# Tensor Completion Based on Triple Tubal Nuclear Norm
## 4 TriTNN-Based Tensor Completion
### An ADMM Solver to Problem (21)
For the proposed TriTNN, the authors believe it can outperform many nuclear norm-based tensor completion models because more spatial-temporal information is exploited. However, generally speaking, it has the following two drawbacks:

1. Computational inefficiency: Compared to TNN and t-TNN, it is more time-consuming since it involves computing TNN, t-TNN and rt-TNN (see Equation (18)). 2. Sample inefficiency: Using the analysis of [23] and [44], to complete an incomplete tensor, TriTNN needs more observations than TNN and t-TNN (limited to the scope of this paper; the authors do not discuss it further). In the future research, the authors are interested in efficient algorithms like [45] to tackle the problem of computational inefficiency. To decrease the sample complexity of TriTNN, it will be helpful to follow the suggestions in [44] to design new atomic norms like [46]. To get better visual completion performances, the authors would like to consider adding smoothness regularization in the model like [47; 48; 49] and adopting different tensorization methods like [50]. It is also helpful for studying new tensor completion models using deep neural networks [51]. For potential extensions of TriTNN, the authors would like to explore the \(p\)-th order (\(p>3\)) extension [52] and extensions to other discrete transforms other than DFT like [53]. Conceptualization, D.W. and A.W. Data curation, D.W. and X.F. Formal analysis, B.W. ([PERSON]). Methodology, D.W. and A.W. Resources, B.W. ([PERSON]) and B.W. ([PERSON]). Software, X.F. and B.W. ([PERSON]). Writing, original draft, D.W., A.W. and X.F. Writing, review and editing, B.W. ([PERSON]) and B.W. ([PERSON]). The authors are grateful to the anonymous referees for their insightful and constructive comments that led to an improved version of the paper. The authors would like to thank [PERSON] and [PERSON] for their long-time support. The authors wish to thank the audiences of White Paper Tao (WPT) for their focus.This work is partially supported by the National Natural Science Foundation of China under Grant Nos. 61702262, 61703209, 91420201 and 61472187. The authors declare no conflict of interest. ## References

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