# Tensor Completion Based on Triple Tubal Nuclear Norm
## 3 The Triple Tubal Nuclear Norm
### The Proposed Row Twist Tubal Nuclear Norm and Triple Tubal Nuclear Norm  
As discussed above, TNN and t-TNN need FFT along the tube and row direction, respectively. Note that, for real-word visual data, like color images, the row and column carry homogeneous information, and they are better treated equally. A simple operation similar to the column twist and column squeeze, called row twist and row squeeze, respectively, is defined. Figure 3: An intuitive illustration of relationships between the original tensor, the column twist tensor, the row twist tensor, the block circulant matrix and the circulant block matricization of a tensor \(\mathbf{\mathcal{T}}\in\mathbb{R}^{3\times 3\times 3}\). Subplots (**a–c**) show the operations on the column twist tensor, the original tensor and the row twist tensor, respectively. It can be seen that \(\text{circ}(\mathbf{\mathcal{T}})\) exploits the tube redundancy in a circulant way while keeping the row and column relationship, \(\text{circ}(\overrightarrow{\mathbf{\mathcal{T}}})\) exploits the row redundancy in a circulant way while keeping the tube and column relationship and \(\text{circ}(\mathbf{\mathcal{T}})\) exploits the column redundancy in a circulant way while keeping the tube and row relationship. **Definition 11** (Tensor row twist and row squeeze). : _Let \(\mathbf{\mathcal{T}}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}\), then the row twist tensor \(\mathbf{\mathcal{T}}_{1}=\underline{\mathbf{\mathcal{T}}}_{1}\) is a tensor of size \(n_{3}\times n_{2}\times n_{1}\) whose horizontal slice \(\mathbf{\mathcal{T}}_{1}(k,\cdot;\cdot)=\mathbf{\mathcal{T}}^{(k)}\). Correspondingly, the row squeeze tensor of \(\mathbf{\mathcal{T}}_{1}\), i.e., \(\mathbf{\mathcal{T}}=\underline{\mathbf{\mathcal{T}}}_{1}\), can be obtained by the reverse process, i.e., \(\mathbf{\mathcal{T}}^{(k)}=\text{RowSqueeze}(\mathbf{\mathcal{T}}_{1}(k,\cdot; \cdot))\). See Figure 4._  
**Definition 12** (Row twist tubal nuclear norm (rt-TNN)). : _The row twist tubal nuclear norm (rt-TNN) of a tensor \(\mathbf{\mathcal{T}}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}\) is defined as the tubal nuclear norm of its row twisted tensor, i.e.,_  
\[\|\mathbf{\mathcal{T}}\|_{\xrightarrow{\star}}=\|\underline{\mathbf{\mathcal{T}}}_{ \xrightarrow{\star}}\|_{\star}=\|\text{\rm bicirc}(\underline{\mathbf{\mathcal{T}} }_{\xrightarrow{\star}})\|_{\star}. \tag{15}\]  
As illustrated in Subplot (c) of Figure 3, from the circulant perspective, rt-TNN essentially exploits the column redundancy in a circulant way while keeping the row and tube relationship. From the computational perspective, FFT is operated along the column direction to compute rt-TNN. The proximal operator of rt-TNN can be derived from the proximal operator of TNN as follows:  
\[\text{RowSqueeze}(\mathfrak{S}_{\text{r}}(\underline{\mathbf{\mathcal{T}}}_{ \xrightarrow{\star}}))=\underset{\mathbf{\mathcal{T}}}{\text{argmin}}\ \frac{1}{2}\|\mathbf{\mathcal{T}}_{0}-\mathbf{\mathcal{T}}\|_{\text{F}}^{2}+\tau\|\mathbf{ \mathcal{T}}\|_{\xrightarrow{\star}}.\]  
It should be noted that each of TNN, t-TNN and rt-TNN only exploits one type of redundancy, i.e., the tube, row and column redundancies, in a circulant way. Real-world data may have more than one type of redundancy, and it is beneficial to exploit such a property. To simultaneously exploit the tube, row and column redundancy in a circulant way while keeping other relationships, we simply combine the TNN, t-TNN and rt-TNN to get the triple tubal nuclear norm. **Definition 13** (Triple tubal nuclear norm).