# Nonconvex Nonlinear Transformation of Low-Rank Approximation for Tensor Completion
## 4 Nonconvex Nonlinear Transformation for Tensor Completion
### Problem Formulation  
The new model combines nonconvex norms and nonlinear transformations, providing a more accurate approximation of the tensor rank. In the image at top left of Figure 1, we compare the rank function and its frequently used surrogates, where the red dashed line representing the L\(\gamma\) norm significantly outperforms the nuclear norm and other nonconvex penalty functions in approximating the green rank function. In the image at bottom left of Figure 1, we demonstrate that nonlinear transformations proposed in recent years can achieve better rank approximation.  
Therefore, to enhance rank approximation, we propose the LRANNT model, which leverages the L\(\gamma\) norm in conjunction with nonlinear transformations. The method for applying these nonlinear transformations is formally outlined in Theorem 1. The corresponding nonconvex nonlinear LRANNT model, which serves as the core of our approach, is presented as follows:  
\[\|\mathcal{X}\|_{\text{LRANNT}}=\|\delta(\mathcal{X})\|_{\gamma}=\sum_{i=1}^{r }\|\varphi(\mathbf{H}_{i})\|_{\gamma}, \tag{1}\]  
where  
\[\|\varphi(\mathbf{H}_{i})\|_{\gamma}=\sum_{t=1}^{\min\{p,q\}}\left(1-e^{- \sigma_{\mathrm{f}}(\varphi(\mathbf{H}_{i}))/\gamma}\right),\forall\mathbf{H} _{i}\in\mathbb{R}^{p\times q}, \tag{2}\]  
Figure 1: Flowchart of the Nonconvex Nonlinear Transformations model. On the left is the transformation evolution based on TNN and the curve comparison of different rank functions. On the right is the nonconvex nonlinear model we proposed. [PERSON], et al., 2011 ([PERSON] et al., 2019); [PERSON], et al., 2019 ([PERSON] et al., 2019); [PERSON], et al. (2019).  
and \(\|\cdot\|_{\gamma}\) denotes the matrix \(\mathrm{L}\gamma\)-norm, \(\mathbf{H}_{i}\) represents the \(i\)-th frontal slice of \(\mathcal{H}\), \(\mathcal{H}=\mathcal{X}\times_{3}\mathbf{M}\), and \(\times 3\) denotes the third mode product between tensor \(\mathcal{X}\) and matrix \(\mathbf{M}\) (see Theorem 3). In this context, applying the nonlinear transformation \(\delta(\cdot)\) effectively diminishes the rank of the leading slices in the resulting tensor.  
Based on the proposed LRANNT, we introduce the LRTC model as follows:  
\[\min_{\mathcal{X},\mathcal{H},\mathbf{M}}\sum_{i=1}^{r}\lVert\varphi(\mathbf{ H}_{i})\rVert_{\gamma},\text{ s.t. }\mathcal{P}_{\Omega}(\mathcal{X})=\mathcal{P}_{\Omega}(\mathcal{O}), \mathcal{X}=\mathcal{H}\times_{3}\mathbf{M}^{\top},\mathbf{M}\mathbf{M}^{\top }=\mathbf{I}_{r\times r}, \tag{3}\]  
where, \(\mathcal{H}\in\mathbb{R}^{n_{1}\times n_{2}\times r}\) and \(\mathbf{M}\in\mathbb{R}^{r\times n_{3}}\) denote the tensors associated with the semi-orthogonal transformation, while \(\mathbf{I}\in\mathbb{R}^{r\times r}\) represents the identity matrix.  
Compared to previous research models, our approach introduces several key innovations as follows:  
* The most notable advantage lies in the combination of nonlinear transformations and nonconvex regularization. Through this innovative combination, we successfully address the suboptimal solutions commonly encountered in traditional convex regularization models. Nonconvex regularization offers greater flexibility in the solution space, enabling a more accurate reconstruction of low-rank tensor structures.
* While traditional convex models rely on well-established optimization techniques, the nonconvex nature of regularization presents challenges in finding the global optimum. To tackle this, we propose an iterative optimization strategy that combines stochastic gradient descent and alternating minimization, effectively navigating the complex landscape of nonconvex optimization. This method not only improves the quality of tensor reconstruction but also ensures robustness in environments with sparse and noisy data.