## 3 Approach
### Supervised Feature Selection for Incomplete Data
#### 3.3.3 Optimization of Objective Function  
In this section, we present the optimization of the above objective function. Although the above objective function is convex, solving problem (24) is still difficult, because the two regularization terms are non-smooth and the two variables \(\mathbf{W}\) and \(\mathbf{V}\) need to be optimized simultaneously. We propose an efficient algorithm to solve this problem by alternatively optimizing variables \(\mathbf{W}\) and \(\mathbf{V}\), respectively.  
(1) Update \(\mathbf{W}\) by Fixing \(\mathbf{V}\)  
Let \(\mathbf{B}=\mathbf{I}\circ\mathbf{V}\mathbf{X}\); thus, the problem (24) becomes  
\[\min_{\mathbf{W}}\lVert\mathbf{Y}-\mathbf{B}\mathbf{W}\rVert_{F}^{2}+\lambda Tr (\mathbf{W}^{\mathsf{T}}\mathbf{Q}\mathbf{W})+\gamma\lVert\mathbf{W}\rVert_ {1}. \tag{25}\]  
Taking the derivative of problem (25) with respect to \(\mathbf{W}\) by setting (26) to zero, we have  
\[\mathbf{B}^{T}\mathbf{B}\mathbf{w}_{i}-\mathbf{B}^{T}\mathbf{y}_{i}+\lambda \mathbf{Q}\mathbf{w}_{i}+\gamma\mathbf{Z}_{i}\mathbf{w}_{i}=0, \tag{26}\]  
where \(\mathbf{Z}_{i}\) (\(i=1,\cdots c\)) is a diagonal matrix, of which the k-th element is denoted by \(\frac{1}{2\lvert w_{i}\rvert}\), where \(\mathbf{Q}\) is a diagonal matrix \(q_{i,i}=\frac{1}{2\lVert(\mathbf{w})_{i}\rVert_{2}}\).  
Hence, we can obtain the solution of \(\mathbf{W}\) :  
\[\mathbf{w}_{i}=(\mathbf{B}^{T}\mathbf{B}+\lambda\mathbf{Q}+\gamma\mathbf{Z}_{ i})^{-1}\mathbf{B}^{T}\mathbf{y}_{i}. \tag{27}\](2) Update **V** by Fixing **W**  
When **W** is fixed, problem (24) becomes  
\[\min_{\textbf{V}}\lVert\textbf{Y}-(\textbf{I}\circ\textbf{V}\textbf{X})\textbf{W }\rVert_{F}^{2}. \tag{28}\]  
Furthermore, according to the [PERSON] norm framework, (27) can be rewritten as  
\[\min_{\textbf{V}}Tr(\textbf{Y}^{T}\textbf{Y}-2\textbf{I}^{T}\circ(\textbf{V} \textbf{X})^{T}\textbf{W}^{T}\textbf{Y})+Tr((\textbf{I}^{T}\circ(\textbf{V} \textbf{X})^{T})\textbf{W}^{T}\textbf{W}((\textbf{V}\textbf{X})\circ\textbf{I })). \tag{29}\]  
By taking the derivative of problem (28) with respect to **V** and setting it to zero, we have  
\[-2(\textbf{I}\circ\textbf{X})^{T}\textbf{W}^{T}\textbf{Y}+2(\textbf{I}\circ \textbf{X})^{T}\textbf{W}^{T}\textbf{W}(\textbf{V}\textbf{X})=0. \tag{30}\]  
Hence, we can obtain the solution of **V** :  
\[\textbf{V}=\textbf{W}^{-1}\textbf{Y}\textbf{X}^{-1}. \tag{31}\]  
Based on the above derivation, the whole SFS-ID procedure for optimizing the problem (24) is summarized in Algorithm 2.  
```
Input: Incomplete dataset \(\textbf{X}\in\mathbb{R}^{n\times d}\), \(\textbf{Y}\in\mathbb{R}^{n\times c}\), regularization parameter \(\lambda\), \(\gamma\). output: \(\textbf{W}\in\mathbb{R}^{d\times c}\). (1) Initialize **W**, **Q**, \(\textbf{Z}_{i}\), **V**. (2) After setting different missing ratios, calculate the indicator matrix **I**. (3) Calculate \(\textbf{B}=\textbf{I}\circ\textbf{V}\textbf{X}\). (4) Fixed **V**, the optimal **W** is formed by (27). (5) Fixed **W**, Calculate the diagonal matrix **V** according to (31). (6) Repeat steps (3)-(5) until convergence. (7) Use the calculated results **W** to select the features.
```  
**Algorithm 2** Proposed Algorithm for Optimizing (24)