# Self-Supervised Nonlinear Transform-Based Tensor Nuclear Norm for Multi-Dimensional Image Recovery

arXiv: https://arxiv.org/abs/2105.14320 · HTML: https://mlsanity.com/paper/2105.14320/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 46 references that are in the map.
- First posted on arXiv 29 May 2021.

## Builds on (most cited first, 14 of 14)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Tensor Robust Principal Component Analysis with A New Tensor Nuclear Norm](https://mlsanity.com/paper/1804.03728/) (2018)
- [Rank Minimization for Snapshot Compressive Imaging](https://mlsanity.com/paper/1807.07837/) (2018)
- [Exact tensor completion using t-SVD](https://mlsanity.com/paper/1502.04689/) (2015)
- [Novel methods for multilinear data completion and de-noising based on tensor-SVD](https://mlsanity.com/paper/1407.1785/) (2014)
- [Tensor Ring Decomposition with Rank Minimization on Latent Space: An Efficient Approach for Tensor Completion](https://mlsanity.com/paper/1809.02288/) (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)
- [PLU: The Piecewise Linear Unit Activation Function](https://mlsanity.com/paper/1809.09534/) (2018)
- [A New Convex Relaxation for Tensor Completion](https://mlsanity.com/paper/1307.4653/) (2013)
- [A variational approach to stable principal component pursuit](https://mlsanity.com/paper/1406.1089/) (2014)
- [A Fast Algorithm for Cosine Transform Based Tensor Singular Value Decomposition](https://mlsanity.com/paper/1902.03070/) (2019)

## Built on by (most cited first, 16 of 16)

- [Low-Rank Tensor Function Representation for Multi-Dimensional Data Recovery](https://mlsanity.com/paper/2212.00262/) (2022)
- [HIR-Diff: Unsupervised Hyperspectral Image Restoration Via Improved Diffusion Models](https://mlsanity.com/paper/2402.15865/) (2024)
- [DDS2M: Self-Supervised Denoising Diffusion Spatio-Spectral Model for Hyperspectral Image Restoration](https://mlsanity.com/paper/2303.06682/) (2023)
- [Tensor Decompositions for Hyperspectral Data Processing in Remote Sensing: A Comprehensive Review](https://mlsanity.com/paper/2205.06407/) (2022)
- [T2LR-Net: An unrolling network learning transformed tensor low-rank prior for dynamic MR image reconstruction](https://mlsanity.com/paper/2209.03832/) (2022)
- [DTR: A Unified Deep Tensor Representation Framework for Multimedia Data Recovery](https://mlsanity.com/paper/2407.05267/) (2024)
- [Self-supervised Deep Hyperspectral Inpainting with the Plug and Play and Deep Image Prior Models](https://mlsanity.com/paper/2501.08195/) (2025)
- [Beyond Motion Cues and Structural Sparsity: Revisiting Small Moving Target Detection](https://mlsanity.com/paper/2509.07654/) (2025)
- [Gaussian Splatting-based Low-Rank Tensor Representation for Multi-Dimensional Image Recovery](https://mlsanity.com/paper/2511.14270/) (2025)
- [Multi-modal and frequency-weighted tensor nuclear norm for hyperspectral image denoising](https://mlsanity.com/paper/2106.12489/) (2021)
- [A New Low-Rank Learning Robust Quaternion Tensor Completion Method for Color Video Inpainting Problem and Fast Algorithms](https://mlsanity.com/paper/2306.09652/) (2023)
- [Exact Tensor Completion Powered by Slim Transforms](https://mlsanity.com/paper/2402.03468/) (2024)
- [Projected Tensor-Tensor Products for Efficient Computation of Optimal Multiway Data Representations](https://mlsanity.com/paper/2409.19402/) (2024)
- [Under-Sampled High-Dimensional Data Recovery via Symbiotic Multi-Prior Tensor Reconstruction](https://mlsanity.com/paper/2504.05992/) (2025)
- [3DeepRep: 3D Deep Low-rank Tensor Representation for Hyperspectral Image Inpainting](https://mlsanity.com/paper/2506.16735/) (2025)
- [Pre-Trained Low-Rank Tensor Decomposition for Multi-Dimensional Image Recovery](https://mlsanity.com/paper/2609.12843/) (2026)

## Abstract (arXiv)

In this paper, we study multi-dimensional image recovery. Recently, transform-based tensor nuclear norm minimization methods are considered to capture low-rank tensor structures to recover third-order tensors in multi-dimensional image processing applications. The main characteristic of such methods is to perform the linear transform along the third mode of third-order tensors, and then compute tensor nuclear norm minimization on the transformed tensor so that the underlying low-rank tensors can be recovered. The main aim of this paper is to propose a nonlinear multilayer neural network to learn a nonlinear transform via the observed tensor data under self-supervision. The proposed network makes use of low-rank representation of transformed tensors and data-fitting between the observed tensor and the reconstructed tensor to construct the nonlinear transformation. Extensive experimental results on tensor completion, background subtraction, robust tensor completion, and snapshot compressive imaging are presented to demonstrate that the performance of the proposed method is better than that of state-of-the-art methods.