Low-rank tensor completion (LRTC) aims to recover a complete low-rank tensor from incomplete observed tensor, attracting extensive attention in various practical applications such as image processing and computer vision. However, current methods often perform well only when there is a sufficient of observed information, and they perform poorly or may fail when the observed information is less than 5\%. In order to improve the utilization of observed information, a new method called the tensor joint rank with logarithmic composite norm (TJLC) method is proposed. This method simultaneously exploits two types of tensor low-rank structures, namely tensor Tucker rank and tubal rank, thereby enhancing the inherent correlations between known and missing elements. To address the challenge of applying two tensor ranks with significantly different directly to LRTC, a new tensor Logarithmic composite norm is further proposed. Subsequently, the TJLC model and algorithm for the LRTC problem are proposed. Additionally, theoretical convergence guarantees for the TJLC method are provided. Experiments on various real datasets demonstrate that the proposed method outperforms state-of-the-art methods significantly. Particularly, the proposed method achieves satisfactory recovery even when the observed information is as low as 1\%, and the recovery performance improves significantly as the observed information increases.

翻译：低秩张量补全（LRTC）旨在从不完全观测张量中恢复完整的低秩张量，在图像处理和计算机视觉等实际应用中受到广泛关注。然而，现有方法通常在观测信息充足时表现良好，在观测信息低于5%时性能不佳甚至可能失效。为了提升观测信息的利用率，本文提出了一种称为基于对数复合范数的张量联合秩（TJLC）方法。该方法同时利用张量Tucker秩和管秩这两种低秩结构，从而增强已知元素与缺失元素之间的内在相关性。针对这两种性质差异显著的张量秩直接应用于LRTC所面临的挑战，进一步提出了一种新的张量对数复合范数。随后，针对LRTC问题提出了TJLC模型和算法，并给出了TJLC方法的理论收敛性保证。在多种真实数据集上的实验表明，所提方法显著优于现有最优方法。特别地，即使在观测信息低至1%的情况下，所提方法仍能实现满意的恢复效果，且恢复性能随观测信息增加而显著提升。