# On the Synergy between Nonconvex Extensions of the Tensor Nuclear Norm for Tensor Recovery
## 4 Experimental Comparison
### Results and Discussion  
We performed recovery from observed tensors with missing rate of 0.4 and 0.8 and standard deviations of noise \(\sigma_{n}\) of 0 and 1, and the results of the combinations of these parameters are shown in (a) to (d) in Figures 1-4. The horizontal axis of each graph is the parameter \(\alpha\) used in Equations (20) and (21) for determining the weights, and the vertical axis is the performance of each method defined by Equation (23). The red, green, blue, and yellow lines show the results of Algorithm 1 with \(\mathbf{w_{\text{Id}}}\) (Id in the legends of the graphs), with \(\mathbf{w_{\text{Obs}}}\) (Obs), with \(\mathbf{w_{\text{Uni}}}\) (Uni), and with the rank-constrained minimization shown in Equation (5) (RC). In the case of Id, Obs, and Uni, the results corresponding to the different values of \(p\) are shown with different line types. Similarly, in the case of RC, the results corresponding to the different target ranks \(r\) are shown with different line types. Figure 1 is the result when the size of the original tensor is \(40\times 40\times 40\) and the rank is \([4,4,4]\). From the graphs (a)-(d) in Figure 1, one can see that  
* In the case of Id, if we choose \(\alpha<2\), the choice of \(p\) does not have much effect on the performance. The slowest degradation of performance due to the change in \(\alpha\) is obtained at \(p=1\). * In the case of Obs, \(p=1/2\) shows the best result across (a)-(d). These results are consistent with the results in previous studies [8,12,18,20]. * Regardless of \(p\), the performance of Id and Obs is the same or better than that of Uni in all cases. * In all cases, RC shows the worst performance unless we can choose the correct rank \(r\). Figure 2 shows the results when we only change the tensor rank to \([5,5,5]\). Note that the size of the tensor is still \(40\times 40\times 40\). The results show a similar trend as in the case of Figure 1. From the results in Figures 1 and 2, for the third-order tensor, we can conclude that the effect of the choice of the weights, \(p\), and the algorithms on performance is rank-independent. To reveal the impact of the changes in the order of the tensor on the common trend in Figures 1 and 2, we performed experiments on the 4 th-order tensor. The results are shown in Figures 3 and 4. In Figures 3 and 4, the sizes of the original tensors are both \(16\times 16\times 16\times 16\), and the ranks are [2, 2, 2, 2] and [3, 3, 3, 3], respectively. In the case of the 4 th-order tensor, there was no change in the common trend of each graph when the rank was varied. The same trend is observed in comparison with Figures 1 and 2. From these observations, we can say that the relationship between the weighting and schatten-\(p\) extension and the performance gap between the proposed algorithm and the rank-constrained minimization that we revealed is a law and is independent of the tensor rank and order. Figure 1: Experimental results for the original tensor with the order 3 and the rank 4. (**a–d**) are the results of varying the missing rate and the standard deviation of the noise \(\sigma_{n}\) during the observation process. The horizontal axis of the graph is the parameter \(\alpha\) used for determining the weights and the vertical axis is the error between the estimated tensor and the original tensor calculated by each method. Red, green, and blue indicate the results for the proposed method (Algorithm 1) with different types of weight vectors—\(\mathbf{w_{ld}}\), \(\mathbf{w_{Obs}}\), and \(\mathbf{w_{Uni}}\), respectively—and yellow is the result of the rank-constrained minimization. The line type corresponds to the value of the parameter \(p\) or \(r\). From these results, we can conclude that  
* It is sufficient to use \(p=1\) if weights that are close to the ideal weights can be estimated in some way. * It is better to set a small \(p\)-value if the weights estimated from the degraded singular values are not reliable.