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.

翻译：本文研究了从少量含噪线性测量中恢复具有潜在低管秩结构的张量问题。传统方法需计算张量奇异值分解（t-SVD），该过程计算量巨大，难以处理大规模张量。针对这一挑战，我们提出一种基于类似Burer-Monteiro（BM）方法分解策略的高效低管秩张量恢复方法。具体而言，我们的核心方法将大型张量分解为两个较小的因子张量，随后通过因子化梯度下降（FGD）求解问题。该策略避免了t-SVD计算，从而降低计算成本与存储需求。我们在无噪声与有噪声情况下提供了严格的理论分析，确保FGD的收敛性。值得注意的是，本方法无需精确估计张量管秩。即便管秩被略微高估，该方法仍能保持稳健性能。系列实验表明，与其他主流方法相比，本方法在多个场景中展现出更快的计算速度与更小的收敛误差。