# On the Synergy between Nonconvex Extensions of the Tensor Nuclear Norm for Tensor Recovery
###### Abstract  
Low-rank tensor recovery has attracted much attention among various tensor recovery approaches. A tensor rank has several definitions, unlike the matrix rank--e.g., the CP rank and the Tucker rank. Many low-rank tensor recovery methods are focused on the Tucker rank. Since the Tucker rank is nonconvex and discontinuous, many relaxations of the Tucker rank have been proposed, e.g., the sum of nuclear norm, weighted tensor nuclear norm, and weighted tensor schatten-\(p\) norm. In particular, the weighted tensor schatten-\(p\) norm has two parameters, the weight and \(p\), and the sum of nuclear norm and weighted tensor nuclear norm are special cases of these parameters. However, there has been no detailed discussion of whether the effects of the weighting and \(p\) are synergistic. In this paper, we propose a novel low-rank tensor completion model using the weighted tensor schatten-\(p\) norm to reveal the relationships between the weight and \(p\). To clarify whether complex methods such as the weighted tensor schatten-\(p\) norm are necessary, we compare them with a simple method using rank-constrained minimization. It was found that the simple methods did not outperform the complex methods unless the rank of the original tensor could be accurately known. If we can obtain the ideal weight, \(p=1\) is sufficient, although it is necessary to set \(p<1\) when using the weights obtained from observations. These results are consistent with existing reports. nuclear norm; optimization; schatten-\(p\) norm; tensor recovery; Tucker decomposition 202121  
On the other hand, the weighted nuclear norm and the schatten-\(p\) norm have been proposed as different surrogates of the matrix rank [17; 18; 19; 20]. Both are a generalization of the nuclear norm and usually perform better than the nuclear norm for low-rank matrix recovery. Following this trend, a weighted tensor nuclear norm and a tensor schatten-\(p\) norm have also been proposed [11; 12]. They are extensions of the weighted nuclear norm and the schatten-\(p\) norm for tensors, respectively, and they generally perform better for low-rank tensor recovery as well. However, for effective use, we need to select appropriate weights and parameters \(p\). The ideal (oracle) weights for the weighted nuclear norm are the inverses of the singular values of the original matrix. This is because the weighted nuclear norm with the oracle weights of the original matrix is identical to the rank. Generally, obtaining the singular values of the original matrix is difficult. Therefore, for practical usage, we need some methods to estimate the singular values of the original matrix to determine the weights [19]. On the other hand, the parameter \(p\) for the schatten-\(p\) norm is generally determined in a heuristic manner and, in most cases, \(p<1\) is employed [8; 12; 18]. We should note that both the weighted tensor nuclear norm and the tensor schatten-\(p\) norm (with \(p<1\)) are in general nonconvex, as is the case with the matrix counterparts. Now, some natural questions arise: Are the effects of the weightings for singular values and the schatten-\(p\) extension synergistic, or does one of them encompass? Is there any chance that a simple rank-constrained minimization, which is also a nonconvex optimization, can compete with these advanced and complicated methods? In this paper, to answer the questions above, we propose a novel general constrained optimization problem combining the weighting and the schatten-\(p\) extension for tensors, and we develop an efficient algorithm to solve it. We performed exhaustive experiments, and the results showed that, if we can use the oracle weights, the combination of \(p=1\) and the weighting is the most effective choice for all cases. We also found that the combination of \(p=1/2\) and the weighting is effective when using the weights estimated from degraded measurements. The rank-constrained minimization problem performs well as long as we know the rank of the original tensor.