# RMRRASL: Robust Matrix Factorization with Robust Adaptive Structure Learning for Feature Selection
## 4 Convergence Analysis
 \tag{47}\]  
_is an auxiliary function of \(\varphi(A_{ij})\)._  
**Lemma 4. **_The function_  
\[ \begin{array}{c}\phi(W_{ij},W_{ij}^{(t)})=\varphi_{ij}(W_{ij},W_{ij}^{(t)}) +\varphi^{\prime}{}_{ij}(W_{ij})(W_{ij}-W_{ij}^{(t)})\\ \qquad\qquad+\frac{[2 ZCZ^{T}W+\mu DistY]_{ij}}{W_{ij}^{(t)}}(W_{ij}-W_{ij}^{(t )})^{2}\end{array}  \tag{48}\]  
_is an auxiliary function of \(\varphi(W_{ij})\)._  
Next, we can prove Lemma 2 using the steps below. **Proof the Lemma 2. ** The Taylor series expansion of \(\varphi(S_{ij})\) can be given as follows:\[\varphi_{ij}(S_{ij})=\varphi_{ij}(S_{ij}^{(t)})+\varphi^{\prime}_{ij}(S_{ij})(S_{ ij}-S_{ij}^{(t)})+\frac{1}{2}\varphi^{\prime\prime}_{ij}(S_{ij})(S_{ij}-S_{ij}^{(t)})^{2} \tag{49}\] \[=\varphi_{ij}(S_{ij}^{(t)})+\varphi^{\prime}_{ij}(S_{ij})(S_{ij}-S_ {ij}^{(t)})+(2[Z^{T}UZ]_{ii}[AA^{T}]_{jj}+2\alpha[M]_{ij}+4\lambda[SS^{T}]_{ij} )(S_{ij}-S_{ij}^{(t)})^{2}\ \.\]  
\(\Box\)  
We can obtain the following inequalities through a series of calculations of the matrix:  
\[ \begin{split}&[Z^{T}UZSAA^{T}]_{ij}=\sum\limits_{l=1}^{k}{[Z^{T}UZS ^{t}]_{il}[AA^{T}]_{lj}}\geq[Z^{T}UZS^{t}]_{ij}[AA^{T}]_{jj}\\ &\geq\left(\sum\limits_{l=1}^{k}{[Z^{T}UZ]_{il}S_{lj}^{t}}\right) [AA^{T}]_{jj}\geq[Z^{T}UZ]_{ii}S_{ij}^{t}[AA^{T}]_{jj}=[Z^{T}UZ]_{ii}[AA^{T}]_{ jj}S_{ij}^{t}\,\end{split}  \tag{50}\]  
\[ \begin{split}&[M^{+}S]_{ij}=\sum\limits_{l=1}^{d}{[M^{+}]_{ii}S_ {lj}^{t}}\geq M_{ij}^{+}S_{ij}^{t}\geq[M_{ij}^{+}-M_{ij}^{-}]S_{ij}^{t}=[M]_{ ij}S_{ij}^{t}.\end{split}  \tag{51}\]  
\[ \begin{split}&[SS^{T}S]_{ij}=\sum\limits_{l=1}^{k}{[SS^{T}]_{il}S_ {lj}^{t}}\geq[SS^{T}]_{ii}S_{ij}^{t}.\end{split}  \tag{52}\]  
To sum up, we can obtain the inequality  
\[\frac{[Z^{T}UZSAA^{T}+\alpha M^{+}S+2\lambda SS^{T}S]_{ij}}{S_{ij}^{(t)}}\geq[ Z^{T}UZS^{t}]_{ij}[AA^{T}]_{jj}+\alpha[M]_{ij}+2\lambda[SS^{T}]_{ii}. \tag{53}\]  
Therefore, we can obtain Equation (54). \[\phi(S_{ij},S_{ij}^{(t)})\geq\varphi_{ij}(S_{ij}). \tag{54}\]  
Similarly, we can prove Lemma 3 and Lemma 4. Finally, according to Lemma 1, we can obtain the update scheme of variables \(S\), \(A\), and \(W\) as follows:  
\[S_{ij}^{t+1}=\underset{S_{ij}}{\text{argmin}}\varphi(S_{ij},S_{ ij}^{t}) \tag{55}\] \[=S_{ij}^{t}-S_{ij}^{t}\frac{\varphi^{\prime}(S_{ij})}{2[Z^{T}UZSAA ^{T}+\alpha M^{+}S+2\lambda SS^{T}S]_{ij}}=S_{ij}^{t}\frac{[Z^{T}UZA^{T}+ \alpha M^{+}S+2\lambda S]_{ij}}{[Z^{T}UZSAA^{T}+\alpha M^{+}S+2\lambda S]_{ij}}\,\] \[A_{ij}^{t+1}=\underset{A_{ij}}{\text{argmin}}\varphi(A_{ij},A_{ ij}^{t})=A_{ij}^{t}-A_{ij}^{t}\frac{\varphi^{\prime}_{ij}(A_{ij})}{2[S^{T}Z^{T} UZSA]_{ij}}=A_{ij}^{t}\frac{[S^{T}Z^{T}UZ]_{ij}}{[S^{T}Z^{T}UZSA]_{ij}}, \tag{56}\]  
\[W_{ij}^{t+1}=\underset{W_{ij}}{\text{argmin}}\varphi(W_{ij},W_{ ij}^{t})=W_{ij}^{t}-W_{ij}^{t}\frac{\varphi^{\prime}_{ij}(W_{ij})}{2[2 ZCZ^{T}W+ \mu DistY]_{ij}}=W_{ij}^{t}\frac{[2 ZCZ^{T}]_{ij}}{[2 ZCZ^{T}W+\mu DistY]_{ij}}. \tag{57}\]  
In summary, Theorem 1 is proven. Next, the convergence of the proposed iterative procedure in Algorithm 1 is proven. For any nonzero vectors \(p\in R^{m}\) and \(q\in R^{m}\), the following inequality can be obtained:  
\[||p||_{2}-\frac{||p||_{2}^{2}}{2||q||_{2}}\leq||q||_{2}-\frac{||q||_{2}^{2}}{2 ||q||_{2}}. \tag{58}\]  
More detailed proof of Equation (58) is similar to that in [36]. Theorem 2: _According to the iterative update approach depicted in Algorithm 1, the objective function value in Equation (12) monotonically decreases in each iteration until it converges to the global optimum [36]._Proof. : First, matrices \(U_{t}\) and \(C_{t}\) are denoted as the \(t\)-th iteration of matrices \(U\) and \(C\). Then, when fixing \(U_{t}\) and \(C_{t}\), matrices \(S_{t+1}\), \(A_{t+1}\), and \(W_{t+1}\) can be updated by solving the following inequality:  
\[\Phi(S_{t+1},A_{t+1},W_{t+1},U_{t},C_{t})\leq\Phi(S_{t},A_{t},W_{t},U_{t},C_{t}). \tag{59}\]  
\(\square\)  
The above inequality can be changed to  
\[\left\{