# Nonconvex Nonlinear Transformation of Low-Rank Approximation for Tensor Completion
## 2 Related Work  
In this section, we review certain main works related to low-rank tensor recovery, namely, the nonlinear transformation model (NTTNN) and nonconvex norm methods, both of which are dedicated to addressing the low-rank tensor recovery problem. In traditional linear models, such as the discrete Fourier transform (DFT), the tensor completion problem can be formulated as:  
\[\min_{\mathcal{W}}\sum_{i=1}^{n_{3}}\lVert\mathbf{W}_{i}\rVert_{*},\text{ s.t. }\mathcal{P}_{\Omega}(\mathcal{X})=\mathcal{P}_{\Omega}(\mathcal{O}), \mathcal{X}=\mathcal{W}\times_{3}\mathbf{F}_{n_{3}}^{-1},\]  
where \(\mathcal{X}\) represents the target tensor, \(\mathcal{O}\) signifies the observation tensor, \(\Omega\) denotes the observation index set, the function \(\mathcal{P}\) preserves entries in \(\Omega\) and zeros-out others, \(\mathbf{F}_{n_{3}}^{-1}\) denotes the inverse DFT matrix [19], \(\lVert\cdot\rVert_{*}\) is the nuclear norm of a matrix, and \(\mathbf{W}_{i}\) represents the \(i\)-th frontal slice of \(\mathcal{W}\). The objective is to recover the low-rank tensor by minimizing the nuclear norm, thereby solving the tensor completion problem. The NTTNN model, proposed by [PERSON] et al. [27], combines element-wise nonlinear transformations \(\varphi\) with learned linear semi-orthogonal transformations \(\mathbf{M}\), forming a composite nonlinear transformation that replaces the original linear transformation. Specifically, the nonlinear composite transformation, denoted as \(\delta(\mathcal{X})\), is calculated by first applying the semi-orthogonal transformation \(\mathbf{M}\) to the tensor \(\mathcal{X}\), and then applying the element-wise nonlinear transformation \(\varphi\) to the result. This process can be expressed as:  
\[\delta(\mathcal{X})=\varphi(\mathcal{X}\times_{3}\mathbf{M}),\]  
where \(\mathbf{M}\) is a learned matrix in \(\mathbb{R}^{r\times n_{3}}\) that satisfies the semi-orthogonal condition \(\mathbf{M}\mathbf{M}^{\top}=\mathbf{I}_{r\times r}\). The transformation \(\varphi\) is applied element-wise to the frontal slices \(\mathbf{H}_{i}\) of the tensor \(\mathcal{H}\), which is obtained by the semi-orthogonal transformation \(\mathbf{M}\) applied to \(\mathcal{X}\). This composite transformation not only improves the rank approximation but also enhances the robustness of the model across various datasets by incorporating learned linear semi-orthogonal transformations. However, as discussed in Section 1, there are the limitations to the nuclear norm that need to be addressed. To address the limitations of convex functions discussed in Section 1, many researchers have proposed nonconvex functions as alternatives. Compared to convex functions, nonconvex functions can better approximate the structure of low-rank tensors, particularly in tensor recovery tasks. Nonconvex norms generally provide a more flexible optimization space, which helps avoid local minima issues commonly found in convex optimization. One such nonconvex norm is the L1/2 norm, introduced by [PERSON] et al. [32], which is defined as:  
\[\lVert\mathbf{X}\rVert_{1/2}=\sum_{i}\left(|\sigma_{i}|^{1/2}\right),\]  
where \(\sigma_{i}\) represents the singular values of the matrix \(\mathbf{X}\). The L1/2 norm applies a nonconvex half-norm to the singular values of the matrix, effectively promoting matrix recovery while avoiding the excessive sparsity that may arise from the L1 norm. Another popular nonconvex function is the maximum concave penalty (MCP), proposed by [PERSON] et al. [33]. The MCP function constrains model parameters in a way that facilitates a more precise rank approximation of tensors. The MCP is defined as:  
\[\mathrm{MCP}(x)=\left\{ \begin{array}{ll}\lambda|x|-\frac{x^{2}}{2},&|x|\leq \lambda\\ \frac{\lambda^{2}}{2},&|x|>\lambda\end{array} \right.,\]  
where \(\lambda\) is a hyperparameter that controls the shape of the nonconvex function.