## 1 Introduction
The aforementioned methods are designed as spectral-based DR methods, in which the spatial relationship between a pixel and its spatial neighborhood is not taken into consideration for DR. Recent investigations show that incorporating spatial information into traditional spectral-based DR methods can further improve the performance of HSI classification [33; 34; 35; 36; 37]. [PERSON] et al. presented a spatially adaptive model to extract the spectral and spatial-contextual information, which significantly enhances the land cover classification performance in both accuracy and computational efficiency [38]. Local pixel NPE (LPNPE) [23] and spatial consistency LLE [39] were proposed to reveal the local manifold structure in HSI data by using the distance between different spatial blocks instead Euclidean distance between pixels. As an extension of LPNPE, spatial and spectral information-based RLDE (SSRLDE) tries to maximize the ratio between local spatial-spectral data scatter and global spatial-spectral data scatter for enhancing the representation ability of embedding features [23]. The spatial-spectral coordination embedding (SSCE) method defines a spatial-spectral coordination distance for neighbor selection, and it can reduce the probability that heterogeneous objects are selected as nearest neighbors [40]. Discriminative spectral-spatial margin (DSSM) exploits spatial-spectral neighbors to obtain the low-dimensional embedding via preserving the local spatial-spectral relationship of HSI data [41]. These spatial-spectral DR methods have difficulty discovering the complex relationships in HSI data due to their pairwise nature. Recently, the spatial information in HSI has been explored to construct a spatial-spectral hypergraph model [42; 43; 44]. [PERSON] et al. proposed an adaptive hyperedge weight estimation scheme to preserve the prominent hyperedges, which is better for improving the classification accuracy [45]. [PERSON] et al. introduced a hypergraph embedding model for feature extraction, which can represent higher order relationships [46]. However, these spatial-spectral hypergraph methods are unsupervised and do not use prior information in HSI data, which is not conducive to extract discriminant features for enhancing the classification performance. Motivated by the above limitations, a new hypergraph embedding method termed spatial-spectral joint regularized sparse hypergraph embedding (SSRHE) is proposed for DR of HSI data. SSRHE explores sparse coefficients and label information of pixels to adaptively select neighbors for constructinga regularized sparse intraclass hypergraph and a regularized sparse interclass hypergraph, which can effectively represent the complex relationships in HSI data. Then, a local spatial neighborhood preserving scatter matrix and a total-scatter matrix are computed to preserve the neighborhood structure in spatial domain and the global structure in spectral domain, respectively. Finally, an optimal objective function is designed to extract spatial-spectral discriminant features, which not only preserves the local spatial structure of HSI data, but also compacts the samples belonging to the same class and separates the samples from different classes simultaneously. Therefore, embedding features achieve a good discriminative power for HSI classification. The rest of this paper is organized as follows. In Section 2, some related works are briefly introduced. Section 3 gives a detailed description of the proposed SSRHE method. In Section 4, experimental results on two real HSI datasets are reported to demonstrate the effectiveness of SSRHE. Finally, Section 5 summarizes this paper and provides some suggestions for future work.