  Tensor-based methods have been widely studied to attack inverse problems in
hyperspectral imaging since a hyperspectral image (HSI) cube can be naturally
represented as a third-order tensor, which can perfectly retain the spatial
information in the image. In this article, we extend the linear tensor method
to the nonlinear tensor method and propose a nonlinear low-rank tensor unmixing
algorithm to solve the generalized bilinear model (GBM). Specifically, the
linear and nonlinear parts of the GBM can both be expressed as tensors.
Furthermore, the low-rank structures of abundance maps and nonlinear
interaction abundance maps are exploited by minimizing their nuclear norm, thus
taking full advantage of the high spatial correlation in HSIs. Synthetic and
real-data experiments show that the low rank of abundance maps and nonlinear
interaction abundance maps exploited in our method can improve the performance
of the nonlinear unmixing. A MATLAB demo of this work will be available at
https://github.com/LinaZhuang for the sake of reproducibility.
