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Quaternion Nuclear Norms Over Frobenius Norms Minimization for Robust Matrix Completion

Yu Guo, Guoqing Chen, Tieyong Zeng, Qiyu Jin, Michael Kwok-Po Ng

TL;DR

This work introduces QNOF, a parameter-free, scale-invariant nonconvex surrogate for the rank of quaternion matrices, defined as $\min_{\dot{X}} \tfrac{1}{2}|| \dot{\mathbf{Y}}-\dot{\mathbf{X}} ||^{2}_{F} + \lambda ( \frac{||\dot{X}||_{*}}{||\dot{X}||_{F}} )$ with $||\dot{X}||_{\mathrm{QNOF}} = \frac{||\dot{X}||_{*}}{||\dot{X}||_{F}}$. Using QSVD, the authors prove that solving QNOF is equivalent to a singular-value $L_1/L_2$ minimization on the diagonal singular-value vector, effectively reducing a quaternion matrix problem to a real-vector problem. They extend QNOF to robust matrix completion, RPCA, and RMC, and develop ADMM-based algorithms with weak convergence guarantees. Extensive experiments on simulated and real color images show that QNOF-based methods outperform state-of-the-art quaternion rank approaches in both accuracy (PSNR/SSIM) and speed, highlighting the practical impact for multi-channel data and color image processing. The work also discusses computational challenges and future directions to reduce the cost of SVD in quaternion settings by developing scalable, quaternion-friendly decompositions.

Abstract

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 $L_1/L_2$ 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.

Quaternion Nuclear Norms Over Frobenius Norms Minimization for Robust Matrix Completion

TL;DR

This work introduces QNOF, a parameter-free, scale-invariant nonconvex surrogate for the rank of quaternion matrices, defined as with . Using QSVD, the authors prove that solving QNOF is equivalent to a singular-value minimization on the diagonal singular-value vector, effectively reducing a quaternion matrix problem to a real-vector problem. They extend QNOF to robust matrix completion, RPCA, and RMC, and develop ADMM-based algorithms with weak convergence guarantees. Extensive experiments on simulated and real color images show that QNOF-based methods outperform state-of-the-art quaternion rank approaches in both accuracy (PSNR/SSIM) and speed, highlighting the practical impact for multi-channel data and color image processing. The work also discusses computational challenges and future directions to reduce the cost of SVD in quaternion settings by developing scalable, quaternion-friendly decompositions.

Abstract

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 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.
Paper Structure (15 sections, 9 theorems, 63 equations, 5 figures, 5 tables, 4 algorithms)

This paper contains 15 sections, 9 theorems, 63 equations, 5 figures, 5 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Quaternion algebraic basics
  3. Quaternion nuclear norm over Frobenius norm minimization
  4. The proposed QNOF model
  5. Solving for QNOF
  6. QNOF for robust matrix completion
  7. QNOF for matrix completion
  8. QNOF for robust principal component analysis
  9. QNOF for robust matrix completion
  10. Convergence analysis
  11. Experimental results
  12. Simulation data
  13. Real image data
  14. Conclusions
  15. Proof of Theorem \ref{['theorem04']}

Key Result

Theorem 2.1

Given a quaternion matrix $\dot{\bm{X}} \in \mathbb{ Q }^{m\times n}$ with rank $r$. There are two unitary quaternion matrices $\dot{\bm{U}} \in \mathbb{ Q }^{m\times m}$ and $\dot{\bm{V}} \in \mathbb{ Q }^{n\times n}$ satisfying $\dot{\bm{X}} = \dot{\bm{U}}\left(\right)\dot{\bm{V}}^{*}$, where $\b

Figures (5)

  • Figure 1: Exact recovery for different ranks ($x$-axis) and noise/missing ($y$-axis) degradation is analyzed with a fixed $n=50$. The first row shows the effect of noise and rank on exact recovery with a fixed proportion of missing entries. The second row illustrates the effect of missing entries and rank on exact recovery with a fixed noise level.
  • Figure 2: CSet12.
  • Figure 3: Comparison of RPCA performance by random impulse noise removal from color images. The random rate is 10%.
  • Figure 4: Comparison of RMC performance by color image inpainting combined with random impulse noise removal. Image missing rate of 50% and noise random rate of 3%.
  • Figure 5: PSNR, SSIM and error $\frac{|| \dot{\bm{X}}^{(k)} - \dot{\bm{X}}^{(k-1)}||_{F}}{|| \dot{\bm{X}}^{(k)} ||_{F}}$ change with iterations as an example of RMC with image missing rate 80% and noise random rate 3%.

Theorems & Definitions (15)

  • Theorem 2.1: QSVD zhang1997quaternions
  • Proposition 3.1: scale invariance
  • Proof 1
  • Proposition 3.2: unitary invariance
  • Proof 2
  • Proposition 3.3: boundedness
  • Proof 3
  • Lemma 3.4
  • Proof 4
  • Theorem 3.5
  • ...and 5 more