  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.
