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Linear spectral unmixing is an essential technique in hyperspectral image processing and interpretation. In recent years, deep learning-based approaches have shown great promise in hyperspectral unmixing, in particular, unsupervised unmixing methods based on autoencoder networks are a recent trend. The autoencoder model, which automatically learns low-dimensional representations (abundances) and reconstructs data with their corresponding bases (endmembers), has achieved superior performance in hyperspectral unmixing. In this article, we explore the effective utilization of spatial and spectral information in autoencoder-based unmixing networks. Important findings on the use of spatial and spectral information in the autoencoder framework are discussed. Inspired by these findings, we propose a spatial-spectral collaborative unmixing network, called SSCU-Net, which learns a spatial autoencoder network and a spectral autoencoder network in an end-to-end manner to more effectively improve the unmixing performance. SSCU-Net is a two-stream deep network and shares an alternating architecture, where the two autoencoder networks are efficiently trained in a collaborative way for estimation of endmembers and abundances. Meanwhile, we propose a new spatial autoencoder network by introducing a superpixel segmentation method based on abundance information, which greatly facilitates the employment of spatial information and improves the accuracy of unmixing network. Moreover, extensive ablation studies are carried out to investigate the performance gain of SSCU-Net. Experimental results on both synthetic and real hyperspectral data sets illustrate the effectiveness and competitiveness of the proposed SSCU-Net compared with several state-of-the-art hyperspectral unmixing methods.