# Fusing Local and Global Information for One-Step Multi-View Subspace Clustering
## 3 Proposed Method
### Formulation  
In this section, we provide the detailed modeling process of the proposed LGOMSC method. For a multi-view data set with \(V\) views, let \(\mathbf{X}^{1},\ldots,\mathbf{X}^{V}\) be the data matrices of the \(V\) views and \(\mathbf{X}^{v}=\left\{x_{1}^{v},\ldots,x_{n}^{v}\right\}\in\mathbb{R}^{d_{v}\times n}\) be the \(v\)-th view data, where \(d_{v}\) is the dimensionality of the \(v\)-th view, and \(n\) is the number of data points. Since each view contains view-specific information about the object, we respectively compute the view-specific subspace \begin{table}
\begin{tabular}{l l} \hline \hline  & **Notation** & **Definition** \\ \hline \(\mathbf{1}_{n}\) & \(n\times n\) Identity matrix \\ \(\mathbf{1}\) & All-ones column vector \\ \(n\) & Number of data sample \\ \(c\) & Number of clusters \\ \(V\) & Number of views \\ \(d_{v}\) & Feature dimension of the \(v\)-th view \\ \(\mathbf{X}^{v}\in\mathbb{R}^{d_{v}\times n}\) & Feature matrix of the \(v\)-th view \\  & Represented as the \(i-\) th row, \(j-\) th column, and \(ij-\) th element of matrix \(\mathbf{X}\), respectively \\ \(\mathbf{X}^{\text{T}}\) & The transpose of a matrix \\ \(\mathbf{S}^{v}\in\mathbb{R}^{n\times n}\) & Subspace representation matrix of the \(v\)-th view \\ \(\mathbf{U}\in\mathbb{R}^{n\times n}\) & Common subspace representation matrix \\ \(\|\cdot\|_{F}\) & The Frobenius norm \\ \(Tr(\cdot)\) & Trace operator of a matrix \\ \(diag(\cdot)\) & Vector of the diagonal elements of a matrix \\ \(rank(\cdot)\) & The rank of a matrix \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations and abbreviations. representation of each view to capture the diversity across views. The objective function of the self-expression model for the multi-view data can be formulated as:  
\[ \begin{split}\min\limits_{\mathbf{S}^{v}}\sum\limits_{v=1}^{V}\left\| \mathbf{X}^{v}-\mathbf{X}^{v}\mathbf{S}^{v}\right\|_{F}^{2}+\lambda_{1}\sum\limits_{v=1}^{ V}\left\|\mathbf{S}^{v}\right\|_{F}^{2}\\ s.t.0\leq s_{ij}^{v}\leq 1,(\mathbf{S}^{v})^{\mathsf{T}}\mathbf{1}=\mathbf{1},diag (\mathbf{S}^{v})=0\end{split}  \tag{1}\]  
where \(\mathbf{S}^{v}=\left\{s_{1}^{v},s_{2}^{v},\ldots,s_{n}^{v}\right\}\in\mathbb{R}^{n \times n}\) is the subspace representation matrix of the \(v\)-th view, \(s_{ij}^{v}\) is the \(j\)-th element of \(s_{i}^{v}\), \(\mathbf{1}\) denotes a column vector with all entries of one, \(diag(\cdot)\) denotes a vector of the diagonal elements of a matrix. Since Model (1) adopts the entire data set to linearly reconstruct each data sample, the subspace representation matrix \(\mathbf{S}^{v}\) captures the global information of the \(v\)-th view data. However, this subspace representation obtained by Model (1) ignores the local structure to construct the subspace representations. In other words, two closed data samples should have similar subspace representations. Hence, to exploit the local information of multi-view data, we integrate the graph learning into Model (1) to compute the subspace representation. Hence, the objection function can be formulated as:  
\[ \begin{split}\min\limits_{\mathbf{S}^{v}}\sum\limits_{v=1}^{V}\left\| \mathbf{X}^{v}-\mathbf{X}^{v}\mathbf{S}^{v}\right\|_{F}^{2}+\lambda_{1}\sum\limits_{v=1}^{ V}\left\|\mathbf{S}^{v}\right\|_{F}^{2}+\sum\limits_{v=1}^{V}\sum\limits_{i=1}^{ n}\sum\limits_{j=1}^{n}\left\|\mathbf{x}_{i}^{v}-\mathbf{x}_{j}^{v}\right\|_{2}^{2}s_{ij}^{v}\\ s.t.0\leq s_{ij}^{v}\leq 1,(\mathbf{S}^{v})^{\mathsf{T}}\mathbf{1}=\mathbf{1},diag (\mathbf{S}^{v})=0\end{split}  \tag{2}\]  
Since multi-view data come from the same object, they should have latent consistency. To characterize this consistency, we adopt a multi-view information fusion term to obtain a common subspace representation matrix \(\mathbf{U}\in\mathbb{R}^{n\times n}\) from the subspace representation matrices \(\{\mathbf{S}^{1},\mathbf{S}^{2}\ldots\mathbf{S}^{V}\}\). This term can be represented as:  
\[