## 3 Ssrhe
### The Regularized Sparse Hypergraph Model  
To discover the complex structure of HSI data, a hypergraph model is exploited to reveal the intrinsic relations between pixels. However, it remains difficult to choose a proper neighborhood size for constructing hypergraphs. Since sparse representation has natural discriminating power to adaptively reveal the inherent relationship of data [47; 48; 49], a sparse hypergraph model is designed based on SR theory. Inspired by the observation that the most compact expression of a certain sample is generally given from similar samples, sparse coefficients are explored to find neighbors adaptively. Suppose \(\mathbf{S}\in R^{N\times N}\) is the sparse coefficient matrix of data samples. The sparse coefficients can be calculated as follows:  
\[\min_{\mathbf{S}}\|s_{i}\|_{1}\qquad s.t.\qquad\|\mathbf{x}_{i}-\mathbf{X} \mathbf{s}_{i}\|\leq\varepsilon,\mathbf{s}_{i}>0 \tag{5}\]  
where \(\varepsilon\) denotes the sparse error, \(\mathbf{s}_{i}=[\mathbf{s}_{i,1},\mathbf{s}_{i,2},...,\mathbf{s}_{i,i-1},0, \mathbf{s}_{i,i+1},...,\mathbf{s}_{i,N}]^{T}\) represents the sparse coefficients of pixel \(\mathbf{x}_{i}\), which can be optimized by using Alternating Direction Method of Multipliers (ADMM) framework [50; 51]. With the sparse coefficient matrix \(\mathbf{S}\), a hypergraph can be constructed with the criterion that nodes \(\mathbf{x}_{i}\) and \(\mathbf{x}_{j}\) are connected in a hyper-edge if sparse coefficient \(\mathbf{s}_{ij}\) is not equal to 0. Since sparse coefficients can reflect the similarity between data, non-zero correspondence coefficient indicates the correlation between pixels, a large value indicates a high similarity. Compared with Euclidean metric, sparse coefficients can more effectively select neighbors of HSI data. According to sparse coefficients and label information of samples, we construct a within-class sparse hypergraph \(\mathbf{G}^{w}(\mathbf{X},\mathbf{E}_{H}^{w},\mathbf{W}_{H}^{w})\) and a between-class sparse hypergraph \(\mathbf{G}^{b}(\mathbf{X},\mathbf{E}_{H}^{b},\mathbf{W}_{H}^{b})\) to characterize the intrinsic structure of HSI data. \(\mathbf{X}\) is denoted as the vertex set, \(\mathbf{E}_{H}^{w}\) and \(\mathbf{E}_{H}^{b}\) are the sets of intraclass hyper-edge and interclass hyper-edge. \(\mathbf{W}_{H}^{w}\) and \(\mathbf{W}_{H}^{b}\) are the weight matrices of hyper-edges of \(\mathbf{G}^{w}\) and \(\mathbf{G}^{b}\), respectively. Figure 2: Flowchart of the proposed SSRHE method. In \(\mathbf{G}^{w}\), the intraclass hyper-edge \(e^{w}\) is formed by connecting pixel \(\mathbf{x}_{i}\) with its corresponding neighbors whose sparse coefficients are non-zero. The subedge weights of \(e^{w}\) and \(e^{b}\) can be defined as  
\[w^{w}_{ij}=\left\{ \begin{array}{cl}&\varphi*s_{ij},\quad if\quad s_{ij}\
eq 0 \quad\text{ and}\quad l_{i}=l_{j}\\ &0,\quad\quad\text{ others}\end{array} \right. \tag{6}\]  
\[w^{b}_{ij}=\left\{ \begin{array}{cl}&s_{ij},\quad if\quad s_{ij}\
eq 0 \quad\text{ and}\quad l_{i}\
eq l_{j}\\ &0,\quad\text{others}\end{array} \right. \tag{7}\]  
in which \(\mathbf{s}_{ij}\) is the sparse coefficient between pixel \(\mathbf{x}_{i}\) and \(\mathbf{x}_{j}\) and parameter \(\varphi\) (\(\varphi>1\)) is used to enhance the contribution of samples from the same class for improving the discriminant power. Figure 3 illustrates the construction of sparse hypergraph on a simple classification model. As shown in Figure 3, a simple graph considers only pairwise relation between two observed samples, while the sparse hypergraph through sparse representation can select neighbors of samples adaptively and represent complex multiple relations among HSI pixels.