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Recently, compressive hyperspectral imaging (CHI) has received increasing interests, which can recover a large range of scenes with a small number of sensors via compressed sensing (CS) theory. However, most of the available CHI methods separate and vectorize hyperspectral cubes into spatial and spectral vectors, which will result in heavy computational and storage burden in the recovery. Moreover, the complexity of real scene makes the sparsifying difficult and thus requires more measurements to achieve accurate recovery. In this paper, these two issues are addressed, and a new CHI approach via sparse tensors and nonlinear CS (NCS) is advanced for accurate maintenance of image structure with limited number of sensors. Based on a multidimensional multiplexing (MDMP) CS scheme, the observed measurements are denoted as tensors and a nonlinear sparse tensor coding is adopted, to develop a new tensor-NCS (T-NCS) algorithm for noniterative recovery of hyperspectral images. Moreover, two recovery schemes are advanced for T-NCS, including example-aided and self-learning CHI approaches. Finally, some experiments are performed on three real hyperspectral data sets to investigate the performance of T-NCS, and the results demonstrate its efficiency and superiority to the counterparts.