# RMRRASL: Robust Matrix Factorization with Robust Adaptive Structure Learning for Feature Selection
## 4 Convergence Analysis  
In order to analyze the convergence of the proposed RMFRASL method, we firstly establish some theorems and definitions below. **Theorem 1**. : _For variables \(S\), \(A\), and \(W\), the value of Equation (12) is nonincreasing under the update rules of Equations (20), (26) and (33)._  
We can refer to the convergence analysis of NMF to solve the above theorem. Therefore, some auxiliary functions are introduced below. **Definition 1**. : \(\phi(m,m^{\prime})\)_is an auxiliary function of \(\ \varphi(m)\) if \(\phi(m,m^{\prime})\) and \(\varphi(m)\) satisfy the following conditions [35]:_  
\[ \begin{array}{l}\phi(m,m^{\prime})\geq\ \varphi(m)\\ \phi(m,m)=\ \varphi(m)\end{array} . \tag{34}\]  
**Lemma 1**. : _If \(\phi(m,m^{\prime})\) is an auxiliary function of \(\ \varphi(m)\), then it is nonincreasing under the following conditions:_  
\[m^{t+1}=\underset{m}{\mathrm{argmin}}\phi(m,m^{t}). \tag{35}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Matrix** & **Complexity** & **Matrix** & **Complexity** \\ \hline \(S\) & \(O(max\,(dn^{2},\,nd^{2}))\) & \(U\) & \(O(dn^{2})\) \\ \(A\) & \(O(max\,(dn^{2},\,kdn))\) & \(C\) & \(O(kn^{2})\) \\ \(W\) & \(O(max\,(nd^{2},\,dn^{2}))\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Computational complexity of each variable. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & **AR** & **CMU PIE** & **Extended YaleB** & **ORL** & **COIL20** \\ \hline RNEFS [20] & 11.57 & 15.43 & 13.89 & 1.698 & 4.187 \\ USFS [21] & 36.12 & 30.09 & 18.27 & 18.29 & 8.524 \\ CNAFS [22] & 11.81 & 14.19 & 11.48 & 4.292 & 5.382 \\ NNSAFS [23] & 34.83 & 52.95 & 43.70 & 4.261 & 8.775 \\ RSR [25] & 3.607 & 4.920 & 4.377 & 1.553 & 1.805 \\ SPNFSR [34] & 15.21 & 18.24 & 16.24 & 7.291 & 8.924 \\ RMFRASL & 2.250 & 2.687 & 2.493 & 1.190 & 1.504 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Running time (s) of each method on different databases. **Proof. **  
\[\varphi(m^{t+1})\leq\phi(m^{t+1},m^{t})\leq\phi(m^{t},m^{t})=\varphi(m^{t}). \tag{36}\]  
In order to apply Lemma 1 to prove Theorem 1, we need to find suitable auxiliary functions for the variables \(S\), \(A\), and \(W\). Therefore, the functions, first-order derivatives, and second-order derivatives of the three variables are specified. \(\square\)  
\[\varphi_{ij}(S_{ij})=[Z^{T}UZ-2A^{T}S^{T}Z^{T}UZ+A^{T}S^{T}Z^{T}UIZSA+\alpha S^{ T}Z^{T}LZS+\lambda S^{T}SS^{T}S-2\lambda S^{T}S]_{ij}, \tag{37}\]  
\[\varphi^{\prime}{}_{ij}(S_{ij})=2[-Z^{T}UZA^{T}+Z^{T}UIZSAA^{T}+\alpha Z^{T}LZS +2\lambda SS^{T}S-2\lambda S]_{ij}, \tag{38}\]  
\[\varphi^{\prime\prime}{}_{ij}(S_{ij})=2[Z^{T}UZ]_{ii}[AA^{T}]_{ij}+2\alpha[M]_ {ij}+4\lambda[SS^{T}]_{ij}, \tag{39}\]  
\[\varphi_{ij}(A_{ij})=[Z^{T}UZ-2A^{T}S^{T}Z^{T}UZ+A^{T}S^{T}Z^{T}UIZSA]_{ij}, \tag{40}\]  
\[\varphi^{\prime}{}_{ij}(A_{ij})=2[-S^{T}Z^{T}UZ+S^{T}Z^{T}UZSA]_{ij}, \tag{41}\]  
\[\varphi^{\prime\prime}{}_{ij}(A_{ij})=2[S^{T}Z^{T}UZS^{T}]_{ii}, \tag{42}\]  
\[\varphi_{ij}(W_{ij})=[ZCZ^{T}-2W^{T}ZCZ^{T}+W^{T}ZCZ^{T}W+\mu DistYW]_{ij}, \tag{43}\]  
\[\varphi^{\prime}{}_{ij}(W_{ij})=[-2 ZCZ^{T}+2 ZCZ^{T}W+\mu DistY]_{ij}, \tag{44}\]  
\[\varphi(m^{t+1})\leq\phi(m^{t+1},m^{t})\leq\phi(m^{t},m^{t})=\varphi(m^{t}). \tag{36}\]0  
where \(\varphi^{\prime}{}_{ij}(\cdot)\) and \(\varphi^{\prime\prime}{}_{ij}(\cdot)\) denote the first-order and second-order derivatives, respectively. Then, we can obtain the three lemmas described below. **Lemma 2. **_The function_  
\[ \begin{array}{c}\phi(S_{ij},S_{ij}^{(t)})=\varphi_{ij}(S_{ij},S_{ij}^{(t)}) +\varphi^{\prime}{}_{ij}(S_{ij})(S_{ij}-S_{ij}^{(t)})\\ \qquad+\frac{[Z^{T}UIZSAA^{T}+\alpha M^{T}S+2\lambda SS^{T}S]_{ij}}{S_{ij}^{(t) }}(S_{ij}-S_{ij}^{(t)})^{2}\end{array}  \tag{46}\]  
_is an auxiliary function of \(\varphi(S_{ij})\)._  
**Lemma 3. **_The function_  
\[ \begin{array}{c}\phi(A_{ij},A_{ij}^{(t)})=\varphi_{ij}(A_{ij},A_{ij}^{(t)}) +\varphi^{\prime}{}_{ij}(A_{ij})(A_{ij}-A_{ij}^{(t)})+\frac{[S^{T}Z^{T}UZSA]_{ ij}}{A_{ij}^{(t)}}(A_{ij}-A_{ij}^{(t)})^{2}\end{array}