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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L_1/L_2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> <mo stretchy="false">/</mo> <msub> <mi>L</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> 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.</p>

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Quaternion Nuclear Norms Over Frobenius Norms Minimization for Robust Matrix Completion

  • 期刊论文

  • 发表日期：

    2026年6月22日

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

摘要

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 \(L_1/L_2\) 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.