###### Abstract  
In the past few years, the sparse representation (SR) graph-based semi-supervised learning (SSL) has drawn a lot of attention for its impressive performance in hyperspectral image classification with small numbers of training samples. Among these methods, the probabilistic class structure regularized sparse representation (PCSSR) approach, which introduces the probabilistic relationship between samples into the SR process, has shown its superiority over state-of-the-art approaches. However, this category of classification methods only apply another SR process to generate the probabilistic relationship, which focuses only on the spectral information but fails to utilize the spatial information. In this paper, we propose using the class adjusted spatial distance (CASD) to measure the distance between each two samples. We incorporate the proposed a CASD-based distance information into PCSSR mode to further increase the discriminability of original PCSSR approach. The proposed method considers not only the spectral information but also the spatial information of the hyperspectral data, consequently leading to significant performance improvement. Experimental results on different datasets demonstrate that compared with state-of-the-start classification models, the proposed method achieves the highest overall accuracies of 99.71%, 97.13%, and 97.07% on Botswana (BOT), Kennedy Space Center (KSC) and the truncated Indian Pines (PINE) datasets, respectively, with a small number of training samples selected from each class. hyperspectral image classification; semi-supervised learning; sparse representation; spatial distance information; regularizer +
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and promising within limited labeled data samples which combines activate learning (AL) with SSL. The generative models such as expectation-maximization algorithms with finite-mixture models [10] have been applied for HSI classification. It is worth mentioning a self-training method based on convolutional neural networks (CNN) proposed by [PERSON] et al. [12]. In their work, authors propose a CNN-based classification framework which uses self-training to gradually assign pseudo labels to unlabeled samples by clustering and employs spatial constraints to regulate self-training process. It is an attractive work that combines the spectral neighborhood information with the spectral information and achieves high performance on several datasets. However, the CNN-based method could be time-consuming at the training stage, and the performance of a self-training model is highly dependent on the initial samples it chooses. Among all SSL methods, the graph-based method [13] has attracted attention from many researchers because it is easy to analyze the mathematical formulation and can obtain a close-form solution. On the other hand, sparse representation (SR) provides us a reliable way, due to its solid foundation in mathematics, to describe the linkage between samples, which could help the graph building. The SR method was first introduced by [PERSON] et al. [13] and [PERSON] et al. [14] to generate the L1-graph. Afterwards, the SR-based graph method was applied in the HSI classification [13; 14; 15; 16; 17]. For example, [PERSON] et al. [15] proposed the L1-graph semi-supervised learning model for hyperspectral image classification, and [PERSON] et al. [17] presented the probabilistic class structure regularized sparse representation (PCSSR) approach which outperforms state-of-the-art algorithms in graph construction in most cases [18; 19]. Different from normal SR methods, the PCSSR approach introduces a probabilistic class structure regularizer into the SR model, where probabilistic class structure reflects the probabilistic relationship between each sample and each class, and further, calculates the distance between each two samples based on their probabilistic relationship. With the distance information provided, the process of the SR algorithm will be guided by it.