## 3 Approach
### \(L_{2,1}\)-Norm Minimization UFS for Incomplete Data
#### 3.2.3 Optimization of Objective Function  
Solving Equation (6) is challenging, because it introduces two variables, \(\mathbf{V}\) and \(\mathbf{W}\). We thus solve the problem by means of an alternative optimization strategy, alternatively optimizing the two variables \(\mathbf{V}\) and \(\mathbf{W}\). (1) Update \(\mathbf{W}\) by Fixing \(\mathbf{V}\)  
Before updating \(\mathbf{W}\), to derive the optimization of the objective function, we must first make the following observation about the \(\mathrm{L}_{2,1}\) norm based on algebraic theory:  
\[\left\|\mathbf{W}\right\|_{2,1}=\sum\limits_{i=1}^{d}\left\|\mathbf{w}_{i} \right\|_{2}=\mathit{Tr}(\mathbf{W}^{T}\mathbf{W}\mathbf{Q})=\mathit{Tr}( \mathbf{W}^{T}\mathbf{Q}\mathbf{W})\]  
where \(\mathbf{Q}\) is a diagonal matrix, of which the \(i\)-th element is defined as \(q_{i,i}=\frac{1}{2\left\|\mathbf{w}_{i}\right\|_{2}}\). Let \(\mathbf{B}=\mathbf{I}\circ\mathbf{X}\); when \(\mathbf{V}\) is fixed, (6) is simplified into (7):  
\[\min_{\mathbf{W}}\mathit{Tr}\Big{(}(\mathbf{B}-\mathbf{I}\circ(\mathbf{X} \mathbf{W}))^{\mathbf{T}}\mathbf{R}(\mathbf{B}-\mathbf{I}\circ(\mathbf{X} \mathbf{W}))\Big{)}+2\beta\mathit{Tr}\Big{(}\mathbf{W}^{T}\mathbf{H}\mathbf{W} \Big{)}+\lambda\mathit{Tr}\Big{(}\mathbf{W}^{T}\mathbf{Q}\mathbf{W}\Big{)} \tag{7}\]  
where \(\mathbf{R}\) and \(\mathbf{Q}\) indicate a diagonal matrix, of which the \(i\)-th element is defined as  
\[r_{i,i}=\frac{1}{2\left\|(\mathbf{B}-\mathbf{I}\circ(\mathbf{X}\mathbf{W}))_{ i,1}\right\|_{2}}, \tag{8}\]  
Figure 1: Instance \(\mathbf{x}_{i}\) is only reconstructed by a few important neighbors in an incomplete data set, rather than being reconstructed by all the instances. \[q_{i,i}=\frac{1}{2\left\|\mathbf{w}_{i}\right\|_{2}}, \tag{9}\]  
and \(\mathbf{H}=(\mathbf{I}\circ\mathbf{X})\mathbf{H}_{v}(\mathbf{I}\circ\mathbf{X}) ^{T},\mathbf{H}_{v}=\mathbf{D}_{v}-\frac{\mathbf{V}+\mathbf{V}^{T}}{2}\) is the Laplacian matrix. \(\mathbf{D}_{v}\) is also a diagonal matrix, of which the \(i\)-th element is \(\sum\limits_{j}\frac{v_{ij}+v_{ij}}{2}\). By taking the derivative of problem (7) with respect to \(\mathbf{W}\) and setting it to zero, we have  
\[-\mathbf{X}^{T}(\mathbf{I}\circ\mathbf{R}\mathbf{B})+(\mathbf{I}\circ\mathbf{ X})^{T}\mathbf{R}\mathbf{X}\mathbf{W}+2\beta\mathbf{H}\mathbf{W}+\lambda\mathbf{Q} \mathbf{W}=0. \tag{10}\]  
Hence, we obtain the following solution of \(\mathbf{W}\):  
\[\mathbf{W}=((\mathbf{I}\circ\mathbf{X})^{T}\mathbf{R}\mathbf{X}+2\beta \mathbf{H}+\lambda\mathbf{Q})^{-1}\mathbf{X}^{T}(\mathbf{I}\circ\mathbf{R} \mathbf{B}). \tag{11}\]  
(2) Update \(\mathbf{V}\) by Fixing \(\mathbf{W}\)  
When \(\mathbf{W}\) is fixed, problem (6) becomes  
\[\min_{\mathbf{V}}\ \gamma\mathcal{T}\mathcal{T}\Big{(}(\mathbf{B}-\mathbf{I} \circ(\mathbf{V}\mathbf{X}))^{T}\mathbf{G}(\mathbf{B}-\mathbf{I}\circ( \mathbf{V}\mathbf{X}))\Big{)}+\beta\mathbf{F}^{\mathbf{W}}\mathbf{V}. \tag{12}\]  
where \(\mathbf{G}\) is a diagonal matrix, of which the \(i\)-th element is defined as  
\[g_{i,i}=\frac{1}{2\left\|(\mathbf{B}-\mathbf{I}\circ(\mathbf{V}\mathbf{X}))_{ i}\right\|_{2}}, \tag{13}\]  
and the \((i,j)\)-th entry of matrix \(\mathbf{F}^{\mathbf{W}}\) is \(\left\|\mathbf{x}_{i}\mathbf{W}-\mathbf{x}_{j}\mathbf{W}\right\|_{2}^{2}\). The derivative of (12) with respect to \(\mathbf{V}\) is  
\[-2\gamma\mathbf{X}^{T}(\mathbf{I}\circ\mathbf{G}\mathbf{B})+2\gamma(\mathbf{ I}\circ\mathbf{X})^{T}\mathbf{G}\mathbf{V}\mathbf{X}+\beta\mathbf{F}^{\mathbf{W}}=0. \tag{14}\]  
Hence, the solution of \(\mathbf{V}\) is  
\[\min_{\mathbf{W}}\mathit{Tr}\Big{(}(\mathbf{B}-\mathbf{I}\circ(\mathbf{X} \mathbf{W}))^{\mathbf{T}}\mathbf{R}(\mathbf{B}-\mathbf{I}\circ(\mathbf{X} \mathbf{W}))\Big{)}+2\beta\mathit{Tr}\Big{(}\mathbf{W}^{T}\mathbf{H}\mathbf{W} \Big{)}+\lambda\mathit{Tr}\Big{(}\mathbf{W}^{T}\mathbf{Q}\mathbf{W}\Big{)} \tag{7}\]0  
We summarize the detailed UPS-ID optimization approach in Algorithm 1.