用四元数核范数提升矩阵补全的鲁棒性，效果优于现有方法。
Quaternion Nuclear Norms Over Frobenius Norms Minimization for Robust Matrix Completion
- 提出四元数核范数对齐弗罗贝尼乌斯范数，实现无参且尺度不变的秩逼近。
- 将问题转化为奇异值L1/L2优化，理论可解且收敛性有保障。
- 适用于多维数据补全，特别适合图像、信号等四元数建模场景。
从不完整或含噪数据中恢复隐藏结构是多个领域面临的普遍挑战，尤其在需要多维数据表示时更为突出。四元数矩阵因其天然建模多维数据的能力，为此类问题提供了有前景的框架。本文提出四元数核范数相对于弗罗贝尼乌斯范数（QNOF）作为四元数矩阵秩的新非凸近似，该方法无参数且尺度不变。利用四元数奇异值分解，我们证明求解QNOF可简化为求解奇异值L₁/L₂问题。此外，将QNOF扩展至鲁棒四元数矩阵补全，采用交替方向乘子法推导出解，并在弱条件下保证收敛性。大量数值实验验证了所提模型的优越性，其性能持续优于当前最先进的四元数方法。
原文摘要 · Abstract (English)
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.
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