Non-negative reduced biquaternion matrix factorization (NRBMF) uses the product of reduced biquaternion (RB) matrices to incorporate the non-negativity constraints of color image pixels into the factorization process. However, NRBMF mainly focuses on reconstruction accuracy and does not explicitly exploit the local geometric structure of image data, which may limit the discriminative ability of the obtained low-dimensional coefficient representations. To address this issue, we propose a graph regularized non-negative reduced biquaternion matrix factorization (GNRBMF) model for color image recognition. The proposed model incorporates a graph Laplacian regularizer into the reduced biquaternion coefficient matrix, encouraging nearby samples in the original space to have similar coefficient representations. Meanwhile, GNRBMF retains the non-negativity property of NRBMF in the reduced biquaternion algebra. To solve the optimization problem, a component-wise alternating projected gradient algorithm is derived, and its convergence properties are analyzed. Experimental results on three color image datasets show that the proposed GNRBMF model achieves competitive or superior recognition performance compared with several methods in most tested settings.

翻译：非负简化四元数矩阵分解（NRBMF）利用简化四元数（RB）矩阵的乘积，将彩色图像像素的非负性约束融入分解过程。然而，NRBMF主要关注重建精度，并未显式利用图像数据的局部几何结构，这可能限制所得低维系数表示的判别能力。为解决此问题，我们提出一种图正则化非负简化四元数矩阵分解（GNRBMF）模型用于彩色图像识别。该模型在简化四元数系数矩阵中引入图拉普拉斯正则化项，促使原始空间中邻近的样本具有相似的系数表示。同时，GNRBMF在简化四元数代数中保留了NRBMF的非负性。为求解优化问题，我们推导了一种分量交替投影梯度算法，并分析了其收敛性。在三个彩色图像数据集上的实验结果表明，在大多数测试场景下，所提出的GNRBMF模型与多种方法相比实现了具有竞争力或更优的识别性能。