# Quaternion tensor left ring decomposition and application for color image inpainting

arXiv: https://arxiv.org/abs/2307.10620 · HTML: https://mlsanity.com/paper/2307.10620/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 52 references that are in the map.
- First posted on arXiv 20 Jul 2023.

## Builds on (most cited first, 15 of 15)

- [Tensor Ring Decomposition](https://mlsanity.com/paper/1606.05535/) (2016)
- [Exact tensor completion using t-SVD](https://mlsanity.com/paper/1502.04689/) (2015)
- [Efficient tensor completion for color image and video recovery: Low-rank tensor train](https://mlsanity.com/paper/1606.01500/) (2016)
- [Efficient Low Rank Tensor Ring Completion](https://mlsanity.com/paper/1707.08184/) (2017)
- [Tensor Ring Decomposition with Rank Minimization on Latent Space: An Efficient Approach for Tensor Completion](https://mlsanity.com/paper/1809.02288/) (2018)
- [Color image recovery using low-rank quaternion matrix completion algorithm](https://mlsanity.com/paper/1909.06567/) (2019)
- [Noisy Tensor Completion via Low-rank Tensor Ring](https://mlsanity.com/paper/2203.08857/) (2022)
- [Tensor-Ring Nuclear Norm Minimization and Application for Visual Data Completion](https://mlsanity.com/paper/1903.08888/) (2019)
- [Quaternion-based bilinear factor matrix norm minimization for color image inpainting](https://mlsanity.com/paper/2005.02886/) (2020)
- [Provable Tensor Ring Completion](https://mlsanity.com/paper/1903.03315/) (2019)
- [Effective Tensor Completion via Element-wise Weighted Low-rank Tensor Train with Overlapping Ket Augmentation](https://mlsanity.com/paper/2109.05736/) (2021)
- [Color Image Inpainting via Robust Pure Quaternion Matrix Completion: Error Bound and Weighted Loss](https://mlsanity.com/paper/2202.02063/) (2022)
- [Quaternion Matrix Completion Using Untrained Quaternion Convolutional Neural Network for Color Image Inpainting](https://mlsanity.com/paper/2305.00416/) (2023)
- [A Low Rank Quaternion Decomposition Algorithm and Its Application in Color Image Inpainting](https://mlsanity.com/paper/2009.12203/) (2020)
- [Quaternion Tensor Train Rank Minimization with Sparse Regularization in a Transformed Domain for Quaternion Tensor Completion](https://mlsanity.com/paper/2209.02964/) (2022)

## Abstract (arXiv)

In recent years, tensor networks have emerged as powerful tools for solving large-scale optimization problems. One of the most promising tensor networks is the tensor ring (TR) decomposition, which achieves circular dimensional permutation invariance in the model through the utilization of the trace operation and equitable treatment of the latent cores. On the other hand, more recently, quaternions have gained significant attention and have been widely utilized in color image processing tasks due to their effectiveness in encoding color pixels by considering the three color channels as a unified entity. Therefore, in this paper, based on the left quaternion matrix multiplication, we propose the quaternion tensor left ring (QTLR) decomposition, which inherits the powerful and generalized representation abilities of the TR decomposition while leveraging the advantages of quaternions for color pixel representation. In addition to providing the definition of QTLR decomposition and an algorithm for learning the QTLR format, the paper further proposes a low-rank quaternion tensor completion (LRQTC) model and its algorithm for color image inpainting based on the defined QTLR decomposition. Finally, extensive experiments on color image inpainting demonstrate that the proposed LRQTC method is highly competitive.