# Transmission Line Fault-Cause Identification Based on Hierarchical Multiview Feature Selection
## 2 Hierarchical Multiview Feature Selection (HMVFS)
### Convergence  
In this subsection, we analyze the convergence of Algorithm 1. We need to guarantee the objective function decreases in each iteration of the optimization algorithm. The following lemma is used to verify its convergence.  
**Lemma 1**.: _For any nonzero values \(a,b\in\mathbb{R}\), the following inequality holds:_  
\[2 ab\leq\ \left(a^{2}+b^{2}\right)\ \Rightarrow a-\frac{a^{2}}{2b}\leq b-\frac{b^{2 }}{2b}. \tag{15}\]  
**Theorem 1**.: _The objective Function (1) monotonically decreases in the iteration of Algorithm 1._  
**Proof.** According to Step 6 and Step 7 in Algorithm 1, we have \(\mathbf{W}_{t+1}\) and \(\mathbf{M}_{t+1}\) as follows:  
\[\mathbf{W}_{t+1}\Leftarrow\underset{\mathbf{W}}{\text{min}}\|\mathbf{X} \mathbf{W}_{t}-\mathbf{Y}-\mathbf{B}\otimes\mathbf{M}_{t}\|_{F}^{2}+\alpha Tr( \mathbf{W}_{t}^{\text{T}}\mathbf{C}_{t}\mathbf{W}_{t})+\beta Tr(\mathbf{W}_{t} ^{\text{T}}\mathbf{D}_{t}\mathbf{W}_{t}), \tag{16}\]  
\[\mathbf{M}_{t+1}\Leftarrow\underset{\mathbf{M}}{\text{min}}\|\mathbf{X} \mathbf{W}_{t+1}-\mathbf{Y}-\mathbf{B}\otimes\mathbf{M}_{t}\|_{F}^{2}. \tag{17}\]  
Firstly, according to (16) and (17), there is  
\[\begin{split}&\|\mathbf{X}\mathbf{W}_{t+1}-\mathbf{Y}-\mathbf{B} \otimes\mathbf{M}_{t+1}\|_{F}^{2}+\alpha Tr(\mathbf{W}_{t+1}^{\text{T}} \mathbf{C}_{t}\mathbf{W}_{t+1})+\beta Tr(\mathbf{W}_{t+1}^{\text{T}}\mathbf{D }_{t}\mathbf{W}_{t+1})\\ &\leq\|\mathbf{X}\mathbf{W}_{t+1}-\mathbf{Y}-\mathbf{B}\otimes \mathbf{M}_{t}\|_{F}^{2}+\alpha Tr(\mathbf{W}_{t+1}^{\text{T}}\mathbf{C}_{t} \mathbf{W}_{t+1})+\beta Tr(\mathbf{W}_{t+1}^{\text{T}}\mathbf{D}_{t}\mathbf{W }_{t+1})\leq\|\mathbf{X}\mathbf{W}_{t}-\mathbf{Y}-\mathbf{B}\otimes\mathbf{M}_ {t}\|_{F}^{2}+\alpha Tr(\mathbf{W}_{t}^{\text{T}}\mathbf{C}_{t}\mathbf{W}_{t}) +\beta Tr(\mathbf{W}_{t}^{\text{T}}\mathbf{D}_{t}\mathbf{W}_{t}).\end{split} \tag{18}\]  
Thus, according to the definition of \(\mathbf{C}\), we have  
\[\begin{split}&\alpha Tr(\mathbf{W}_{t+1}^{\text{T}}\mathbf{C}_{t} \mathbf{W}_{t+1})=\sum\limits_{i=1}^{d}\frac{\|(\mathbf{w}_{i})_{t+i}\|_{2}^{ 2}}{2\|(\mathbf{w}_{i})_{t}\|_{2}}\\ &=\alpha\sum\limits_{i=1}^{d}\left\|(\mathbf{w}_{i})_{t+1}\right\| _{2}-\alpha\sum\limits_{i=1}^{d}\left\|(\mathbf{w}_{i})_{t+1}\right\|_{2}- \sum\limits_{i=1}^{d}\frac{\left\|(\mathbf{w}_{i})_{t+1}\right\|_{2}^{2}}{2 \|(\mathbf{w}_{i})_{t}\|_{2}})=\alpha\Phi(\mathbf{W}_{t+1})-\alpha(\sum \limits_{i=1}^{d}\left\|(\mathbf{w}_{i})_{t+1}\right\|_{2}-\sum\limits_{i=1}^{ d}\frac{\left\|(\mathbf{w}_{i})_{t+1}\right\|_{2}^{2}}{2\|(\mathbf{w}_{i})_{t}\|_{2 }})\end{split} \tag{19}\]  
We also perform the same transformation with \(Tr(\mathbf{W}_{t+1}^{\text{T}}\mathbf{D}_{t}\mathbf{W}_{t+1})\), \(Tr(\mathbf{W}_{t}^{\text{T}}\mathbf{C}_{t}\mathbf{W}_{t})\) and \(Tr(\mathbf{W}_{t}^{\text{T}}\mathbf{D}_{t}\mathbf{W}_{t})\). We can rewrite (18) as  
\[\begin{split}&\Psi(\mathbf{W}_{t+1},\mathbf{M}_{t+1})+\alpha \Phi(\mathbf{W}_{t+1})+\beta\Omega(\mathbf{W}_{t+1})-\alpha(\sum\limits_{i=1 }^{d}\left\|(\mathbf{w}_{i})_{t+1}\right\|_{2}-\sum\limits_{i=1}^{d}\frac{\left\| (\mathbf{w}_{i})_{t+1}\right\|_{2}^{2}}{2\left\|(\mathbf{w}_{i})_{t}\right\|_{ 2}})-\beta(\sum\limits_{v}^{\text{m}}\left\|(\mathbf{W}_{v})_{t+1}\right\|_{F} -\sum\limits_{v}^{\text{m}}\frac{\left\|(\mathbf{W}_{v})_{t+1}\right\|_{2}^{2}} {2\left\|(\mathbf{W}_{v})_{t}\right\|_{F}})\\ &\leq\Psi(\mathbf{W}_{t},\mathbf{M}_{t})+\alpha\Phi(\mathbf{W}_{t })+\beta\Omega(\mathbf{W}_{t})-\alpha(\sum\limits_{i=1}^{d}\left\|(\mathbf{w} )_{t}\right\|_{2}-\sum\limits_{i=1}^{d}\frac{\left\|(\mathbf{w}_{i})_{t}\right\| _{2}^{2}}{2\left\|(\mathbf{w}_{i})_{t}\right\|_{2}})-\beta(\sum\limits_{v}^{ \text{m}}\left\|(\mathbf{W}_{v})_{t}\right\|_{F}-\sum\limits_{v}^{\text{m}} \frac{\left\|(\mathbf{W}_{v})_{t}\right\|_{F}^{2}}{2\left\|(\mathbf{W}_{v})_