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Hyperspectral unmixing is a significant task in remote sensing image analysis. Existing learning-based methods for hyperspectral unmixing generally are in the form of an autoencoder and take geometric distances, such as spectral angle distance (SAD) as loss functions. These methods ignored the distribution similarity between the observation and the reconstruction, which might help improve the unmixing performance. Besides, the autoencoder is trained by directly comparing the difference between the observation and the reconstruction, and the difference between their features has been neglected. Based on the above considerations, we propose a joint metric neural network for hyperspectral unmixing, by introducing the Wasserstein distance and feature matching as regularization terms and SAD as the underlying loss. The proposed neural network consists of two parts: an autoencoder is used for endmember extraction and abundance estimation, while a discriminator is used to compute the Wasserstein distance. The Wasserstein distance can stably provide useful gradient information that promotes the autoencoder to reach a solution with better unmixing performance. The feature matching is adapted to an intermediate layer of the discriminator for enforcing the features of the observation and the reconstruction to be equal, which can lead to further improvement of the unmixing performance. The model analysis and the regularization parameter analysis are conducted to demonstrate the effectiveness of our method. Experimental results on four real-world hyperspectral data sets show that our method outperforms the state-of-the-art methods, especially in terms of abundance estimation.