# Color Image Inpainting via Robust Pure Quaternion Matrix Completion: Error Bound and Weighted Loss

arXiv: https://arxiv.org/abs/2202.02063 · HTML: https://mlsanity.com/paper/2202.02063/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 75 references that are in the map.
- First posted on arXiv 4 Feb 2022.

## Builds on (most cited first, 21 of 21)

- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [Quaternion Knowledge Graph Embeddings](https://mlsanity.com/paper/1904.10281/) (2019)
- [Beyond Fully-Connected Layers with Quaternions: Parameterization of Hypercomplex Multiplications with 1/n Parameters](https://mlsanity.com/paper/2102.08597/) (2021)
- [Deep Quaternion Networks](https://mlsanity.com/paper/1712.04604/) (2017)
- [Quaternion Convolutional Neural Networks](https://mlsanity.com/paper/1903.00658/) (2019)
- [Low-Rank Matrix Completion: A Contemporary Survey](https://mlsanity.com/paper/1907.11705/) (2019)
- [Multi-channel Weighted Nuclear Norm Minimization for Real Color Image Denoising](https://mlsanity.com/paper/1705.09912/) (2017)
- [Quaternion Convolutional Neural Networks for Heterogeneous Image Processing](https://mlsanity.com/paper/1811.02656/) (2018)
- [Quaternion Generative Adversarial Networks](https://mlsanity.com/paper/2104.09630/) (2021)
- [Color image recovery using low-rank quaternion matrix completion algorithm](https://mlsanity.com/paper/1909.06567/) (2019)
- [Quaternion-based bilinear factor matrix norm minimization for color image inpainting](https://mlsanity.com/paper/2005.02886/) (2020)
- [Adaptive Multinomial Matrix Completion](https://mlsanity.com/paper/1408.6218/) (2014)
- [Sparse Representation of a Polytope and Recovery of Sparse Signals and Low-rank Matrices](https://mlsanity.com/paper/1306.1154/) (2013)
- [Deep Octonion Networks](https://mlsanity.com/paper/1903.08478/) (2019)
- [Probabilistic low-rank matrix completion on finite alphabets](https://mlsanity.com/paper/1412.2632/) (2014)
- [Low Rank Matrix Completion with Exponential Family Noise](https://mlsanity.com/paper/1502.06919/) (2015)
- [Low Rank Pure Quaternion Approximation for Pure Quaternion Matrices](https://mlsanity.com/paper/2012.15138/) (2020)
- [Weighted Truncated Nuclear Norm Regularization for Low-Rank Quaternion Matrix Completion](https://mlsanity.com/paper/2101.02443/) (2021)
- [Color Image Classification via Quaternion Principal Component Analysis Network](https://mlsanity.com/paper/1503.01657/) (2015)
- [Convex Optimization Learning of Faithful Euclidean Distance Representations in Nonlinear Dimensionality Reduction](https://mlsanity.com/paper/1406.5736/) (2014)
- [Low Rank Quaternion Matrix Recovery via Logarithmic Approximation](https://mlsanity.com/paper/2107.01380/) (2021)

## Built on by (most cited first, 8 of 8)

- [High Dimensional Statistical Estimation under Uniformly Dithered One-bit Quantization](https://mlsanity.com/paper/2202.13157/) (2022)
- [Quantizing Heavy-tailed Data in Statistical Estimation: (Near) Minimax Rates, Covariate Quantization, and Uniform Recovery](https://mlsanity.com/paper/2212.14562/) (2022)
- [Quantized Low-Rank Multivariate Regression with Random Dithering](https://mlsanity.com/paper/2302.11197/) (2023)
- [Error Bound of Empirical ℓ2 Risk Minimization for Noisy Standard and Generalized Phase Retrieval Problems](https://mlsanity.com/paper/2205.13827/) (2022)
- [Quaternion Nuclear Norms Over Frobenius Norms Minimization for Robust Matrix Completion](https://mlsanity.com/paper/2504.21468/) (2025)
- [A New Low-Rank Learning Robust Quaternion Tensor Completion Method for Color Video Inpainting Problem and Fast Algorithms](https://mlsanity.com/paper/2306.09652/) (2023)
- [Quaternion tensor left ring decomposition and application for color image inpainting](https://mlsanity.com/paper/2307.10620/) (2023)
- [Low Rank Support Quaternion Matrix Machine](https://mlsanity.com/paper/2512.08327/) (2025)

## Abstract (arXiv)

In this paper, we study color image inpainting as a pure quaternion matrix completion problem. In the literature, the theoretical guarantee for quaternion matrix completion is not well-established. Our main aim is to propose a new minimization problem with an objective combining nuclear norm and a quadratic loss weighted among three channels. To fill the theoretical vacancy, we obtain the error bound in both clean and corrupted regimes, which relies on some new results of quaternion matrices. A general Gaussian noise is considered in robust completion where all observations are corrupted. Motivated by the error bound, we propose to handle unbalanced or correlated noise via a cross-channel weight in the quadratic loss, with the main purpose of rebalancing noise level, or removing noise correlation. Extensive experimental results on synthetic and color image data are presented to confirm and demonstrate our theoretical findings.