# Consensus Guided Multi-View Unsupervised Feature Selection with Hybrid Regularization
## 3 Proposed Method
### Formulation  
Within the multi-view feature selection methodological framework [14], cluster labels are conventionally necessitated to supervise the learning regimen of dimensionality reduction operators through the following formulation :  
\[\min_{\mathbf{W}^{v}}\sum_{v=1}^{V}\left(\|\mathbf{X}^{v}\mathbf{W}^{v}-\mathbf{Y}^{*}\|_{F}^{ 2}+\alpha\|\mathbf{W}^{v}\|_{2,1}\right) \tag{1}\]  
The formulation incorporates an equilibrium coefficient \(\alpha\) to balance objective components, where \(\mathbf{Y}^{*}\in\mathbb{R}^{n\times k}\) is expected to indicate the ground-truth cluster assignments of all samples. The feature projection matrix \(\mathbf{W}^{v}\in\mathbb{R}^{d^{v}\times k}\) establishes an optimized mapping between the feature space and label space for the \(v\)-th view. Furthermore, structured sparsity patterns are induced through \(L_{2,1}\)-norm regularization on \(\mathbf{W}^{(v)}\), serving to mitigate feature redundancy and suppress noise artifacts through column-wise feature suppression. After solving \(\mathbf{W}^{v}\), the importance of each feature in diverse views can be derived by \(\|w_{i}^{v}\|_{2}\). This metric enables systematic selection of the top-\(r\) discriminative features through descending norm-based ranking, as features with higher norms typically capture more significant variation or discriminative power in the data representation. However, in unsupervised learning scenarios where ground-truth labels \(Y^{*}\) are inherently inaccessible, effective feature projection learning necessitates the exploitation of inherent structural relationships within the data distribution. To address this problem, we introduce ensemble clustering to capture complementary cluster structure information and design a multi-view objective function for more robust and comprehensive representation learning. Figure 1: The framework of the proposed CGMvFS. Specifically, in order to capture diverse intrinsic data structures of each view, Kmeans is applied to generate \(B\) basic partitions \(\{\mathbf{P}_{i}^{v}\}_{i=1}^{B}\), where \(\mathbf{P}_{b}^{v}\in\mathbb{R}^{n\times k_{b}^{v}}\) is a partition matrix that indicating \(k_{b}^{v}\) clusters in the \(v\)-th view space [28]. The \((i,j)\)-th entry of \(\mathbf{P}_{b}^{v}\) is defined as follows:  
\[p_{ij}= \begin{cases}1,\text{ if }\mathbf{x}_{i}\in C_{j}\\ 0,\text{ otherwise}\end{cases}  \tag{2}\]  
By concatenating \(B\) basic partitions, the ensemble partition matrix [28; 29]\(\mathbf{P}^{v}=[\mathbf{P}_{1}^{v}\ \mathbf{P}_{2}^{v}\ \cdots\ \mathbf{P}_{B}^{v}]\) is constructed to provide sufficient intra-view cluster structural information. Then, for the \(v\)-th perspective, the co-association matrix formulated can be analytically using the subsequent mathematical derivation:  
\[\mathbf{M}^{v}=\frac{1}{B}\mathbf{P}^{v}(\mathbf{P}^{v})^{\top} \tag{3}\]  
The \((i,j)\)-th entry of the co-association matrix \(\mathbf{M}^{v}\) quantifies the probability that \(\mathbf{x}_{i}^{v}\) and \(\mathbf{x}_{j}^{v}\) are co-clustered according to multiple basic partitions in the \(v\)-th view. When two samples are frequently co-assigned to identical clusters through various partitions, their corresponding entry \(M_{ij}^{v}\) converges toward 1, infrequent co-occurrences result in values approaching 0. This formulation effectively encodes similarity relationships between samples through different view-specific perspectives.