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：本文研究了从少量损坏的线性测量中恢复具有潜在低管秩结构的张量的问题。传统方法求解该问题需要计算张量奇异值分解（t-SVD），这一过程计算量极大，使其难以应用于大规模张量处理。为应对这一挑战，我们提出了一种基于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，我们的核心方法是将大型张量分解为两个较小的因子张量，然后通过因子化梯度下降（FGD）求解问题。该策略无需计算t-SVD，从而降低了计算成本和存储需求。我们提供了严格的理论分析，确保FGD在无噪声和有噪声情况下均能收敛。此外，值得注意的是，我们的方法无需精确估计张量管秩。即使在管秩略微高估的情况下，我们的方法仍能表现出稳健的性能。一系列实验表明，与其他主流方法相比，我们的方法在多个场景下展现出更快的计算速度和更小的收敛误差。