Accelerated Tensor Robust Principal Component Analysis via Factorized Tensor Norm Minimization

In this paper, we aim to develop an efficient algorithm for the solving Tensor Robust Principal Component Analysis (TRPCA) problem, which focuses on obtaining a low-rank approximation of a tensor by separating sparse and impulse noise. A common approach is to minimize the convex surrogate of the tensor rank by shrinking its singular values. Due to the existence of various definitions of tensor ranks and their corresponding convex surrogates, numerous studies have explored optimal solutions under different formulations. However, many of these approaches suffer from computational inefficiency primarily due to the repeated use of tensor singular value decomposition in each iteration. To address this issue, we propose a novel TRPCA algorithm that introduces a new convex relaxation for the tensor norm and computes low-rank approximation more efficiently. Specifically, we adopt the tensor average rank and tensor nuclear norm, and further relax the tensor nuclear norm into a sum of the tensor Frobenius norms of the factor tensors. By alternating updates of the truncated factor tensors, our algorithm achieves efficient use of computational resources. Experimental results demonstrate that our algorithm achieves significantly faster performance than existing reference methods known for efficient computation while maintaining high accuracy in recovering low-rank tensors for applications such as color image recovery and background subtraction.