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Sparse unmixing (SU) algorithms use the existing spectral library as prior knowledge to analyze the endmembers and estimate abundance maps. The majority of SU algorithms use loss functions based on the $L_{2,1}$ -norm or $F$ -norm to minimize reconstruction error. They have different advantages and shortcomings. In short, $F$ -norm has a differentiable characteristic, and it is easy to minimize as a loss function. However, it is very sensitive to heavy noise and outliers. While the $L_{2,1}$ -norm emphasizes the reconstruction error on each band and is robust to noise with different intensities in different bands. But the $L_{2,1}$ -norm is nondifferentiable at zero-point. This article introduces an adaptive loss function based on the $\sigma $ -norm for SU, which combines the advantages of the $L_{2,1}$ -norm and $F$ -norm. The adaptive loss function is related to a nonnegative parameter $\sigma $. By adjusting the parameter $\sigma $ , the adaptive loss function can approach the $F$ -norm or $L_{2,1}$ -norm. To the best of our knowledge, it is the first time to apply an adaptive loss function to SU. Moreover, the adaptive loss function is globally differentiable, and we propose an optimization algorithm for the adaptive loss function and verify its convergence. Experiments on real-world and synthetic HSIs show that the adaptive loss function effectively enhances the performance of the SU algorithms. [ABSTRACT FROM AUTHOR]

Semisupervised band selection (BS) technique plays an important role in processing hyperspectral images (HSIs) because of its superiority of using the limited labeled data and plentiful unlabeled data to select the discriminative and informative feature, which copes with the problem of high-dimensional and scare labeled samples of HSIs. Among semisupervised BS models, graph-based methods are superior to others in many situations and have received more and more attention. However, traditional graph-based models construct a similarity matrix and select valuable bands independently. The similarity matrix remains constant, which will damage the local manifold structure and lead to a suboptimal result. To solve this problem, we propose a semisupervised band selection with an optimal graph (BSOG) approach, which performs BS and local structure learning simultaneously. Instead of fixing the input similarity matrix, the similarity matrix is updated constantly to learn a better local structure. Besides, the learned similarity matrix is adaptive. Then, the optimal band subset can be selected by analyzing the obtained projection matrix $\boldsymbol W$. An efficient and simple optimization algorithm is proposed to solve this model. Experiments on four real HSIs data validate the effectiveness of the proposed model. [ABSTRACT FROM AUTHOR]