Recently, tensor singular value decomposition (t-SVD), based on the tensor-tensor product (t-product), has become a powerful tool for processing third-order tensor data. However, constrained by the fact that the basic element is the fiber (i.e., vector) in the t-product, higher-order tensor data (i.e., order $d>3$ ) are usually unfolded into third-order tensors to satisfy the classical t-product setting, which leads to the destroying of high-dimensional structure. By revisiting the basic element in the t-product, we suggest a generalized t-product called element-based tensor-tensor product (elt-product) as an alternative of the classic t-product, where the basic element is a $(d-2)$ th-order tensor instead of a vector. The benefit of the elt-product is that it can better preserve high-dimensional structures and that it can explore more complex interactions via higher-order convolution instead of first-order convolution in classic t-product. Starting from the elt-product, we develop new tensor-SVD and low-rank tensor metrics (e.g., rank and nuclear norm). Equipped with the suggested metrics, we present a tensor completion model for high-order tensor data and prove the exact recovery guarantees. To harness the resulting nonconvex optimization problem, we apply an alternating direction method of the multiplier (ADMM) algorithm with a theoretical convergence guarantee. Extensive experimental results on the simulated and real-world data (color videos, light-field images, light-field videos, and traffic data) demonstrate the superiority of the proposed model against the state-of-the-art baseline models.

中文翻译：

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从基本元角度重新审视高阶张量奇异值分解


最近，基于张量-张量积 （t-product） 的张量奇异值分解 （t-SVD） 已成为处理三阶张量数据的强大工具。然而，受制于基本元素是 t 积中的纤维（即向量）这一事实，高阶张量数据（即 $d>3$ 阶）通常被展开为三阶张量以满足经典的 t 积设置，从而导致高维结构的破坏。通过重新审视 t 积中的基本元素，我们提出了一种称为基于元素的张量-张量积 （elt-product） 的广义 t 积，作为经典 t 积的替代方案，其中基本元素是 $（d-2）$ 的 th-order 张量，而不是向量。elt-product 的好处是它可以更好地保留高维结构，并且它可以通过高阶卷积而不是经典 t 产品中的一阶卷积来探索更复杂的交互。从 elt-product 开始，我们开发了新的 tensor-SVD 和低秩张量度量（例如，rank 和 nuclear norm）。配备建议的指标，我们为高阶张量数据提供了一个张量完成模型，并证明了确切的恢复保证。为了利用由此产生的非凸优化问题，我们应用了具有理论收敛保证的乘子交替方向法 （ADMM） 算法。对仿真和真实世界数据（彩色视频、光场图像、光场视频和交通数据）的大量实验结果表明，与最先进的基线模型相比，所提出的模型具有优势。