# RMRRASL: Robust Matrix Factorization with Robust Adaptive Structure Learning for Feature Selection
## 4 Convergence Analysis
\begin{array}{l}\mbox{tr}((Z-ZS^{(t+1)}A^{(t+1)})^{T}\!U^{(t)}(Z-ZS^{ (t+1)}A^{(t+1)}))+\alpha\,tr(S^{(t+1)^{T}}Z^{T}LZS^{(t+1)})\\ +\beta\mbox{tr}((Z^{T}-Z^{T}W^{(t+1)})^{T}C^{(t)}(Z^{T}-Z^{T}W^{(t+1)}))\\ \leq\left\{\begin{array}{l}\mbox{tr}((Z-ZS^{(t)}A^{(t)})^{T}U^{(t)}(Z-ZS^{(t )}A^{(t)}))+\alpha\,tr(S^{(t)^{T}}Z^{T}LZS^{(t)})\\ +\beta\mbox{tr}((Z^{T}-Z^{T}W^{(t)})^{T}C^{(t)}(Z^{T}-Z^{T}W^{(t)}))\end{array} \right.\end{array} \right. \tag{60}\]  
On the basis of the definitions of matrices \(U_{t}\) and \(C_{t}\), the following inequalities can be obtained:  
\[\left\{ \begin{array}{l}\sum\limits_{i}\frac{||(Z-ZS^{(t+1)}A^{(t+1)})^{i}|| _{2}^{2}}{2||(Z-ZS^{(t)}A^{(t)})^{i}||_{2}}+\mbox{str}(S^{(t+1)^{T}}Z^{T}LZS^{ (t+1)})+\beta\sum\limits_{i}\frac{||(Z^{T}-Z^{T}W^{(t+1)})^{i}||_{2}^{2}}{2|| (Z-Z^{T}W^{(t)})^{i}||_{2}}\\ \leq\left\{\sum\limits_{i}\frac{||(Z-ZS^{(t)}A^{(t)})^{i}||_{2}^{2}}{2||(Z-ZS^{ (t)}A^{(t)})^{i}||_{2}}+\mbox{str}(S^{(t)^{T}}Z^{T}LZS^{(t)})+\beta\sum\limits _{i}\frac{||(Z^{T}-Z^{T}W^{(t)})^{i}||_{2}^{2}}{2||(Z^{T}-Z^{T}W^{(t)})^{i}||_ {2}}\right.\end{array} \right., \tag{61}\]  
\[\left\{ \begin{array}{l}\sum\limits_{i}||(Z-ZS^{(t+1)}A^{(t+1)})^{i}||_{2}- (\sum\limits_{i}||(Z-ZS^{(t+1)}A^{(t+1)})^{i}||_{2}-\sum\limits_{i}\frac{||( Z-ZS^{(t+1)}A^{(t+1)})^{i}||_{2}^{2}}{2||(Z-ZS^{(t)}A^{(t)})^{i}||_{2}})\\ +\mbox{str}(S^{(t+1)^{T}}Z^{T}LZS^{(t+1)})\\ +\beta\sum\limits_{i}||(Z^{T}-Z^{T}W^{(t)})^{i}||_{2}-\beta(\sum\limits_{i}|| (Z^{T}-Z^{T}W^{(t+1)})^{i}||_{2}-\sum\limits_{i}\frac{||(Z^{T}-Z^{T}W^{(t+1)} )^{i}||_{2}^{2}}{2||(Z^{T}-Z^{T}W^{(t)})^{i}||_{2}})\\ \leq\left\{\begin{array}{l}\sum\limits_{i}||(Z-ZS^{(t)}A^{(t)})^{i}||_{2}-( \sum\limits_{i}||(Z-ZS^{(t)}A^{(t)})^{i}||_{2}-\sum\limits_{i}\frac{||(Z-ZS^{ (t)}A^{(t)})^{i}||_{2}^{2}}{2||(Z-ZS^{(t)}A^{(t)})^{i}||_{2}})\\ +\mbox{str}(S^{(t)^{T}}Z^{T}LZS^{(t)})\\ +\beta\sum\limits_{i}||(Z^{T}-Z^{T}W^{(t)})^{i}||_{2}-\beta(\sum\limits_{i}|| (Z^{T}-Z^{T}W^{(t)})^{i}||_{2}-\sum\limits_{i}\frac{||(Z^{T}-Z^{T}W^{(t)})^{i }||_{2}^{2}}{2||(Z^{T}-Z^{T}W^{(t)})^{i}||_{2}})\end{array}\right.. \tag{62}\]  
According to Equation (58), the following inequalities can be obtained:  
\[\begin{array}{l}\sum\limits_{i}||(Z-ZS^{(t+1)}A^{(t+1)})^{i}||_{2}-\sum \limits_{i}\frac{||(Z-ZS^{(t)}A^{(t+1)})^{i}||_{2}^{2}}{2||(Z-ZS^{(t)}A^{(t)}) ^{i}||_{2}^{2}}\\ \leq\sum\limits_{i}||(Z-ZS^{(t)}A^{(t)})^{i}||_{2}-\sum\limits_{i}\frac{||(Z-ZS ^{(t)}A^{(t)})^{i}||_{2}^{2}}{2||(Z-ZS^{(t)}A^{(t)})^{i}||_{2}}\\ \leq\sum\limits_{i}||(Z^{T}-Z^{T}W^{(t)})^{i}||_{2}-\sum\limits_{i}\frac{||(Z^{ T}-Z^{T}W^{(t)})^{i}||_{2}^{2}}{2||(Z-Z^{T}W^{(t)})^{i}||_{2}}\end{array}\right.,\end{array}  \tag{63}\]  
\[ \begin{array}{l}\sum\limits_{i}||(Z^{T}-Z^{T}W^{(t+1)})^{i}||_{2}-\sum \limits_{i}\frac{||(Z^{T}-Z^{T}W^{(t+1)})^{i}||_{2}^{2}}{2||(Z^{T}-Z^{T}W^{(t)} )^{i}||_{2}}\\ \leq\sum\limits_{i}||(Z^{T}-Z^{T}W^{(t)})^{i}||_{2}-\sum\limits_{i}\frac{||(Z^{ T}-Z^{T}W^{(t)})^{i}||_{2}^{2}}{2||(Z^{T}-Z^{T}W^{(t)})^{i}||_{2}}\end{array}  \tag{64}\]  
Combining Equation (61) to Equation (64), the following inequality can be obtained:  
\[\left\{\sum\limits_{i}||(Z-ZS^{(t+1)}A^{(t+1)})^{i}||_{2}+\mbox{str}(S^{(t+1)^{ T}}Z^{T}LZS^{(t+1)})+\beta\sum\limits_{i}||(Z^{T}-Z^{T}W^{(t+1)})^{i}||_{2}}\right. \tag{65}\] \[\leq\left\{\sum\limits_{i}||(Z-ZS^{(t)}A^{(t)})^{i}||_{2}+\mbox{str }(S^{(t)^{T}}Z^{T}LZS^{(t)})+\beta\sum\limits_{i}||(Z^{T}-Z^{T}W^{(t)})^{i}||_{2} \right.\]  
From Equation (65), we can clearly see that Equation (12) is monotonically decreasing during iterations and has a lower bound. Therefore, the proposed iterative optimization algorithm will converge. Moreover, numerical experiments verified that the objective function values of our proposed method tend to converge.