## 3 Ssrhe
### The Regularized Sparse Hypergraph Model
The weight matrices corresponding to hyper-edge regions are defined as  
\[\omega^{w}_{i}=w(e^{w}_{i})=\sum_{\mathbf{x}_{j}\in e^{w}_{i}}w^{w}_{ij} \tag{8}\]  
\[\omega^{b}_{i}=w(e^{b}_{i})=\sum_{\mathbf{x}_{j}\in e^{b}_{i}}w^{b}_{ij} \tag{9}\]  
The incidence matrix \(\mathbf{H}^{w}=[H^{w}_{ij}:h(e^{w}_{j},\mathbf{x}_{i})]\in R^{|\mathbf{E}^{w} |\times|\mathbf{X}|}\) of intraclass hypergraph is calculated by  
\[H^{w}_{ij}=\left\{\begin{array}{cl}&\exp(-\frac{(\mathbf{x}_{i}-\mathbf{x}_{ j})(\mathbf{x}_{i}-\mathbf{x}_{j})^{T}}{2t^{2}}),\quad if\quad\mathbf{x}_{i}, \mathbf{x}_{j}\in e^{w}_{j}\\ &0,\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad  
\[H^{b}_{ij}=\left\{ \begin{array}{ccc}\exp(-\frac{(\mathbf{x}_{i}-\mathbf{x}_{j})( \mathbf{x}_{i}-\mathbf{x}_{j})^{T}}{2^{2}}),&if&\mathbf{x}_{i},\mathbf{x}_{j} \in e^{b}_{j}\\ 0,&others\end{array} \right. \tag{13}\]  
Based on \(\mathbf{H}^{b}\) and \(w(e^{b})\), the degree of vertex \(\mathbf{x}_{i}\) and interclass hyper-edge \(e^{b}_{i}\) can be defined as  
\[\theta^{b}_{i}=d(\mathbf{x}_{i},e^{b}_{j})=\sum_{j}w(e^{b}_{j})h(e^{b}_{j}, \mathbf{x}_{i})=\sum_{j}w(e^{b}_{j})H^{b}_{ij} \tag{14}\]  
\[\theta^{b}_{i}=d(e^{b}_{i})=\sum_{j}h(e^{b}_{i},\mathbf{x}_{j})=\sum_{j}H^{b}_ {ij} \tag{15}\]  
In low-dimensional embedding space, the pixels from the same class should be as compact as possible for the intraclass sparse hypergraph, whereas pixels from different classes should be as separated as possible with the interclass sparse hypergraph. Therefore, the objective functions with intraclass and interclass sparse hypergraph constraint are constructed as  
\[ \begin{array}{l}\arg\min_{P}\frac{1}{2}\sum_{e^{w}_{k}\in E^{w}}\sum_{x_{i},x_{j}\in e^{b}_{k}}\frac{w(\theta^{b}_{i})h(e^{w}_{k},x_{i})h(e^{w}_{k},x_{i} )}{d(e^{w}_{k})}\big{\|}P^{T}x_{i}-P^{T}x_{j}\big{\|}^{2}\\ =\sum_{k=1}^{n}\sum_{i,j=1}^{n}\frac{w(\theta^{b}_{i})H^{w}_{k}H^{w}_{k}}{ \theta^{b}_{k}}tr(P^{T}x_{i}x_{i}T^{P}-P^{T}x_{j}x_{i}^{T}P)\\ =\left\{\sum_{k=1}^{n}\sum_{i=1}^{n}\frac{w(\theta^{b}_{i})H^{w}_{k}H^{w}_{k}} {\theta^{b}_{k}}tr(P^{T}x_{i}x_{i}T^{P})\sum_{j=1}^{n}\frac{H^{w}_{k}H^{w}_{k} }{\theta^{b}_{k}}\right\}-\left\{\sum_{k=1}^{n}\sum_{i,j=1}^{n}\frac{tr[P^{T}x _{j}w(e^{w}_{k}H^{w}_{k}H^{w}_{k}x_{i}T^{P}]}{\theta^{b}_{k}}\right\}\\ =tr(P^{T}XD^{w}_{w}X^{T}P)-tr\left[P^{T}XH^{w}W^{w}(D^{w}_{e})^{-1}(H^{w})^{T} X^{T}P\right]\\ =tr\left\{P^{T}X[D^{w}_{v}-H^{w}W^{w}(D^{w}_{e})^{-1}(H^{w})^{T}]X^{T}P\right\} \\ =tr(P^{T}XL^{w}X^{T}P)\\ \end{array}  \tag{16}\]  
\[ \begin{array}{l}\arg\max_{P}\frac{1}{2}\sum_{k^{c}\in E^{b}}\sum_{x_{i},x_{j }\in e^{b}_{k}}\frac{w(e^{b}_{k})h(e^{b}_{k^{c}_{i}})h(e^{b}_{k^{c}_{i}},x_{j })}{d(e^{b}_{i})}\big{\|}P^{T}x_{i}-P^{T}x_{j}\big{\|}^{2}\\ =\sum_{k=1}^{n}\sum_{i,j=1}^{n}\frac{w(\theta^{b}_{i})H^{b}_{k}H^{b}_{k}}{ \theta^{b}_{k}}tr(P^{T}x_{i}x_{i}T^{P}-P^{T}x_{j}x_{i}^{T}P)\\ =\left\{\sum_{k=1}^{n}\sum_{i=1}^{n}\frac{w(\theta^{b}_{i})H^{b}_{k}}{\theta^{b }_{k}}tr(P^{T}x_{i}x_{i}T^{P})\sum_{j=1}^{n}\frac{H^{b}_{k}\theta^{b}_{k}}{ \theta^{b}_{k}}\right\}-\left\{\sum_{k=1}^{n}\sum_{i,j=1}^{n}\frac{tr[P^{T}x_{j }w(e^{b}_{k}H^{b}_{k}H^{b}_{k^{c}_{i}}T^{P}]}{\theta^{b}_{k}}\right\}\\ =tr(P^{T}XD^{b}_{v}X^{T}P)-tr\left[P^{T}XH^{b}W^{b}(D^{b}_{e})^{-1}(H^{b})^{T} X^{T}P\right]\\ =tr\left\{P^{T}X[D^{b}_{v}-H^{b}W^{b}(D^{b}_{e})^{-1}(H^{b})^{T}]X^{T}P\right\} \\ =tr(P^{T}XL^{b}X^{T}P)\end{array}  \tag{17}\]  
where \(\mathbf{L}^{w}=\mathbf{D}^{w}_{v}-\mathbf{H}^{w}\mathbf{W}^{w}(\mathbf{D}^{w}_{ e})^{-1}(\mathbf{H}^{w})^{T}\) is the intraclass sparse hypergraph Laplacian matrix and \(\mathbf{L}^{b}=\mathbf{D}^{b}_{v}-\mathbf{H}^{b}\mathbf{W}^{b}\left(\mathbf{D}^{b }_{e}\right)^{-1}\left(\mathbf{H}^{b}\right)^{T}\) denotes the interclass sparse hypergraph Laplacian matrix.