# An Effective Multi-Label Feature Selection Model Towards Eliminating Noisy Features
## 3 The Methodology: ELC
### Property Analysis  
The feature subset \(\hat{\mathbf{F}}=\left\{\hat{\mathbf{f}}_{1},\hat{\mathbf{f}}_{2},\ldots,\hat {\mathbf{f}}_{k}\right\}\) that is selected by ELC can be considered as maximally maintaining feature-label space consistency. \(\hat{\mathbf{F}}\) is expected to be constituted by the remarkable features and exclude the noisy ones. In this subsection, we will further analyze the properties of ELC and reveal its underlying characteristics. Suppose that each feature in \(\mathbf{F}\) has been standardized to have mean zero and unit length. Then, the following things hold for Equation (1):  
\[\left\|\hat{\mathbf{Y}}^{T}\hat{\mathbf{Y}}-\mathbf{S}\right\|_{F}^{2}=\left\| \frac{1}{n^{2}}\left(\mathbf{Y}^{T}(\mathbf{X}\mathbf{W})(\mathbf{X}\mathbf{W} )^{T}\mathbf{Y}\right)-\mathbf{S}\right\|_{F}^{2}.\]  
This is the objective of ELC. For more clearly illustrating its properties, let \(\hat{\mathcal{S}}=n^{2}\mathbf{S}\) and \(\mathcal{H}=\mathbf{Y}^{T}(\mathbf{X}\mathbf{W})(\mathbf{X}\mathbf{W})^{T} \mathbf{Y}\). Then,  
\[\left\|\hat{\mathbf{Y}}^{T}\hat{\mathbf{Y}}-\mathbf{S}\right\|_{F}^{2}=\frac{1 }{n^{2}}\left(tr(\mathcal{H}^{T}\mathcal{H})+tr(\hat{\mathcal{S}}^{T}\hat{ \mathcal{S}})-2 tr(\hat{\mathcal{S}}^{T}\mathcal{H})\right).\]  
Three terms are involved in this equation. Clearly, \(tr(\hat{\mathcal{S}}^{T}\hat{\mathcal{S}})\) represents the label correlation information extracted from the label space and is constant in the selection process. Thus, it is easy toconclude that \(\min_{\mathbf{W}}\left\|\hat{\mathbf{Y}}^{T}\hat{\mathbf{Y}}-\mathbf{S}\right\|_{F}^ {2}\) is equivalent to \(\min_{\mathbf{W}}tr(\mathcal{H}^{T}\mathcal{H})\) and \(\max_{\mathbf{W}}tr(\hat{\mathcal{S}}^{T}\mathcal{H})\). Then, two properties of ELC are given as follows:  
**Property 1**. : _Label correlation information can be maximally embedded in feature selection by ELC._  
Proof. : \(tr(\hat{\mathcal{S}}^{T}\mathcal{H})=tr\left((\mathbf{X}\mathbf{W})^{T} \mathbf{Y}\hat{\mathbf{S}}\mathbf{Y}^{T}(\mathbf{X}\mathbf{W})\right)=\sum_{i= 1}^{k}\mathbf{\hat{f}}_{i}^{T}(\mathbf{Y}\hat{\mathbf{S}}\mathbf{Y}^{T}) \mathbf{\hat{f}}_{i}=\sum_{i=1}^{k}\mathbf{\hat{f}}_{i}^{T}\left(\sum\limits_{ c1=1}^{l}\sum\limits_{c2=1}^{l}\mathbf{y}_{c1}s_{c1,c2}\mathbf{y}_{c2}^{T} \right)\mathbf{\hat{f}}_{i}\), where \(s_{c1,c2}\) is the correlation degree of the labels \(\mathbf{y}_{c1}\) and \(\mathbf{y}_{c2}\), and \(\mathbf{X}\mathbf{W}\) indicates the selected features. Then, the following things holds: \(\min_{\mathbf{W}}\left\|\hat{\mathbf{Y}}^{T}\hat{\mathbf{Y}}-\mathbf{S}\right\| _{F}^{2}\propto\max_{\mathbf{W}}\sum_{i=1}^{k}\mathbf{\hat{f}}_{i}^{T}\left( \sum_{c1=1}^{l}\sum_{c2=1}^{l}\mathbf{y}_{c1}s_{c1,c2}\mathbf{y}_{c2}^{T} \right)\mathbf{\hat{f}}_{i}\), \(\sum_{c1=1}^{l}\sum_{c2=1}^{l}\mathbf{y}_{c1}s_{c1,c2}\mathbf{y}_{c2}^{T}\) can be regarded as the correlation information of pairwise labels. Therefore, ELC can maximally embed label correlations in its feature selection process. Label correlation information is important for multi-label learning. For example, the images about seas may share some common labels for recognition, such as ship, fish, and seagull, and their close correlations may help us distinguish the image category and find their shared features. The existing multi-label learning methods are categorized on the basis of the label correlation orders they consider [39]. Their correlation modeling capabilities directly affect their discriminative performance. As demonstrated in Property 1, ELC can measure the pairwise label correlations. Furthermore, it can also preserve this correlation information in its constructed feature subspace, which is crucial for ELC to eliminate noisy features. In other words, the features that can maximally preserve label correlation information are preferred by ELC. This strategy facilitates ELC building a low-dimensional feature space that is consistent with the label space and also suitable for multi-label learning.