# Tensor train rank minimization with nonlocal self-similarity for tensor completion

arXiv: https://arxiv.org/abs/2004.14273 · HTML: https://mlsanity.com/paper/2004.14273/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Apr 2020.

## Built on by (most cited first, 5 of 5)

- [Nonlocal Patch-Based Fully-Connected Tensor Network Decomposition for Remote Sensing Image Inpainting](https://mlsanity.com/paper/2109.05889/) (2021)
- [Fully-Connected Tensor Network Decomposition for Robust Tensor Completion Problem](https://mlsanity.com/paper/2110.08754/) (2021)
- [Coarse to Fine Two-Stage Approach to Robust Tensor Completion of Visual Data](https://mlsanity.com/paper/2106.10422/) (2021)
- [Patch Tracking-based Streaming Tensor Ring Completion for Visual Data Recovery](https://mlsanity.com/paper/2105.14620/) (2021)
- [Hyperspectral Image Recovery Constrained by Multi-Granularity Non-Local Self-Similarity Priors](https://mlsanity.com/paper/2508.01435/) (2025)

## Abstract (arXiv)

The tensor train (TT) rank has received increasing attention in tensor completion due to its ability to capture the global correlation of high-order tensors (textrmorder >3). For third order visual data, direct TT rank minimization has not exploited the potential of TT rank for high-order tensors. The TT rank minimization accompany with ket augmentation, which transforms a lower-order tensor (e.g., visual data) into a higher-order tensor, suffers from serious block-artifacts. To tackle this issue, we suggest the TT rank minimization with nonlocal self-similarity for tensor completion by simultaneously exploring the spatial, temporal/spectral, and nonlocal redundancy in visual data. More precisely, the TT rank minimization is performed on a formed higher-order tensor called group by stacking similar cubes, which naturally and fully takes advantage of the ability of TT rank for high-order tensors. Moreover, the perturbation analysis for the TT low-rankness of each group is established. We develop the alternating direction method of multipliers tailored for the specific structure to solve the proposed model. Extensive experiments demonstrate that the proposed method is superior to several existing state-of-the-art methods in terms of both qualitative and quantitative measures.