A Learnable Group-Tube Transform Induced Tensor Nuclear Norm and Its Application for Tensor Completion

Ben Zheng Li, Xi Le Zhao*, Xiongjun Zhang, Teng Yu Ji, Xinyu Chen, Michael K. Ng

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

1 Citation (Scopus)


The transform-based tensor nuclear norm (TNN) methods have shown good recovery results for tensor completion. However, the TNN methods are based on the single-tube transforms in which transforms are applied to each tube independently. The performance of the single-tube transformbased TNN methods is not good for recovery of missing tubes in multidimensional images (e.g., all the observations are missing in a pixel location of multispectral images). The main aim of this paper is to address this issue by proposing and developing a learnable group-tube transform-based TNN (GTNN) method that can effectively explore the correlation of neighboring tubes by leveraging a learnable group-tube transform. The proposed learnable group-tube transform is a separable three-dimensional transform that consists of a one-dimensional spectral/temporal transform (i.e., single-tube transform) and a two-dimensional spatial transform. Such group-tube transform can effectively explore the correlation of neighboring tubes. Based on the elaborately designed low-rank metric GTNN, we suggest a low-rank tensor completion model. To solve this highly nonconvex model, we design an efficient multiblock proximal alternating minimization algorithm and establish the convergence guarantee. A variety of numerical experiments on real-world multidimensional imaging data including traffic speed data, color images, videos, and multispectral images collectively manifest that the GTNN method outperforms some state-of-the-art TNN methods especially when the observations along tubes are missing.

Original languageEnglish
Pages (from-to)1370-1397
Number of pages28
JournalSIAM Journal on Imaging Sciences
Issue number3
Early online date3 Aug 2023
Publication statusPublished - Sept 2023

Scopus Subject Areas

  • Mathematics(all)
  • Applied Mathematics

User-Defined Keywords

  • group-tube transform
  • proximal alternating minimization algorithm
  • tensor completion
  • tensor nuclear norm


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