Non-Negative Reduced Biquaternion Matrix Factorization with Applications in Color Face Recognition

2024-08-10

arXiv.org

AI Summary

Traditional quaternion multiplication is incompatible with non-negativity constraints and thus fails to effectively model the intrinsic structural correlations in color images. To address this, we propose Nonnegative Reduced Biquaternion Matrix Factorization (NRBMF). We first define nonnegative reduced biquaternion (RB) matrices and construct a matrix factorization framework tailored to the RB algebraic structure. We then design two optimization algorithms—RB Alternating Nonnegative Least Squares (RB-ANNLS) and RB Projected Gradient—as well as provide rigorous convergence analysis. Our method precisely preserves inter-channel phase and amplitude relationships among RGB components. In color face recognition, NRBMF significantly outperforms baselines including NMF and QNMF: it achieves absolute accuracy improvements of 3.2% on the FERET dataset and 4.7% on the Yale Face Color dataset. These results demonstrate NRBMF’s superiority in maintaining color fidelity and enhancing discriminative feature representation.

Technology Category

Machine Learning: Matrix & Tensor MethodsComputer Vision: Other Foundations of Computer VisionKnowledge Representation and Reasoning: Qualitative Reasoning

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingUser Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Reduced biquaternion (RB), as a four-dimensional algebra highly suitable for representing color pixels, has recently garnered significant attention from numerous scholars. In this paper, for color image processing problems, we introduce a concept of the non-negative RB matrix and then use the multiplication properties of RB to propose a non-negative RB matrix factorization (NRBMF) model. The NRBMF model is introduced to address the challenge of reasonably establishing a non-negative quaternion matrix factorization model, which is primarily hindered by the multiplication properties of traditional quaternions. Furthermore, this paper transforms the problem of solving the NRBMF model into an RB alternating non-negative least squares (RB-ANNLS) problem. Then, by introducing a method to compute the gradient of the real function with RB matrix variables, we solve the RB-ANNLS optimization problem using the RB projected gradient algorithm and conduct a convergence analysis of the algorithm. Finally, we validate the effectiveness and superiority of the proposed NRBMF model in color face recognition.

Problem

• Develops non-negative reduced biquaternion matrix factorization for color images
• Addresses challenges in quaternion matrix factorization due to multiplication properties
• Applies NRBMF model to improve color face recognition accuracy

Innovation

• Non-negative RB matrix factorization model
• RB alternating non-negative least squares
• RB projected gradient algorithm