# On the Synergy between Nonconvex Extensions of the Tensor Nuclear Norm for Tensor Recovery
## 2 Low-Rank Tensor Completion
Moreover, the situation in which we know the rank of each unfolded matrix of the original tensor \(\hat{r}_{m}\) is unrealistic. The weighted tensor schatten-\(p\) norm (WTSPN) is proposed as a representation of a nonconvex but continuous tensor rank,  
\[\|\cdot\|_{\mathbf{w},\boldsymbol{\gamma},p}\colon\mathbb{R}^{n_{1}\times \cdots\times n_{N}}\to\mathbb{R}_{+}\colon\mathcal{X}\mapsto\sum_{m=1}^{N} \gamma_{m}\|\text{unfold}_{m}(\mathcal{X})\|_{\mathbf{w}_{m},p}^{p}, \tag{6}\]  
where \(\|\cdot\|_{\mathbf{w},p}^{p}\) is a weighted schatten-\(p\) norm raised to the power \(p\) (WSPN) [20]; \(\mathbf{w}=[\mathbf{w}_{1}^{\top},\cdots,\mathbf{w}_{N}^{\top}]^{\top}\) is the weight vector of the WSPNs; \(\gamma_{m}\) is a positive constant satisfying \(\sum_{m=1}^{N}\gamma_{m}=1\); \(\boldsymbol{\gamma}=[\gamma_{1},\cdots,\gamma_{N}]^{\top}\), and \(\text{unfold}(\cdot)\) is an unfolding operator. The WTSPN is generally a nonconvex function that is consistent with the weighted tensor nuclear norm [11] when \(p=1\), the tensor schatten-\(p\) norm [12] when \(\mathbf{w}\) is uniform (all elements of \(\mathbf{w}\) are the same value), and the sum of nuclear norm [1; 3] when \(p=1\) and \(\mathbf{w}\) is uniform. The \(m\)th mode tensor unfolding operator of the \(n\)th order tensor \(\text{unfold}_{m}:\mathbb{R}^{n_{1}\times\cdots\times n_{N}}\to\mathbb{R}^{n_{m} \times L_{m}}\) is defined as a map from the tensor elements \((i_{1},\cdots,i_{N})\) to the corresponding matrix elements \((i_{m},j_{m})\), where \(I_{m}=\prod_{ \begin{subarray}{c}k=1\\ k\
eq m\end{subarray} }^{N}i_{k}\),  
\[j_{m}=1+\sum_{ \begin{subarray}{c}k=1\\ k\
eq m\end{subarray} }^{N}(i_{k}-1)\prod_{ \begin{subarray}{c}l=1\\ l\
eq m\end{subarray} }^{k-1}i_{l}. \tag{7}\]  
The WSPN is described as  
\[\|\cdot\|_{\mathbf{w},p}^{p}\colon\mathbb{R}^{n_{p}\times n_{h}}\to\mathbb{R} _{+}\colon\mathbf{X}\mapsto\sum_{k=1}^{n_{m}}w_{k}\sigma_{k}(\mathbf{X})^{p}, \tag{8}\]  
where \(0<p\), \(n_{m}=\min(n_{p},n_{h})\), \(\sigma_{k}(\mathbf{X})\in\mathbb{R}_{+}\)\((k=1,\cdots,n_{m})\) is the \(k\)th largest singular value of \(\mathbf{X}\), and \(\mathbf{w}=[w_{1},\cdots,w_{n_{m}}]^{\top}\in\mathbb{R}_{+}^{n_{m}}\) is a weight vector that satisfies \(0\leq w_{1}\leq w_{2}\leq\cdots\leq w_{n_{m}}\). The WSPN is a generalization of the nuclear norm and the weighted nuclear norm [17, 19], which are often used in low-rank matrix recovery. As mentioned in Section 1, the proper weights of the WPSPN and the proper value of \(p\) have not been investigated in detail. Revealing them is one of the objectives of this paper. On the other hand, when recovering the observation model of Equation (1) with the WPSPN as the regularization term and the \(\ell_{2}\) norm as the fidelity term, even if the noise variance does not change, the optimal hyperparameter (balancing the regularization term and the fidelity term) varies according to the parameters of the regularization term, \(\mathbf{w}\), and \(p\). This makes a fair comparison difficult. In addition, a parameter that is so difficult to tune is not desirable for practical use. Therefore, in the next section, we propose a method to solve this problem.