# A New Convex Relaxation for Tensor Completion

arXiv: https://arxiv.org/abs/1307.4653 · HTML: https://mlsanity.com/paper/1307.4653/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Jul 2013.

## Built on by (most cited first, 9 of 9)

- [Guaranteed Tensor Recovery Fused Low-rankness and Smoothness](https://mlsanity.com/paper/2302.02155/) (2023)
- [Self-Supervised Nonlinear Transform-Based Tensor Nuclear Norm for Multi-Dimensional Image Recovery](https://mlsanity.com/paper/2105.14320/) (2021)
- [A new perspective on low-rank optimization](https://mlsanity.com/paper/2105.05947/) (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)
- [Multi-Tensor Network Representation for High-Order Tensor Completion](https://mlsanity.com/paper/2109.04022/) (2021)
- [Tensor Factorization via Transformed Tensor-Tensor Product for Image Alignment](https://mlsanity.com/paper/2212.05719/) (2022)
- [Low-Rank Tensor Learning by Generalized Nonconvex Regularization](https://mlsanity.com/paper/2410.18402/) (2024)
- [Infrared Small Target Detection Using Double-Weighted Multi-Granularity Patch Tensor Model With Tensor-Train Decomposition](https://mlsanity.com/paper/2310.05347/) (2023)

## Abstract (arXiv)

We study the problem of learning a tensor from a set of linear measurements. A prominent methodology for this problem is based on a generalization of trace norm regularization, which has been used extensively for learning low rank matrices, to the tensor setting. In this paper, we highlight some limitations of this approach and propose an alternative convex relaxation on the Euclidean ball. We then describe a technique to solve the associated regularization problem, which builds upon the alternating direction method of multipliers. Experiments on one synthetic dataset and two real datasets indicate that the proposed method improves significantly over tensor trace norm regularization in terms of estimation error, while remaining computationally tractable.