=1}^{c}\sum_{j=1}^{c}\bigl{(}\varphi_{j}-t_{ij}\bigr{)}}^{p}. \tag{10}\]  
where  
\[h_{i}= \begin{cases}v_{i},&i\
eq q\\ x_{d},&i=q\end{cases} , \tag{11}\]  
Here \(m\) and \(p\) are the fuzzy coefficients and the typical matrix fuzzy coefficient, respectively. With the help of the viewpoint, its convergence speed has been improved, and its robustness has also been strengthened. Unfortunately, the above six algorithms, FCM, SCAD1, FRFCM, V-FCM, ESSC, and DVPFCM algorithms, all have the problem that they are sensitive to the initialization of cluster centers. An improper initial value may cause the result to converge to a local optimal value or cause a slow clustering process, which has a great impact on the clustering result and also contributes to the unstable result of the algorithm. Moreover, these algorithms also have other problems. The FCM algorithm is the most classic algorithm, but the actual efficiency and accuracy of processing high-dimensional data of it are usually not ideal. Although the V-FCM algorithm combines viewpoints to simplify the algorithm process and improve the convergence speed of the algorithm, its anti-noise ability is still weak. The ESSC algorithm proposes a weighted subspace clustering objective function based on entropy, which greatly increases the intra-cluster compactness and inter-cluster separation. However, more parameters need to be manually set, which increases the uncertainty of the algorithm.