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

Low-Tubal-Rank Tensor Recovery via Factorized Gradient Descent

We study matrix sensing, which is the problem of reconstructing a low-rank
matrix from a few linear measurements. It can be formulated as an
overparameterized regression problem, which can be solved by factorized
gradient descent when starting from a small random initialization.
Linear neural networks, and in particular matrix sensing by factorized
gradient descent, serve as prototypical models of non-convex problems in modern
machine learning, where complex phenomena can be disentangled and studied in
detail. Much research has been devoted to studying special cases of asymmetric
matrix sensing, such as asymmetric matrix factorization and symmetric positive
semi-definite matrix sensing.
Our key contribution is introducing a continuous differential equation that
we call the $\textit{perturbed gradient flow}$. We prove that the perturbed
gradient flow converges quickly to the true target matrix whenever the
perturbation is sufficiently bounded. The dynamics of gradient descent for
matrix sensing can be reduced to this formulation, yielding a novel proof of
asymmetric matrix sensing with factorized gradient descent. Compared to
directly analyzing the dynamics of gradient descent, the continuous formulation
allows bounding key quantities by considering their derivatives, often
simplifying the proofs. We believe the general proof technique may prove useful
in other settings as well.

矩阵感知是从少量线性测量中重建低秩矩阵的问题，我们引入了连续微分方程，称其为 “扰动梯度流”，通过边界足够有界的累计误差，证明扰动梯度流迅速收敛到真实目标矩阵，从而提供了一种基于梯度下降的非对称矩阵感知的新证明方法。