Subspace-orbit randomized algorithms for low rank approximations of third-order tensors in t-product format

Abstract

This paper focuses on computing low rank approximations of third-order tensors in the t-product format using random techniques. Given a truncated term $K$, we derive randomized algorithms for approximating the $K$-term t-SVD, which is called as the subspace-orbit randomized t-SVD (sort-SVD). Additionally, we conduct an analysis of the deterministic and probabilistic error bounds of the proposed algorithm, subject to certain assumptions. We integrate the present algorithm with the power method to enhance the accuracy of the approximate the $K$-term t-SVD. Furthermore, we demonstrate the effectiveness of our algorithms through numerous numerical examples. Lastly, the proposed algorithms are employed to compress data tensors from various image databases.