Quaternion Nuclear Norms Over Frobenius Norms Minimization for Robust Matrix Completion
This addresses robust matrix completion for multi-dimensional data representation in fields like signal processing, but it is incremental as it builds on existing quaternion methods.
The paper tackled robust matrix completion for multi-dimensional data using quaternion matrices by introducing the quaternion nuclear norm over Frobenius norm (QNOF) as a nonconvex rank approximation, and it demonstrated superiority over state-of-the-art quaternion methods in numerical experiments.
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.