Quaternion Nuclear Norms Over Frobenius Norms Minimization for Robust Matrix Completion

2025-04-30

AI Summary

To address structural recovery from incomplete and noisy multidimensional data, this paper proposes a robust quaternion matrix completion method. The core contribution is the first parameter-free, scale-invariant quaternion nuclear norm-to-Frobenius norm ratio (QNOF), serving as a nonconvex surrogate for rank; we rigorously prove its equivalence to singular value ℓ₁/ℓ₂ ratio optimization. The method integrates quaternion singular value decomposition with the alternating direction method of multipliers (ADMM) framework, ensuring weak convergence. Extensive experiments demonstrate that our approach significantly outperforms existing quaternion matrix completion algorithms under diverse noise types and missing patterns—achieving higher reconstruction accuracy and superior robustness. This work establishes a novel paradigm for multidimensional signal processing via geometrically principled quaternion representations.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

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.

Problem

• Recovering hidden structures from incomplete noisy data
• Proposing quaternion nuclear norm for robust matrix completion
• Extending QNOF to guarantee weak convergence conditions

Innovation

• Quaternion nuclear norm over Frobenius norm
• Parameter-free and scale-invariant approximation
• Alternating direction multiplier method solution