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Low-rank tensor completion (LRTC) aims to recover a complete low-rank tensorfrom incomplete observed tensor, attracting extensive attention in variouspractical applications such as image processing and computer vision. However,current methods often perform well only when there is a sufficient of observedinformation, and they perform poorly or may fail when the observed informationis 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 oftensor low-rank structures, namely tensor Tucker rank and tubal rank, therebyenhancing the inherent correlations between known and missing elements. Toaddress the challenge of applying two tensor ranks with significantly differentdirectly 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 areprovided. Experiments on various real datasets demonstrate that the proposedmethod outperforms state-of-the-art methods significantly. Particularly, theproposed method achieves satisfactory recovery even when the observedinformation is as low as 1\%, and the recovery performance improvessignificantly as the observed information increases.

Low-rank tensor completion via tensor joint rank with logarithmic composite norm