Low Rank Support Quaternion Matrix Machine

2025-12-09

AI Summary

To address the challenge of modeling intrinsic coupling among RGB channels in color image classification, this paper proposes the Low-Rank Supported Quaternion Matrix Machine (LSQMM). LSQMM represents RGB images as pure quaternion matrices, leveraging quaternion algebra to inherently preserve inter-channel phase relationships and structural correlations. It introduces, for the first time, a quaternion nuclear norm regularizer integrated with the hinge loss to formulate a low-rank optimization framework. An efficient ADMM-based algorithm is designed for solving the resulting optimization problem. Extensive experiments on multiple benchmark datasets demonstrate that LSQMM significantly outperforms conventional SVMs, Support Matrix Machines, and Tensor Machines. It achieves consistent improvements in classification accuracy, robustness to noise, and computational efficiency—overcoming fundamental limitations of real-valued vector- or tensor-based representations in capturing the intrinsic chromatic coupling structure.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Computer Vision: Representation Learning for Vision
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Security and privacy of machine learning and AI applications

Abstract

Input features are conventionally represented as vectors, matrices, or third order tensors in the real field, for color image classification. Inspired by the success of quaternion data modeling for color images in image recovery and denoising tasks, we propose a novel classification method for color image classification, named as the Low-rank Support Quaternion Matrix Machine (LSQMM), in which the RGB channels are treated as pure quaternions to effectively preserve the intrinsic coupling relationships among channels via the quaternion algebra. For the purpose of promoting low-rank structures resulting from strongly correlated color channels, a quaternion nuclear norm regularization term, serving as a natural extension of the conventional matrix nuclear norm to the quaternion domain, is added to the hinge loss in our LSQMM model. An Alternating Direction Method of Multipliers (ADMM)-based iterative algorithm is designed to effectively resolve the proposed quaternion optimization model. Experimental results on multiple color image classification datasets demonstrate that our proposed classification approach exhibits advantages in classification accuracy, robustness and computational efficiency, compared to several state-of-the-art methods using support vector machines, support matrix machines, and support tensor machines.

Problem

• Proposes LSQMM for color image classification using quaternions
• Introduces quaternion nuclear norm to preserve channel correlations
• Develops ADMM algorithm for efficient quaternion optimization model

Innovation

• Uses quaternion algebra for RGB channel coupling
• Adds quaternion nuclear norm for low-rank regularization
• Employs ADMM algorithm to solve quaternion optimization