We study the problem of recovering a low-tubal-rank tensor $\mathcal{X}\_\star\in \mathbb{R}^{n \times n \times k}$ from noisy linear measurements under the t-product framework. A widely adopted strategy involves factorizing the optimization variable as $\mathcal{U} * \mathcal{U}^\top$, where $\mathcal{U} \in \mathbb{R}^{n \times R \times k}$, followed by applying factorized gradient descent (FGD) to solve the resulting optimization problem. Since the tubal-rank $r$ of the underlying tensor $\mathcal{X}_\star$ is typically unknown, this method often assumes $r < R \le n$, a regime known as over-parameterization. However, when the measurements are corrupted by some dense noise (e.g., Gaussian noise), FGD with the commonly used spectral initialization yields a recovery error that grows linearly with the over-estimated tubal-rank $R$. To address this issue, we show that using a small initialization enables FGD to achieve a nearly minimax optimal recovery error, even when the tubal-rank $R$ is significantly overestimated. Using a four-stage analytic framework, we analyze this phenomenon and establish the sharpest known error bound to date, which is independent of the overestimated tubal-rank $R$. Furthermore, we provide a theoretical guarantee showing that an easy-to-use early stopping strategy can achieve the best known result in practice. All these theoretical findings are validated through a series of simulations and real-data experiments.

翻译：本研究探讨了在t-积框架下从含噪线性测量中恢复低管秩张量$\mathcal{X}_\star\in \mathbb{R}^{n \times n \times k}$的问题。广泛采用的策略是将优化变量分解为$\mathcal{U} * \mathcal{U}^\top$的形式（其中$\mathcal{U} \in \mathbb{R}^{n \times R \times k}$），随后应用分解梯度下降法（FGD）求解所得优化问题。由于目标张量$\mathcal{X}_\star$的真实管秩$r$通常未知，该方法常假设$r < R \le n$，即过参数化机制。然而，当测量值被稠密噪声（如高斯噪声）污染时，采用常用谱初始化的FGD所产生的恢复误差会随着高估的管秩$R$线性增长。为解决此问题，我们证明即使管秩$R$被显著高估，采用小初始化仍能使FGD实现近乎极小极大最优的恢复误差。通过四阶段分析框架，我们解析了这一现象并建立了迄今最精确的误差界，该误差界独立于高估的管秩$R$。此外，我们提供了理论保证，表明易于实施的早停策略在实践中可获得当前最优结果。所有理论发现均通过一系列仿真和真实数据实验得到验证。