# Framelet Representation of Tensor Nuclear Norm for Third-Order Tensor Completion

arXiv: https://arxiv.org/abs/1909.06982 · HTML: https://mlsanity.com/paper/1909.06982/

## Key facts

- Cited by 37 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Sep 2019.

## Built on by (most cited first, 30 of 37)

- [Low-Rank Tensor Function Representation for Multi-Dimensional Data Recovery](https://mlsanity.com/paper/2212.00262/) (2022)
- [DDS2M: Self-Supervised Denoising Diffusion Spatio-Spectral Model for Hyperspectral Image Restoration](https://mlsanity.com/paper/2303.06682/) (2023)
- [Fusformer: A Transformer-based Fusion Approach for Hyperspectral Image Super-resolution](https://mlsanity.com/paper/2109.02079/) (2021)
- [Self-Supervised Nonlinear Transform-Based Tensor Nuclear Norm for Multi-Dimensional Image Recovery](https://mlsanity.com/paper/2105.14320/) (2021)
- [Revisiting Nonlocal Self-Similarity from Continuous Representation](https://mlsanity.com/paper/2401.00708/) (2024)
- [Nonlinear Transform Induced Tensor Nuclear Norm for Tensor Completion](https://mlsanity.com/paper/2110.08774/) (2021)
- [Tensor Recovery Based on A Novel Non-convex Function Minimax Logarithmic Concave Penalty Function](https://mlsanity.com/paper/2206.13506/) (2022)
- [T2LR-Net: An unrolling network learning transformed tensor low-rank prior for dynamic MR image reconstruction](https://mlsanity.com/paper/2209.03832/) (2022)
- [H2TF for Hyperspectral Image Denoising: Where Hierarchical Nonlinear Transform Meets Hierarchical Matrix Factorization](https://mlsanity.com/paper/2304.11141/) (2023)
- [To Fold or Not to Fold: Graph Regularized Tensor Train for Visual Data Completion](https://mlsanity.com/paper/2306.11123/) (2023)
- [Cross-Frequency Implicit Neural Representation with Self-Evolving Parameters](https://mlsanity.com/paper/2504.10929/) (2025)
- [Iterative Low-rank Network for Hyperspectral Image Denoising](https://mlsanity.com/paper/2509.00356/) (2025)
- [Two New Low Rank Tensor Completion Methods Based on Sum Nuclear Norm](https://mlsanity.com/paper/2108.03002/) (2021)
- [Low-rank tensor completion via tensor joint rank with logarithmic composite norm](https://mlsanity.com/paper/2309.16208/) (2023)
- [Adaptive Block Sparse Regularization under Arbitrary Linear Transform](https://mlsanity.com/paper/2401.15292/) (2024)
- [DTR: A Unified Deep Tensor Representation Framework for Multimedia Data Recovery](https://mlsanity.com/paper/2407.05267/) (2024)
- [Haar Nuclear Norms with Applications to Remote Sensing Imagery Restoration](https://mlsanity.com/paper/2407.08509/) (2024)
- [Low-Rank Tensor Learning by Generalized Nonconvex Regularization](https://mlsanity.com/paper/2410.18402/) (2024)
- [Beyond Motion Cues and Structural Sparsity: Revisiting Small Moving Target Detection](https://mlsanity.com/paper/2509.07654/) (2025)
- [When Bayesian Tensor Completion Meets Multioutput Gaussian Processes: Functional Universality and Rank Learning](https://mlsanity.com/paper/2512.21486/) (2025)
- [Multi-modal and frequency-weighted tensor nuclear norm for hyperspectral image denoising](https://mlsanity.com/paper/2106.12489/) (2021)
- [Tensor Full Feature Measure and Its Nonconvex Relaxation Applications to Tensor Recovery](https://mlsanity.com/paper/2109.12257/) (2021)
- ["Sparse + Low-Rank'' Tensor Completion Approach for Recovering Images and Videos](https://mlsanity.com/paper/2110.09298/) (2021)
- [Low-Rank Tensor Completion Based on Bivariate Equivalent Minimax-Concave Penalty](https://mlsanity.com/paper/2201.12709/) (2022)
- [Tensor Robust PCA with Nonconvex and Nonlocal Regularization](https://mlsanity.com/paper/2211.02404/) (2022)
- [Robust Data Clustering with Outliers via Transformed Tensor Low-Rank Representation](https://mlsanity.com/paper/2307.09055/) (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)
- [A novel non-convex minimax p-th order concave penalty function approach to low-rank tensor completion](https://mlsanity.com/paper/2502.19979/) (2025)
- [MNT-TNN: Spatiotemporal Traffic Data Imputation via Compact Multimode Nonlinear Transform-based Tensor Nuclear Norm](https://mlsanity.com/paper/2503.22955/) (2025)

## Abstract (arXiv)

The main aim of this paper is to develop a framelet representation of the tensor nuclear norm for third-order tensor completion. In the literature, the tensor nuclear norm can be computed by using tensor singular value decomposition based on the discrete Fourier transform matrix, and tensor completion can be performed by the minimization of the tensor nuclear norm which is the relaxation of the sum of matrix ranks from all Fourier transformed matrix frontal slices. These Fourier transformed matrix frontal slices are obtained by applying the discrete Fourier transform on the tubes of the original tensor. In this paper, we propose to employ the framelet representation of each tube so that a framelet transformed tensor can be constructed. Because of framelet basis redundancy, the representation of each tube is sparsely represented. When the matrix slices of the original tensor are highly correlated, we expect the corresponding sum of matrix ranks from all framelet transformed matrix frontal slices would be small, and the resulting tensor completion can be performed much better. The proposed minimization model is convex and global minimizers can be obtained. Numerical results on several types of multi-dimensional data (videos, multispectral images, and magnetic resonance imaging data) have tested and shown that the proposed method outperformed the other testing methods.