# Nonconvex Nonlinear Transformation of Low-Rank Approximation for Tensor Completion
## 3 Notations and Preliminaries
### Preliminaries  
This section presents key definitions for the transformations and operations employed in the proposed model, including the nonlinear composite transform, tensor product, and mode-\(k\) product. These operations are fundamental to the mathematical framework underlying our approach and play a crucial role in the modeling process. **Theorem 1** (composite transform \(\delta\)[27]). : _The nonlinear composite transform \(\delta\) is a combination of a linear transformation and a nonlinear operation. It is defined as the composition of a linear transformation \(\mathbf{M}\) and an element-wise nonlinear function \(\varphi\), applied to the transformed tensor. Specifically, the transform is expressed as_  
\[\delta(\mathcal{X})=(\varphi\circ\mathbf{M})(\mathcal{X})=\varphi(\mathcal{X} \times_{3}\mathbf{M}),\]_where \(\mathbf{M}\in\mathbb{R}^{r\times n_{3}}\) is a learned linear semi-orthogonal matrix that satisfies \(\mathbf{M}\mathbf{M}^{\top}=\mathbf{I}_{r\times r}\). The function \(\varphi\) represents an element-wise nonlinear operation applied to the frontal slices of the transformed tensor \(\mathcal{H}\), which is derived from \(\mathcal{X}\) through the linear transformation \(\mathbf{M}\)._  
**Theorem 2** (tensor product [19]). : _Let \(A\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}\) and \(B\in\mathbb{R}^{n_{2}\times l\times n_{3}}\) be two tensors. The tensor product \(A*B\) is defined as a tensor of size \(n_{1}\times l\times n_{3}\), and it can be computed using the unfold and fold operations as follows:_  
\[A*B=\text{fold}(\mathbf{b}\circ(\text{unfold}(A))\cdot\text{unfold}(B)),\]  
_where \(\mathbf{b}\circ\) denotes a circular operation applied to the unfolding of \(A\), and the unfold and fold operations are defined as_  
\[\text{unfold}(A)=\Big{[}A^{(1)},A^{(2)},\ldots,A^{(n_{3})}\Big{]},\quad\text{ fold}(\text{unfold}(A))=A,\]  
_This operation satisfies the property that \(C=A*B\) is equivalent to \(C=AB\) when the proper tensor contraction is applied, which can be utilized in various tensor decompositions and transformations._  
**Theorem 3** (mode-k product [35]). : _The mode-k product describes an operation between a matrix \(\mathbf{D}\) and a tensor \(\mathcal{H}\), where the matrix is multiplied by the tensor along a specific mode. Formally, the mode-k product is defined as_  
\[\mathcal{X}=\mathcal{H}\times_{k}\mathbf{D}=\text{fold}_{k}(\mathbf{D}\mathbf{ H}_{(k)}),\]  
_where \(\mathbf{D}\) is a matrix, \(\mathcal{H}\) is the tensor, and \(\mathbf{H}_{(k)}\) denotes the mode-k matricization of \(\mathcal{H}\). The operation reorganizes the elements of the resulting matrix into a tensor through the mode-k folding process, which is essential for tensor decompositions and transformations._  
**Lemma 1** ([36; 37; 38; 39]). : _Consider the singular value decomposition (SVD) of a matrix \(\mathbf{A}\in\mathbb{R}^{m\times n}\), with singular values \(\sigma_{1},\sigma_{2},\ldots,\sigma_{t}\), where \(t=\min(m,n)\) is the rank. The gradient of the function \(\varphi(\sigma_{k})\) with respect to each singular value \(\sigma_{k}\) is \(\
abla\varphi(\sigma_{k})\). To quantify the difference between the recovered and target matrices, the following loss function is introduced:_  
\[\mathcal{L}(\mathbf{A})=\frac{1}{2}\|\mathbf{A}-\mathbf{B}\|_{F}^{2}\]  
_where \(\mathbf{B}\) is the target matrix and \(\|\cdot\|_{F}\) denotes the Frobenius norm. To optimize this loss function, a weighted singular value thresholding (WSVT) technique based on the singular values is employed. This method optimizes the matrix rank by imposing gradient constraints and singular value adjustments, ensuring that the gradients satisfy the condition \(\
abla\varphi(\sigma_{1})\geq\
abla\varphi(\sigma_{2})\geq\cdots\geq\
abla \varphi(\sigma_{t})\), thereby facilitating low-rank matrix recovery._  
**Lemma 2** ([29]). : _Based on the properties of the trace operation, we simplify the optimization problem by utilizing the singular value decomposition (SVD) of the matrix.