# Quaternion Nuclear Norms Over Frobenius Norms Minimization for Robust Matrix Completion

arXiv: https://arxiv.org/abs/2504.21468 · HTML: https://mlsanity.com/paper/2504.21468/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 52 references that are in the map.
- First posted on arXiv 30 Apr 2025.

## Builds on (most cited first, 14 of 14)

- [Weighted Schatten p-Norm Minimization for Image Denoising and Background Subtraction](https://mlsanity.com/paper/1512.01003/) (2015)
- [Nonconvex Nonsmooth Low-Rank Minimization via Iteratively Reweighted Nuclear Norm](https://mlsanity.com/paper/1510.06895/) (2015)
- [Color image recovery using low-rank quaternion matrix completion algorithm](https://mlsanity.com/paper/1909.06567/) (2019)
- [A Scale Invariant Approach for Sparse Signal Recovery](https://mlsanity.com/paper/1812.08852/) (2018)
- [Quaternion-based bilinear factor matrix norm minimization for color image inpainting](https://mlsanity.com/paper/2005.02886/) (2020)
- [Limited-angle CT reconstruction via the L1/L2 minimization](https://mlsanity.com/paper/2006.00601/) (2020)
- [Color Image Inpainting via Robust Pure Quaternion Matrix Completion: Error Bound and Weighted Loss](https://mlsanity.com/paper/2202.02063/) (2022)
- [Quaternion Nuclear Norm minus Frobenius Norm Minimization for color image reconstruction](https://mlsanity.com/paper/2409.07797/) (2024)
- [Analysis and algorithms for some compressed sensing models based on L1/L2 minimization](https://mlsanity.com/paper/2007.12821/) (2020)
- [Minimizing L1 over L2 norms on the gradient](https://mlsanity.com/paper/2101.00809/) (2021)
- [Analysis of The Ratio of ℓ1 and ℓ2 Norms in Compressed Sensing](https://mlsanity.com/paper/2004.05873/) (2020)
- [Selecting Regularization Parameters for nuclear norm type minimization problems](https://mlsanity.com/paper/2204.06206/) (2022)
- [Separable Quaternion Matrix Factorization for Polarization Images](https://mlsanity.com/paper/2207.14039/) (2022)
- [On Partly Smoothness, Activity Identification and Faster Algorithms of L1 over L2 Minimization](https://mlsanity.com/paper/2401.15405/) (2024)

## Built on by (most cited first, 1 of 1)

- [Low-Rank Tensor Completion Based on Fractional Regularization with Ky Fan p-k Norm](https://mlsanity.com/paper/2606.19046/) (2026)

## Abstract (arXiv)

Recovering hidden structures from incomplete or noisy data remains a pervasive challenge across many fields, particularly where multi-dimensional data representation is essential. Quaternion matrices, with their ability to naturally model multi-dimensional data, offer a promising framework for this problem. This paper introduces the quaternion nuclear norm over the Frobenius norm (QNOF) as a novel nonconvex approximation for the rank of quaternion matrices. QNOF is parameter-free and scale-invariant. Utilizing quaternion singular value decomposition, we prove that solving the QNOF can be simplified to solving the singular value L1/L2 problem. Additionally, we extend the QNOF to robust quaternion matrix completion, employing the alternating direction multiplier method to derive solutions that guarantee weak convergence under mild conditions. Extensive numerical experiments validate the proposed model's superiority, consistently outperforming state-of-the-art quaternion methods.