This paper considers the problem of recovering a tensor with an underlying low-tubal-rank structure from a small number of corrupted linear measurements. Traditional approaches tackling such a problem require the computation of tensor Singular Value Decomposition (t-SVD), that is a computationally intensive process, rendering them impractical for dealing with large-scale tensors. Aim to address this challenge, we propose an efficient and effective low-tubal-rank tensor recovery method based on a factorization procedure akin to the Burer-Monteiro (BM) method. Precisely, our fundamental approach involves decomposing a large tensor into two smaller factor tensors, followed by solving the problem through factorized gradient descent (FGD). This strategy eliminates the need for t-SVD computation, thereby reducing computational costs and storage requirements. We provide rigorous theoretical analysis to ensure the convergence of FGD under both noise-free and noisy situations. Additionally, it is worth noting that our method does not require the precise estimation of the tensor tubal-rank. Even in cases where the tubal-rank is slightly overestimated, our approach continues to demonstrate robust performance. A series of experiments have been carried out to demonstrate that, as compared to other popular ones, our approach exhibits superior performance in multiple scenarios, in terms of the faster computational speed and the smaller convergence error.

本文提出了基于Burer-Monteiro方法的高效低管秩张量恢复算法，通过张量分解和因式梯度下降的策略，避免了传统计算张量奇异值分解的高计算成本和存储要求，同时在多个场景中展示了与其他方法相比更快的计算速度和更小的收敛误差。

低秩张量恢复的因式化梯度下降法