## 3 Approach
### \(L_{2,1}\)-Norm Minimization UFS for Incomplete Data
#### 3.2.2 The Objection Function of UFS on Incomplete Data  
In this paper, to enhance the robustness of the reconstruction error of data points and select the discriminative features, we apply the L\({}_{2,1}\)-norm loss function and a reconstruction error term. More specifically, in the incomplete data set, instance \(\mathbf{x}_{i}\) is only reconstructed by a few important neighbors, rather than being reconstructed by all the instances (as illustrated in Figure 1).  
\begin{table}
\begin{tabular}{c c} \hline
**Notation** & **Definition** \\ \hline \(\mathbf{X}\in\mathbb{R}^{n\times d}\) & Incomplete data set of \(n\) instances and \(d\) features \\ \hline \(\mathbf{Y}\in\mathbb{R}^{n\times c}\) & The label set \(c\) is the total number of classes \\ \hline \(\mathbf{I}\in\mathbb{R}^{n\times d}\) & The index matrix indicating whether the information in \(\mathbf{X}\) is complete or not \\ \hline \(\mathbf{W}\in\mathbb{R}^{d\times\mathbf{d}}\) & The feature weight coefficient matrix \\ \hline \(\mathbf{V}\in\mathbb{R}^{n\times n}\) & The reconstruction weight matrix \\ \hline \(\left\|\cdot\right\|_{F}\) & Frobenius norm of a matrix \\ \hline \(\left\|\cdot\right\|_{2,1}\) & L\({}_{2,1}\) norm of a matrix \\ \hline \(\left\|\cdot\right\|_{2}\) & \(L_{2}\) norm of a function \\ \hline \(\circ\) & Hadamard product \\ \hline \end{tabular}
\end{table}
Table 1: The notations used in this article.  
Based on the observed cases in which each instance has a different missing value in an incomplete data set, we cannot apply an optimization algorithm to obtain the weight matrices \(\mathbf{W}\) and \(\mathbf{V}\). In order to take into account the observed information in an incomplete data set, the indicator matrix \(\mathbf{I}\) is employed. To minimize the reconstruction error and the residual loss term and to preserve the incomplete data manifold structure, we can express the problem as follows:  
\[\min_{\mathbf{W},\mathbf{V}}\ \left\|\mathbf{I}\circ(\mathbf{X}-\mathbf{X} \mathbf{W})\right\|_{2,1}+\gamma\left\|\mathbf{I}\circ(\mathbf{X}-\mathbf{V} \mathbf{X})\right\|_{2,1}+\beta\sum_{i,j}v_{ij}\|\mathbf{I}_{i}\circ(\mathbf{ x}_{i}\mathbf{W})-\mathbf{I}_{j}\circ(\mathbf{x}_{j}\mathbf{W})\right\|_{2}^{2}+ \lambda\left\|\mathbf{W}\right\|_{2,1}, \tag{6}\]  
where \(\circ\) is the Hadamard product, which enables us to formulate the objective function of the proposed unsupervised feature selection for incomplete data (UFS-ID) method.The first term in problem (6) is used to minimize the loss function of \(\mathbf{W}\) and the second term is used to minimize the reconstruction error of data instances. To preserve the incomplete data structure, the third term, for structure embedding,is added into the problem (6), and the fourth term of problem (6) is used to force the matrix \(\mathbf{W}\) to havesparsity and robustness. Since these four terms of the objective function in (6) are related to each other, they can jointly improve the performance of feature selection in an incomplete data set.  
**Remark 1**.: \(\lambda\) _is a nonnegative tuning parameter used to control the sparsity of the feature weight coefficient matrix \(\mathbf{W}\), \(\beta\) is a nonnegative tuning parameter used to balance the structure embedding in the problem (6), and \(\gamma\) is a nonnegative parameter used to control the reconstruction error of data instances. In our experiments, we observed that the results of the proposed approach were sensitive to the regularization parameters \(\gamma\), \(\beta\), and \(\lambda\), so these parameters were all tuned in the grid \(\{10^{-3},10^{-2},10^{-1},10,10^{2},10^{3}\}\) in our experiments._