Revisiting High-Order Tensor Singular Value Decomposition From Basic Element Perspective

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.