The power of small initialization in noisy low-tubal-rank tensor recovery
This addresses the issue of noise sensitivity in tensor recovery for applications like signal processing, though it is incremental as it improves an existing method.
The paper tackles the problem of recovering a low-tubal-rank tensor from noisy linear measurements, showing that using small initialization in factorized gradient descent achieves a nearly minimax optimal recovery error independent of overestimated rank, with sharp error bounds validated by simulations and experiments.
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.