# Nonconvex Nonlinear Transformation of Low-Rank Approximation for Tensor Completion
## 1 Introduction  
With the development of data collection techniques, tensors, as multidimensional arrays, are widely applied in a range of fields, including video [1], hyperspectral images (MSI) [2], magnetic resonance images (MRI) [3, 4], etc. However, the acquisition of tensor data is often compromised by missing values or corruptions. These issues may arise due to sensor limitations, hardware malfunctions, or interference from noise, which degrade the quality of high-dimensional data and negatively impact subsequent analyses and decision-making processes. Consequently, the problem of tensor completion (TC) [5] has gained significant attention. Tensor completion seeks to recover the true tensor from a limited set of observed entries, thereby mitigating the impact of missing or corrupted data and improving the utility of high-dimensional datasets. Since tensors exhibit correlations and redundancies, tensor completion can be achieved based on its low-rank property, which can be assessed using the rank function. Low-rank matrix completion (LRMC) methods are well-established and can be directly applied to solve low-rank tensor completion (LRTC) problems [6, 7, 8]. However, matrix-based methods essentially flatten the tensor into a matrix for processing, which neglects the multidimensional structure of the tensor, resulting in suboptimal completion performance. To overcome the limitations of matrix-based methods, researchers have started to explore tensor-based approaches. The essence of the LRTC problem lies in the attempt to minimize the rank of the target tensor while maintaining coherence with the observed incomplete tensor [9, 10]. However, a unified definition of tensor rank has not yet been established. The CANDECOMP/PARAFAC(CP) rank is determined by the count ofthe smallest rank-1 tensor [11]. This approach extends the concept of matrix rank to higher-dimensional structures and has proven effective in solving tensor completion problems [12; 13; 14]. Nevertheless, solving the CP rank of a tensor is NP-hard [15], and there is still no optimization algorithm that solves the CP rank directly. Additionally, the Tucker-rank has been suggested by researchers as a direct way to minimize the Tucker-rank problem [16]. However, this solution is still NP-hard. To address this challenge, the tensor nuclear norm (TNN) was introduced as a convex surrogate for tensor rank [17], which sums the matrix nuclear norms of each frontal slice of the Fourier-transformed tensor. TNN effectively captures the low-rank characteristics of tensors [18]. The TNN-based LRTC model aims to minimize the TNN of the tensor, subject to the constraint that the observed entries match the corresponding values in the incomplete tensor. As mentioned above, TNN is computed based on the one-dimensional fast discrete Fourier transform (1-D fast DFT) [19] along the third mode of the tensor. However, the discrete cosine transform (DCT) was then proposed to obtain a better rank approximation of the transformed tensor and less computational cost [20; 21]. Subsequently, researchers proposed that any invertible or semi-invertible transformation could replace the DFT for computation [22; 23], thereby improving the performance of the LRTC task. Additionally, transformation-based learning methods were introduced and shown to exhibit better robustness across various types of data [24; 25; 26]. However, these methods are all based on the low-rank property of tensor slices under linear transformations. Due to the limitations of TNN methods that rely on linear transformations to enforce low-rank structure in tensor slices, TNN based on nonlinear transformations (NTTNN) was proposed to better approximate the frontal slices of the transformed tensor [27]. Despite the good performance achieved by transformation-based tensor nuclear norms, the inherent issues with nuclear norms still manifest in related tensor completion work.