# Nonlinear Transform Induced Tensor Nuclear Norm for Tensor Completion

arXiv: https://arxiv.org/abs/2110.08774 · HTML: https://mlsanity.com/paper/2110.08774/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 63 references that are in the map.
- First posted on arXiv 17 Oct 2021.

## Builds on (most cited first, 16 of 16)

- [Tensor Ring Decomposition](https://mlsanity.com/paper/1606.05535/) (2016)
- [Orthogonal Convolutional Neural Networks](https://mlsanity.com/paper/1911.12207/) (2019)
- [Global Convergence of ADMM in Nonconvex Nonsmooth Optimization](https://mlsanity.com/paper/1511.06324/) (2015)
- [Orthogonal Deep Neural Networks](https://mlsanity.com/paper/1905.05929/) (2019)
- [Bayesian CP Factorization of Incomplete Tensors with Automatic Rank Determination](https://mlsanity.com/paper/1401.6497/) (2014)
- [Can We Gain More from Orthogonality Regularizations in Training Deep CNNs?](https://mlsanity.com/paper/1810.09102/) (2018)
- [Efficient tensor completion for color image and video recovery: Low-rank tensor train](https://mlsanity.com/paper/1606.01500/) (2016)
- [Novel methods for multilinear data completion and de-noising based on tensor-SVD](https://mlsanity.com/paper/1407.1785/) (2014)
- [Non-local Meets Global: An Iterative Paradigm for Hyperspectral Image Restoration](https://mlsanity.com/paper/2010.12921/) (2020)
- [Tensor Ring Decomposition with Rank Minimization on Latent Space: An Efficient Approach for Tensor Completion](https://mlsanity.com/paper/1809.02288/) (2018)
- [Global Convergence of Block Coordinate Descent in Deep Learning](https://mlsanity.com/paper/1803.00225/) (2018)
- [Framelet Representation of Tensor Nuclear Norm for Third-Order Tensor Completion](https://mlsanity.com/paper/1909.06982/) (2019)
- [Dictionary Learning with Low-rank Coding Coefficients for Tensor Completion](https://mlsanity.com/paper/2009.12507/) (2020)
- [Hyperspectral Denoising Using Unsupervised Disentangled Spatio-Spectral Deep Priors](https://mlsanity.com/paper/2102.12310/) (2021)
- [Tensor Q-Rank: New Data Dependent Definition of Tensor Rank](https://mlsanity.com/paper/1910.12016/) (2019)
- [A Fast Algorithm for Cosine Transform Based Tensor Singular Value Decomposition](https://mlsanity.com/paper/1902.03070/) (2019)

## Built on by (most cited first, 13 of 13)

- [Tensor Recovery Based on A Novel Non-convex Function Minimax Logarithmic Concave Penalty Function](https://mlsanity.com/paper/2206.13506/) (2022)
- [Spectral-Spatial Extraction through Layered Tensor Decomposition for Hyperspectral Anomaly Detection](https://mlsanity.com/paper/2503.05183/) (2025)
- [Tensor Factorization via Transformed Tensor-Tensor Product for Image Alignment](https://mlsanity.com/paper/2212.05719/) (2022)
- [Low-rank tensor completion via tensor joint rank with logarithmic composite norm](https://mlsanity.com/paper/2309.16208/) (2023)
- [Low-Rank Tensor Learning by Generalized Nonconvex Regularization](https://mlsanity.com/paper/2410.18402/) (2024)
- [Projected Tensor-Tensor Products for Efficient Computation of Optimal Multiway Data Representations](https://mlsanity.com/paper/2409.19402/) (2024)
- [MNT-TNN: Spatiotemporal Traffic Data Imputation via Compact Multimode Nonlinear Transform-based Tensor Nuclear Norm](https://mlsanity.com/paper/2503.22955/) (2025)
- [Federated Structured Sparse PCA for Anomaly Detection in IoT Networks](https://mlsanity.com/paper/2503.23981/) (2025)
- [3DeepRep: 3D Deep Low-rank Tensor Representation for Hyperspectral Image Inpainting](https://mlsanity.com/paper/2506.16735/) (2025)
- [LRTuckerRep: Low-rank Tucker Representation Model for Multi-dimensional Data Completion](https://mlsanity.com/paper/2508.03755/) (2025)
- [StoTAM: Stochastic Alternating Minimization for Tucker-Structured Tensor Sensing](https://mlsanity.com/paper/2601.13522/) (2026)
- [Quaternion Nonlinear Transform-Induced Nuclear Norm for Low-Rank Tensor Completion](https://mlsanity.com/paper/2605.01467/) (2026)
- [Exponential-Family Tensor Completion via Nonconvex Dual Total-Variation Regularization](https://mlsanity.com/paper/2606.30958/) (2026)

## Abstract (arXiv)

The linear transform-based tensor nuclear norm (TNN) methods have recently obtained promising results for tensor completion. The main idea of this type of methods is exploiting the low-rank structure of frontal slices of the targeted tensor under the linear transform along the third mode. However, the low-rankness of frontal slices is not significant under linear transforms family. To better pursue the low-rank approximation, we propose a nonlinear transform-based TNN (NTTNN). More concretely, the proposed nonlinear transform is a composite transform consisting of the linear semi-orthogonal transform along the third mode and the element-wise nonlinear transform on frontal slices of the tensor under the linear semi-orthogonal transform, which are indispensable and complementary in the composite transform to fully exploit the underlying low-rankness. Based on the suggested low-rankness metric, i.e., NTTNN, we propose a low-rank tensor completion (LRTC) model. To tackle the resulting nonlinear and nonconvex optimization model, we elaborately design the proximal alternating minimization (PAM) algorithm and establish the theoretical convergence guarantee of the PAM algorithm. Extensive experimental results on hyperspectral images, multispectral images, and videos show that the our method outperforms linear transform-based state-of-the-art LRTC methods qualitatively and quantitatively.