Quaternion tensor low rank approximation

In this paper, we propose novel approaches for low-rank approximation of quaternion tensors. The first method employs quasi-norms to approximate a low-rank tensor using the QT-product, which generalizes the L-product to N-mode quaternions. The second method leverages Non-Convex norms to approximate both the Tucker and TT-rank for tensor completion. We demonstrate that the proposed methods provide more accurate tensor approximations compared to traditional convex relaxations of rank, such as the nuclear norm. Furthermore, we establish theoretical guarantees supporting the effectiveness of our models. To validate their performance, we conduct extensive numerical experiments, illustrating the superiority of our methods in inpainting and denoising applications. The results confirm that incorporating Non-Convex surrogate functions and quaternion tensor representations leads to enhanced reconstruction accuracy and robustness, making them valuable tools for high-dimensional data processing.