# On the Synergy between Nonconvex Extensions of the Tensor Nuclear Norm for Tensor Recovery
## 4 Experimental Comparison  
### Setting  
In Section 1, we posed two questions:  
* Are the effects of the weighting and \(p\)-squared on singular values synergistic? Or does one encompass the other?
* Is simple rank-constrained minimization insufficient?  
To answer these questions, we performed some experiments using an artificial tensor. Each element of an \(N\)th order artificial tensor \(\mathcal{X}\in\mathbb{R}^{n_{1}\times\cdots\times n_{N}}\) is generated by using the Tucker model:  
\[\mathcal{X}(i_{1},\cdots,i_{N})=\sum_{1\leq j_{1}\leq r_{1},\cdots,1\leq j_{N }\leq r_{N}}\mathcal{S}(j_{1},\cdots,j_{N})\prod_{k}^{N}\mathbf{U}_{k}(i_{k}, j_{k}), \tag{19}\]  
where \([r_{1},\cdots,r_{N}]\), \(\mathcal{S}\in\mathbb{R}^{n_{1}\times\cdots\times r_{N}}\) and \(\mathbf{U}_{k}\in\mathbb{R}^{n_{k}\times r_{k}}\) are the matrix ranks of \(\text{unfold}_{m}(\mathcal{X})\)\((m=1,\cdots,N)\), the core tensor, and the factor tensor. Each element of \(\mathcal{S}\) and \(\mathbf{U}_{k}\) is generated uniformly over the intervals \([0,1]\) and \([-0.5,0.5]\), respectively. Finally, we normalized the difference between the maximum and minimum elements of \(\mathcal{X}\) to \(1\).  
As mentioned in Section 1, the widely used weights used as the ideal weights for the singular values are the inverses of the singular values of the unfolded original tensor \(\mathcal{X}_{\text{org}}\). Throughout this paper, we define our _ideal weights_ as  
\[(\mathbf{w}_{\text{ld}}(\alpha))_{i,j}=R\frac{\sigma_{i}(\text{unfold}_{j}( \mathcal{X}_{\text{org}}))^{-\alpha}}{\sum_{k}\sigma_{k}(\text{unfold}_{j}( \mathcal{X}_{\text{org}}))^{-\alpha}}, \tag{20}\]  
where \(R\) is the smaller of the row and column dimensions of \(\text{unfold}_{j}(\mathcal{X}_{\text{org}})\). Since the ideal weights are not always optimal in terms of the recovery performance, we introduce a parameter \(\alpha\) to bring additional flexibility to the setting of the weights. However, the true singular values are not available in practical applications.  
A method that does not require the true singular values is to use the singular values obtained from observations for estimating the ideal weights. One of these methods is as follows:  
\[(\mathbf{w}_{\text{Obs}}(\alpha))_{i,j}=R\frac{\sigma_{i}(\text{unfold}_{j}( \tilde{\mathcal{Y}}))^{-\alpha}}{\sum_{k}\sigma_{k}(\text{unfold}_{j}(\tilde{ \mathcal{Y}}))^{-\alpha}} \tag{21}\]  
where \(\tilde{\mathcal{Y}}\) is a tensor the missing elements of the observation tensor \(\mathcal{Y}\) filled in using the average of the observed entries of the observation tensor \(\mathcal{Y}\). We refer to these weights as _observation weights_.  
The tensor schatten-\(p\) norm with no weights is a special case of the WPSPN. Therefore, we can use the WPSPN with the following special weights as the tensor schatten-\(p\) norm:  
\[(\mathbf{w}_{\text{Uni}})_{i,j}=1. \tag{22}\]In the following experiments, we use three types of weights (the ideal weights \(\mathbf{w_{\text{Id}}}\), observation weights \(\mathbf{w_{\text{Obs}}}\) and uniform weights \(\mathbf{w_{\text{Uni}}}\)) to reveal the relationship between the weighting and schatten-\(p\) extension on the performance of WTSPN. The weight determination parameter \(\alpha\) varies in increments of 0.25 in the range \([1,4]\). The parameters of schatten-\(p\) are chosen from \(p=\{1/2,2/3,1\}\), and each element of \(\gamma\) is set to \(1/N\), where \(N\) is the order of the target tensor. In all cases, the parameter of ADMM is set to \(\lambda=100\).  
We compare the performance of Algorithm 1 using \(\mathbf{w_{\text{Id}}}\), \(\mathbf{w_{\text{Obs}}}\), and \(\mathbf{w_{\text{Uni}}}\) as well as rank-constrained minimization. The following error is used to evaluate the performance of each method:  
\[\text{error}(\hat{\mathcal{X}},\mathcal{Y})=\frac{1}{\prod_{m=1}^{N}I_{m}} \|\hat{\mathcal{X}}-\mathcal{Y}\|_{2}, \tag{23}\]  
where \(\hat{\mathcal{X}}\) is the estimated tensor obtained by each method.