# Tensor Full Feature Measure and Its Nonconvex Relaxation Applications to Tensor Recovery

arXiv: https://arxiv.org/abs/2109.12257 · HTML: https://mlsanity.com/paper/2109.12257/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 68 references that are in the map.
- First posted on arXiv 25 Sep 2021.

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

- [Tensor Robust Principal Component Analysis with A New Tensor Nuclear Norm](https://mlsanity.com/paper/1804.03728/) (2018)
- [Exact tensor completion using t-SVD](https://mlsanity.com/paper/1502.04689/) (2015)
- [Bayesian CP Factorization of Incomplete Tensors with Automatic Rank Determination](https://mlsanity.com/paper/1401.6497/) (2014)
- [Novel methods for multilinear data completion and de-noising based on tensor-SVD](https://mlsanity.com/paper/1407.1785/) (2014)
- [Square Deal: Lower Bounds and Improved Relaxations for Tensor Recovery](https://mlsanity.com/paper/1307.5870/) (2013)
- [Framelet Representation of Tensor Nuclear Norm for Third-Order Tensor Completion](https://mlsanity.com/paper/1909.06982/) (2019)
- [Robust Low-rank Tensor Recovery: Models and Algorithms](https://mlsanity.com/paper/1311.6182/) (2013)
- [Low-rank tensor completion: a Riemannian manifold preconditioning approach](https://mlsanity.com/paper/1605.08257/) (2016)
- [Multi-dimensional imaging data recovery via minimizing the partial sum of tubal nuclear norm](https://mlsanity.com/paper/1712.05870/) (2017)
- [Tensor N-tubal rank and its convex relaxation for low-rank tensor recovery](https://mlsanity.com/paper/1812.00688/) (2018)
- [Deep Plug-and-play Prior for Low-rank Tensor Completion](https://mlsanity.com/paper/1905.04449/) (2019)
- [Total Variation Regularized Tensor RPCA for Background Subtraction from Compressive Measurements](https://mlsanity.com/paper/1503.01868/) (2015)
- [Semiblind Hyperspectral Unmixing in the Presence of Spectral Library Mismatches](https://mlsanity.com/paper/1507.01661/) (2015)
- [Generalized Singular Value Thresholding](https://mlsanity.com/paper/1412.2231/) (2014)

## Abstract (arXiv)

Tensor sparse modeling as a promising approach, in the whole of science and engineering has been a huge success. As is known to all, various data in practical application are often generated by multiple factors, so the use of tensors to represent the data containing the internal structure of multiple factors came into being. However, different from the matrix case, constructing reasonable sparse measure of tensor is a relatively difficult and very important task. Therefore, in this paper, we propose a new tensor sparsity measure called Tensor Full Feature Measure (FFM). It can simultaneously describe the feature information of each dimension of the tensor and the related features between two dimensions, and connect the Tucker rank with the tensor tube rank. This measurement method can describe the sparse features of the tensor more comprehensively. On this basis, we establish its non-convex relaxation, and apply FFM to low rank tensor completion (LRTC) and tensor robust principal component analysis (TRPCA). LRTC and TRPCA models based on FFM are proposed, and two efficient Alternating Direction Multiplier Method (ADMM) algorithms are developed to solve the proposed model. A variety of real numerical experiments substantiate the superiority of the proposed methods beyond state-of-the-arts.