基于高阶近邻矩阵和二分图的非负矩阵分解方法

【作者】 张鹏飞；

【导师】 郭莉；

【作者基本信息】 青岛大学 ， 软件工程（专业学位）， 2025， 硕士

【摘要】 近年来,非负矩阵分解(Nonnegative Matrix Factorization,NMF)因其揭示数据潜在结构的能力而备受关注。然而,现有NMF方法在数据关联性上普遍存在探索不足的问题,它们仅考虑一阶近邻关系,没有考虑二阶甚至高阶的近邻关系,这可能不足以充分挖掘数据的局部几何结构。实际上,高阶近邻能够挖掘更复杂的关联关系和数据的潜在结构。此外,这些方法往往未能充分利用系数矩阵固有的簇结构,导致其在聚类、分类等任务中的效能受限。本文提出一种新颖高效的非负矩阵分解分解方法,称为基于高阶近邻矩阵和二分图的非负矩阵分解方法,有效解决了上述缺陷。具体而言,该方法通过跨阶半正定近邻图来寻求因子矩阵,从而为数据提供全面且互补的近邻信息,同时在二分图结构中强化因子矩阵的簇结构,使得系数矩阵具有清晰的簇结构,因而更适用于聚类应用。此外,我们引入系数矩阵的正交约束条件,不仅增强了模型的可解释性,且导致数据表示的直接聚类解释。本文的主要贡献总结如下:(1)本文利用由跨阶邻居构成的高阶近邻图来进行非负矩阵分解,这为数据提供了全面且互补的信息。其次,模型还对系数矩阵施加了正交性约束,使其更适应于聚类。(2)受经典子空间聚类模型的启发,低秩约束不等同于表示矩阵有清晰的聚类结构和突出的聚类能力,所以通过二分图划分来保留聚类信息,这使得系数矩阵具有清晰的簇结构并进一步提高聚类能力。(3)本文采用乘性迭代准则实现变量交替优化,从而确保目标函数序列的收敛性。此外,通过建立更新算子与KKT条件之间的等价关系,给出了收敛性的严谨理论证明。(4)本文在两种人脸数据集(Jaffe、PIX)、一个手写数字数据集(Semeion)以及两个物体图像数据集(COIL20、COIL100)共五个数据集上,与七种对比方法(WNMF、RMNMF、CNMF、KNMF、ONMF、OPMC、GLS-NMF)进行了对比实验,大量的实验结果证实了该方法的有效性。此外,对收敛性、参数敏感性、簇结构、数据重组等进行了可视化来进一步验证所提方法的理论性质。最后设计了消融实验,证实了增加高阶邻接矩阵和二分图的学习能够提升算法的性能。更多还原

【Abstract】 In recent years,Nonnegative Matrix Factorization(NMF)has garnered significant at-tention for its ability to reveal latent data structures.However,existing NMF methods gen-erally suffer from insufficient exploration of data correlations,as they only consider first-order neighborhood relationships while neglecting higher-order(second-order or beyond)proximities,which may fail to fully capture the local geometric structure of data.In reality,higher-order neighborhoods can uncover more complex associations and latent data struc-tures.Moreover,these methods often underutilize the inherent cluster structure of coeffi-cient matrices,limiting their effectiveness in tasks like clustering and classification.This paper proposes a novel and efficient NMF method called Fine-Grained Bipartite Nonnega-tive Matrix Factorization for Clustering(Figer-NMF),which effectively addresses these limitations.Specifically,this study:Seeks factor matrices through cross-order positive semi-definite neighbor graph to provide comprehensive and complementary proximity in-formation;Enhances the cluster structure of factor matrices within a bipartite graph frame-work,yielding coefficient matrices with distinct cluster structures that are better suited for clustering applications;Incorporates orthogonality constraints on coefficient matrices to improve model interpretability while enabling direct cluster interpretation of data represen-tations.The main contributions of this paper are summarized as follows:(1)This study proposes a higher-order neighbor graph constructed from cross-order neighbors for non-negative matrix factorization,offering comprehensive and complemen-tary information for the data.Furthermore,an orthogonality constraint is imposed on the coefficient matrix to enhance its suitability for clustering.(2)Inspired by classical subspace clustering models,this study recognizes that low-rank constraints alone do not guarantee a clear cluster structure or strong clustering capa-bility in the representation matrix.Therefore,this study preserves clustering information via bipartite graph partitioning,ensuring that the coefficient matrix exhibits a distinct clus-ter structure and further improving clustering performance.(3)A multiplicative iterative algorithm ensures convergence of the objective function sequence.This study provides rigorous theoretical proofs by establishing equivalence be-tween update rules and KKT conditions.(4)Experiments on five datasets(Jaffe and PIX face datasets,Semeion handwritten digits,COIL20 and COIL100 object images)demonstrate superiority over seven baseline methods(WNMF,RMNMF,CNMF,KNMF,ONMF,OPMC,GLS-NMF).Additional vis-ualizations(convergence curves,parameter sensitivity,cluster structures,data reorganiza-tions)and ablation studies confirm the benefits of higher-order neighbor matrices and bi-partite graph learning.更多还原