# Fusing Local and Global Information for One-Step Multi-View Subspace Clustering
## 3 Proposed Method
### Formulation
\begin{split}\min\limits_{\mathbf{U}}\sum\limits_{v=1}^{V}\left\|\bm {S}^{v}-\mathbf{U}\right\|_{F}^{2}\\ s.t.\mathbf{U}\geq 0,\mathbf{U}^{\mathsf{T}}\mathbf{1}=\mathbf{1}\end{split}  \tag{3}\]  
Through minimizing Model (3), this common subspace representation matrix \(\mathbf{U}\) can make these the subspace representation matrices \(\{\mathbf{S}^{1},\mathbf{S}^{2}\ldots\mathbf{S}^{V}\}\) to have latent consistency. Hence, we add this multi-view information fusion term into Model (2) as:  
\[ \begin{split}\min\limits_{\mathbf{S}^{v},\mathbf{U},\mathbf{F}_{v}=1}^{V}\left\| \mathbf{X}^{v}-\mathbf{X}^{v}\mathbf{S}^{v}\right\|_{F}^{2}+\lambda_{1}\sum\limits_{v=1}^{ V}\left\|\mathbf{S}^{v}\right\|_{F}^{2}+\sum\limits_{v=1}^{V}\sum\limits_{i=1}^{n} \sum\limits_{j=1}^{n}\left\|\mathbf{x}_{i}^{v}-\mathbf{x}_{j}^{v}\right\|_{2}^{2}s_{ij }^{v}\\ +\sum\limits_{v=1}^{V}\left\|\mathbf{S}^{v}-\mathbf{U}\right\|_{F}^{2}\\ s.t.0\leq s_{ij}^{v}\leq 1,\mathbf{1}^{\mathsf{T}}\mathbf{s}_{i}^{v}=1,diag (\mathbf{S}^{v})=0,\mathbf{U}\geq 0,\mathbf{U}^{\mathsf{T}}\mathbf{1}=\mathbf{1}\end{split}  \tag{4}\]  
After obtaining the common subspace structure, we can get the affinity matrix \(\mathbf{W}=1/2(\mathbf{U}+\mathbf{U}^{\mathsf{T}})\) and perform spectral clustering on such a subspace affinity matrix. However, the constructions of subspace representation and clustering are divided into two individual procedures, which ignore their interactions. To address this problem, we consider introducing a rank constraint [46] on the Laplacian matrix \(\mathbf{L_{U}}=\mathbf{D}-\mathbf{W}\), where the degree matrix \(\mathbf{D}\) is defined as a diagonal matrix whose \(i\)-th diagonal element is \(d_{ii}=\sum_{j=1}^{n}w_{ij}\). If the common subspace representation matrix \(\mathbf{U}\) is non-negative, then the Laplacian matrix has the following theorem. Theorem 1: _The number of connected components in the graph with \(\mathbf{U}\) is equal to the multiplicity of zero eigenvalue of the Laplacian matrix \(\mathbf{L_{U}}\)[47]._  
According to Theorem 1, we consider making the number of zero eigenvalues of the Laplacian matrix \(\mathbf{L_{U}}\) to be equal to the number of clustering clusters, i.e., \(rank(\mathbf{L_{U}})=n-c\)By adding the rank constraint \(rank(\mathbf{L_{U}})=n-c\) into Model (4), the common subspace representation matrix \(\mathbf{U}\) will have the ideal property. Therefore, we can directly obtain the cluster result from \(\mathbf{U}\) without discretization. However, it is difficult to directly solve the rank constraint \(rank(\mathbf{L_{U}})=n-c\). It is known that \(rank(\mathbf{L_{U}})=n-c\) is equivalent to \(\sum\limits_{i=1}^{c}\sigma_{i}(\mathbf{L_{U}})=0\), where \(\sigma_{i}(\mathbf{L_{U}})\) denotes the \(i\)-th smallest eigenvalues of \(\mathbf{L_{U}}\). Since \(\mathbf{L_{U}}\) is positive semi-definite, \(\sigma_{i}(\mathbf{L_{U}})\geq 0\). According to Ky Fan's Theorem [48], \(\sum\limits_{i=1}^{c}\sigma_{i}(\mathbf{L_{U}})=\min_{\mathbf{F}^{\text{T}}\mathbf{F}- \mathbf{\text{T}}}\mathbf{r}(\mathbf{F^{\text{T}}}\mathbf{L_{U}}\mathbf{F})\). Therefore, to hold \(\sum\limits_{i=1}^{c}\sigma_{i}(\mathbf{L_{U}})=0\), the objective function of the proposed LGOMSC is formulated as:  
\[ \begin{split}\min\limits_{\mathbf{S^{v}},\mathbf{U},\mathbf{F}_{v}=1}\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}\\ +\sum\limits_{v=1}^{V}\left\|\mathbf{S^{v}}-\mathbf{U}\right\|_{F}^{2}+2 \lambda\mathbf{T}\mathbf{r}(\mathbf{F^{\text{T}}}\mathbf{L_{U}}\mathbf{F})\\ s.t.0\leq s_{ij}^{v}\leq 1,\mathbf{1^{\text{T}}}\mathbf{s}_{i}^{v}=1,diag( \mathbf{S^{v}})=0,\mathbf{U}\geq 0,\mathbf{U^{\text{T}}}\mathbf{1}=\mathbf{1},\mathbf{F^{\text{T}}}\mathbf{F}= \mathbf{I}_{c}\end{split}