Efficient Tensor Robust Principal Analysis via Right Invertible Matrix Based Tensor Products

In this paper, we extend the definition of tensor products from using invertible transformations to utilising right-invertible matrices, exploring the algebraic properties of these new tensor products. On the basis of this new definition, we define the concepts of tensor rank and tensor nuclear norm and investigate their properties, ensuring consistency with their matrix counterparts. We then derive a singular value thresholding (SVT) formula to approximately solve the subproblems in the alternating direction method of multipliers (ADMM), which is a key component of our proposed tensor robust principal component analysis (TRPCA) algorithm. We conduct a complexity analysis of the proposed algorithm, demonstrating its computational efficiency, and apply it to grayscale video denoising and motion detection problems, where it shows significant improvements in efficiency while maintaining a similar level of quality. This work provides a promising approach for handling large-scale data, offering new insights and solutions for advanced data analysis tasks.