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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报

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双自适应权重非负矩阵分解鲁棒半监督学习

李春忠1,   靖凯立2,   周硕兵1,   口洋洋1   

  1. 1. 安徽财经大学统计与应用数学学院，蚌埠 233030

    2. 西安交通大学数学与统计学院，西安 710049
  • 收稿日期:2024-06-04 接受日期:2025-01-04 出版日期:2025-10-15 发布日期:2025-10-15
  • 基金资助:
    安徽省高校自然科学基金 (KJ2021A0481; KJ2021A0473).

Robust Semi-supervised Learning with Double Adaptive Weighted Non-negative Matrix Factorization

LI Chunzhong1,   JING Kaili2,   ZHOU Shuobing1,   KOU Yangyang1   

  1. 1. School of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu 233030

    2. School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049
  • Received:2024-06-04 Accepted:2025-01-04 Online:2025-10-15 Published:2025-10-15
  • Supported by:
    The Natural Science Foundation of Colleges and Universities in Anhui Province (KJ2021A0481; KJ2021A0473).

摘要：

高维数据建模在机器学习和模式识别领域是非常有价值的研究内容。高维数据在数据分析过程中存在的“维数灾难”问题制约了机器学习模型的有效介入，子空间和非负矩阵分解方法从空间变换的角度提供了一种有效策略。非负矩阵分解在无监督和半监督学习中通过改进损失函数和增加先验的方式提高算法的鲁棒性和普适性。构造了一种基于双自适应权重学习的非负矩阵分解的损失函数，分别在高维空间和低维空间上根据数据集的类结构信息进行学习，利用加权$L_{2,1}$范数提高模型鲁棒性，利用权重学习的策略学习低维空间上的相似性度量，从而获得比较好的算法鲁棒性。在Benchmark数据集和高光谱图像上的实验验证了新算法的优越性。

关键词: 非负矩阵分解, 自适应权重, 半监督学习, 鲁棒

Abstract:

High-dimensional data modeling in the fields of machine learning and pattern recognition is ubiquitous and of great value. The ``curse of dimensionality" problem that exists in the high-dimensional data analysis process constrains the effective intervention of many machine learning models. Many effective methods for subspace and non-negative matrix reconstruction have been proposed. Non-negative matrix reconstruction can improve algorithm construction in unsupervised and semi-supervised learning by improving the loss function and adding priors. This paper proposes a non-negative matrix factorization loss function based on adaptive dual-weight learning. The proposed loss-function learns based on the class structure information of the data set in high-dimensional space and low-dimensional space, uses weighted $L_{2,1}$ norm to improve model robustness, and uses weighted learning strategies to learn approximation in low-dimensional space. This results in better algorithmic robustness. Experimental results on some benchmark datasets and hyperspectral images demonstrate the superiority of the new algorithm.

Key words: non-negative matrix factorization, self-adaptation, semi-supervised learning, robustness

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