# Tensor Completion Based on Triple Tubal Nuclear Norm
## 4 TriTNN-Based Tensor Completion
### An ADMM Solver to Problem (21)
Although all of them are based on the tubal nuclear norm, the main difference lies in that TNN and t-TNN only use information of one orientation, whereas TriTNN uses information of three orientations. Now, we compare the proposed model with Tubal-Alt-Min [37]. It is based on tensor tubal rank and employs the tensor factorization strategy for tensor completion. The differences between TriTNN and Tubal-Alt-Min are: (a) TriTNN preserves the low-rank structure by summing three tubal nuclear norms, whereas Tubal-Alt-Min adopts low-rank tensor factorization to characterize the low-rank property of a tensor. In this way, they are two different kinds of models for tensor completion (since Tubal-Alt-Min and the proposed TriTNN are quite different algorithms and the main goal of this paper is to improve upon TNN, we do not compare Tubal-Alt-Min in the experiment section.) (b) Since TriTNN is based on the tubal nuclear norm, it is formulated as a convex optimization problem (21). Benefiting from convexity, each local minimum of Problem (21) must be a global minimum. However, Tubal-Alt-Min is formulated as a non-convex optimization problem, thus it may produce sub-optimal solutions. ## 5 Experiments

In this section, extensive experiments will be conducted to explore the effectiveness of the proposed Algorithm 1. All the codes are implemented in MATLAB language, and all experiments are carried out in Windows 10 based on an Intel Core(TM) 2.60-GHz CPU with 12 G RAM. To explore the effectiveness of the proposed TriTNN-based model, we compare with the following nuclear norm-based tensor completion models:

* The tensor nuclear norm-based model with ADMM solver: high accuracy low-rank tensor completion (HaLRTC, denoted by SNNin this paper) (code available: [[http://www.cs.rochester.edu/u/jiu/publications.html](http://www.cs.rochester.edu/u/jiu/publications.html)]([http://www.cs.rochester.edu/u/jiu/publications.html](http://www.cs.rochester.edu/u/jiu/publications.html))) [2], The tensor nuclear norm is defined as the weighted sum of nuclear norms of the unfolding matrices along each mode (thus, we denote this model as SNN): \[\|\mathbf{\mathcal{T}}\|_{SNN}:=\alpha_{1}\|\mathbf{T}_{(1)}\|_{*}+\alpha_{2}\|\mathbf{T}_{ (2)}\|_{*}+\alpha_{3}\|\mathbf{T}_{(3)}\|_{*},\] where \(\alpha_{1},\alpha_{2},\alpha_{3}\) are positive parameters and \(\mathbf{T}_{(i)}\in\mathbb{R}^{n_{i}\times\prod_{j\
eq i}n_{i}}\), \(i=1,2,3\), is the unfolding matrix of tensor \(\mathbf{\mathcal{T}}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}\) along the \(i\)-th mode [2]. * The latent tensor nuclear norm-based model (LatentNN) (code available: [[https://github.com/ryotat/tensor](https://github.com/ryotat/tensor)]([https://github.com/ryotat/tensor](https://github.com/ryotat/tensor))) [21]. The latent tensor nuclear norm is defined as:\[\|\boldsymbol{\mathcal{T}}\|_{latent}:=\inf_{\boldsymbol{\mathcal{X}}+\boldsymbol{ \mathcal{Y}}+\boldsymbol{Z}=\boldsymbol{\mathcal{T}}}\|\boldsymbol{X}_{(1)}\|_ {*}+\|\boldsymbol{Y}_{(2)}\|_{*}+\|\boldsymbol{Z}_{(3)}\|_{*},\] where \(\boldsymbol{X}_{(1)},\boldsymbol{Y}_{(2)}\) and \(\boldsymbol{Z}_{(3)}\) are the first-mode, second-mode and third-mode unfoldings of latent tensors \(\boldsymbol{\mathcal{X}},\boldsymbol{\mathcal{Y}}\) and \(\boldsymbol{\mathcal{Z}}\), respectively. * The square nuclear norm-based model (SquareNN) (code available: [[https://sites.google.com/site/mucun1988/publi](https://sites.google.com/site/mucun1988/publi)]([https://sites.google.com/site/mucun1988/publi](https://sites.google.com/site/mucun1988/publi))) [23]. The square nuclear norm of a tensor is defined as the nuclear norm of the most balanced unfolding of a tensor (see [7; 23]). * The most related tubal nuclear norm-based model (TNN) (code available: [[https://github.com/jamiezeminzhang/](https://github.com/jamiezeminzhang/)]([https://github.com/jamiezeminzhang/](https://github.com/jamiezeminzhang/))) [30] and the twist tubal nuclear norm-based model (t-TNN) [16].