# Tensor Q-Rank: New Data Dependent Definition of Tensor Rank

arXiv: https://arxiv.org/abs/1910.12016 · HTML: https://mlsanity.com/paper/1910.12016/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Oct 2019.

## Built on by (most cited first, 10 of 10)

- [Nonlinear Transform Induced Tensor Nuclear Norm for Tensor Completion](https://mlsanity.com/paper/2110.08774/) (2021)
- [OTLRM: Orthogonal Learning-based Low-Rank Metric for Multi-Dimensional Inverse Problems](https://mlsanity.com/paper/2412.11165/) (2024)
- [H2TF for Hyperspectral Image Denoising: Where Hierarchical Nonlinear Transform Meets Hierarchical Matrix Factorization](https://mlsanity.com/paper/2304.11141/) (2023)
- [Handling The Non-Smooth Challenge in Tensor SVD: A Multi-Objective Tensor Recovery Framework](https://mlsanity.com/paper/2311.13958/) (2023)
- [Optimal Matrix-Mimetic Tensor Algebras via Variable Projection](https://mlsanity.com/paper/2406.06942/) (2024)
- [DTR: A Unified Deep Tensor Representation Framework for Multimedia Data Recovery](https://mlsanity.com/paper/2407.05267/) (2024)
- [Robust Data Clustering with Outliers via Transformed Tensor Low-Rank Representation](https://mlsanity.com/paper/2307.09055/) (2023)
- [Projected Tensor-Tensor Products for Efficient Computation of Optimal Multiway Data Representations](https://mlsanity.com/paper/2409.19402/) (2024)
- [3DeepRep: 3D Deep Low-rank Tensor Representation for Hyperspectral Image Inpainting](https://mlsanity.com/paper/2506.16735/) (2025)
- [Beyond Low-rankness: Guaranteed Matrix Recovery via Modified Nuclear Norm](https://mlsanity.com/paper/2507.18327/) (2025)

## Abstract (arXiv)

Recently, the textitTensor Nuclear Norm~(TNN) regularization based on t-SVD has been widely used in various low tubal-rank tensor recovery tasks. However, these models usually require smooth change of data along the third dimension to ensure their low rank structures. In this paper, we propose a new definition of data dependent tensor rank named textittensor Q-rank by a learnable orthogonal matrix Q, and further introduce a unified data dependent low rank tensor recovery model. According to the low rank hypothesis, we introduce two explainable selection method of Q, under which the data tensor may have a more significant low tensor Q-rank structure than that of low tubal-rank structure. Specifically, maximizing the variance of singular value distribution leads to Variance Maximization Tensor Q-Nuclear norm~(VMTQN), while minimizing the value of nuclear norm through manifold optimization leads to Manifold Optimization Tensor Q-Nuclear norm~(MOTQN). Moreover, we apply these two models to the low rank tensor completion problem, and then give an effective algorithm and briefly analyze why our method works better than TNN based methods in the case of complex data with low sampling rate. Finally, experimental results on real-world datasets demonstrate the superiority of our proposed model in the tensor completion problem with respect to other tensor rank regularization models.