Tensor Ring Discriminant Analysis Used for Dimension Reduction of Remote Sensing Feature Tensor

Effective feature dimension reduction (DR) from high-dimensional remote sensing images has been a significant challenge for remote sensing object recognition. Directly adopting vector-based DR method ignores remote sensing data's inherent tensor structure information, leading to the undersample problem (USP). In addition, the existing tensor-based DR methods either require an exponential storage space increasing with the orders of the input tensor (i.e., Tucker-form methods) or are dependent on the permutation of tensor modes limiting the discriminant capability of the DR results (i.e., tensor train (TT) form methods). To conquer these problems, unlike the existing Tucker or TT form feature representation, the novel tensor ring (TR) subspace learning theory is proposed systematically and rigorously to extend the traditional vector and tensor subspace learning to the TR subspace. Then, by embedding the Fisher criterion into TR subspace, the TR discriminant analysis (TRDA) is proposed to achieve DR for remote sensing tensors with flexible tensor rank and lower storage cost. To train TRDA under different computing resources, nonrecursive and exact TRDA training methods are presented to obtain the global suboptimal and local optimal solutions, respectively. Furthermore, to adapt to the case of multisource data and unlabeled data, the multiple TRDA (MTRDA) and semi-supervised TRDA (S-TRDA) are further proposed to refine multisource features in multiple TR subspaces and absorb useful information using adaptive scatter tensor, respectively. Using optical, hyperspectral, and SAR datasets, experimental results demonstrate that the proposed TRDA can obtain better recognition accuracy and smaller storage cost than the typical vector and tensor-based DR methods.