  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.
