# VSFCM: A Novel Viewpoint-Driven Subspace Fuzzy C-Means Algorithm
## 3 Proposed VSFCM Algorithm
### The Mechanism of the VSFCM Algorithm
For this reason, we can naturally give the following constraints (\(i=1,\ldots,c\), \(j=1,\ldots,n\), \(k=1,\ldots,l\)):  
\[(x_{jk}-h_{ik})^{2}-\eta(v_{0k}-h_{ik})^{2}\geq 0. \tag{27}\]  
Therefore, we can get  
\[\eta=\alpha_{0}\underset{i,j,k}{\text{min}}\frac{(x_{jk}-h_{ik})^{2}}{(v_{0k}- h_{ik})^{2}}. \tag{28}\]  
Here \(\alpha_{0}\) is a constant, and \(\alpha_{0}\in[0,1]\). Secondly, we show the solution process of \(h_{ik}\). Starting from \(\frac{\partial J^{{}^{\prime}}}{\partial h_{ik}}=0\), we have (\(i=1,\ldots,c\), \(k=1,\ldots,l\))  
\[\frac{\partial J^{{}^{\prime}}}{\partial h_{ik}}=2\sum\limits_{j=1}^{n}u_{ij} ^{m}w_{ik}^{t}\Big{[}(1-\eta)h_{ik}-x_{jk}+\eta v_{0k}\Big{]}=0. \tag{29}\]  
It can be further calculated, and we can get  
\[(1-\eta)\sum\limits_{j=1}^{n}u_{ij}^{m}w_{ik}^{t}h_{ik}=\sum\limits_{j=1}^{n} u_{ij}^{m}w_{ik}^{t}\Big{(}x_{jk}-\eta v_{0k}\Big{)}. \tag{30}\]  
Regarding the solution of \(h_{ik}\), there are two cases:  
Case 1: From (17) we know that \(h_{ik}=x_{ck}\) when \(i=q\) (\(k=1,\ldots,l\)). Case 2: When \(i\
eq q\), we can get from (30):  
\[h_{ik}=v_{ik}=\frac{\sum\limits_{j=1}^{n}\Big{(}x_{jk}-\eta v_{0k}\Big{)}u_{ij }^{m}}{\sum\limits_{j=1}^{n}(1-\eta)u_{ij}^{m}}. \tag{31}\]In summary, we can get  
\[h_{ik}=\left\{ \begin{array}{ll}x_{ek},&i=q,\\ \frac{\sum\limits_{j=1}^{n}{(x_{jk}-\eta v_{0k})u_{ij}^{m}}}{\sum\limits_{j=1}^{n }{(1-\eta)u_{ij}^{m}}},&i\
eq q.\end{array} \right.. \tag{32}\]  
Finally, we provide the solving process of \(w_{ik}\). Starting from \(\frac{\partial{f^{\prime}}}{\partial{w_{ik}}}=0\) (\(i=1,\ldots,c\), \(k=1,\ldots,l\)), we can get  
\[\frac{\partial{f^{\prime}}}{\partial{w_{ik}}}=tw_{ik}^{t-1}{\left(\sum\limits _{j=1}^{n}{u_{ij}^{m}}{[(x_{jk}-h_{ik})^{2}-\eta(v_{0k}-h_{ik})^{2}]}+\varphi_{ i}\right)}-\zeta_{i}=0. \tag{33}\]  
Similarly, for the convenience of expression, we set  
\[T_{ik}=\sum\limits_{j=1}^{n}{u_{ij}^{m}}{[(x_{jk}-h_{ik})^{2}-\eta(v_{0k}-h_{ ik})^{2}]}+\varphi_{i}. \tag{34}\]  
From (33), we obtain  
\[w_{ik}={\left(\frac{\zeta_{i}}{tT_{ik}}\right)}^{\frac{1}{l-1}}. \tag{35}\]  
Note that  
\[1=\sum\limits_{k=1}^{l}{w_{ik}}=\sum\limits_{k=1}^{l}{\left(\frac{\zeta_{i}}{ tT_{ik}}\right)}^{\frac{1}{l-1}}. \tag{36}\]  
From (36), we have  
\[{\left(\frac{\zeta_{i}}{t}\right)}^{\frac{1}{l-1}}=\frac{1}{\sum\limits_{p=1 }^{l}{T_{ip}}^{-\frac{1}{l-1}}}. \tag{37}\]  
Substituting (37) into (35), we obtain  
\[w_{ik}=\frac{T_{ik}^{-\frac{1}{l-1}}}{\sum\limits_{p=1}^{l}{T_{ip}}^{-\frac{1 }{l-1}}}. \tag{38}\]  
Among them \(t\in(1,+\infty)\), \(\varphi_{i}\) is a penalty term parameter, which reflects the contribution of each attribute to the cluster centers. Moreover, the selection of its value is critical to the performance of the clustering. In actual processing, we can define \(\varphi_{i}\) as the ratio of the sum of the previous part of (19) and the fuzzy feature weight:  
\[\varphi_{i}=K\frac{\sum\limits_{j=1}^{n}{u_{ij}^{m}D_{ij}}}{\sum\limits_{k=1 }^{l}{w_{ik}^{t}}} \tag{39}\]  
Here \(K\) is a positive constant. So far, the derivation process of cluster centers, membership degree matrix, and weight matrix of the VSFCM algorithm have been fully explained.