Factor-Bounded Nonnegative Matrix Factorization

2021 ◽  
Vol 15 (6) ◽  
pp. 1-18
Author(s):  
Kai Liu ◽  
Xiangyu Li ◽  
Zhihui Zhu ◽  
Lodewijk Brand ◽  
Hua Wang
Keyword(s):  
Matrix Factorization ◽  
Clustering Algorithm ◽  
Nonnegative Matrix Factorization ◽  
Nonnegative Matrix ◽  
Optimization Methods ◽  
Auxiliary Function ◽  
Image Clustering ◽  
Real World Datasets ◽  
The Relationship ◽  
Matrix Factors

Nonnegative Matrix Factorization (NMF) is broadly used to determine class membership in a variety of clustering applications. From movie recommendations and image clustering to visual feature extractions, NMF has applications to solve a large number of knowledge discovery and data mining problems. Traditional optimization methods, such as the Multiplicative Updating Algorithm (MUA), solves the NMF problem by utilizing an auxiliary function to ensure that the objective monotonically decreases. Although the objective in MUA converges, there exists no proof to show that the learned matrix factors converge as well. Without this rigorous analysis, the clustering performance and stability of the NMF algorithms cannot be guaranteed. To address this knowledge gap, in this article, we study the factor-bounded NMF problem and provide a solution algorithm with proven convergence by rigorous mathematical analysis, which ensures that both the objective and matrix factors converge. In addition, we show the relationship between MUA and our solution followed by an analysis of the convergence of MUA. Experiments on both toy data and real-world datasets validate the correctness of our proposed method and its utility as an effective clustering algorithm.

Robust semi-supervised nonnegative matrix factorization for image clustering

2021 ◽  
Vol 111 ◽  
pp. 107683 ◽  
Author(s):  
Siyuan Peng ◽  
Wee Ser ◽  
Badong Chen ◽  
Zhiping Lin
Keyword(s):  
Matrix Factorization ◽  
Nonnegative Matrix Factorization ◽  
Nonnegative Matrix ◽  
Image Clustering

A Generalized Divergence Measure for Nonnegative Matrix Factorization

2007 ◽  
Vol 19 (3) ◽  
pp. 780-791 ◽  
Author(s):  
Raul Kompass
Keyword(s):  
Matrix Factorization ◽  
Nonnegative Matrix Factorization ◽  
Nonnegative Matrix ◽  
Auxiliary Function ◽  
Divergence Measure ◽  
Factorization Problem ◽  
Special Cases ◽  
Leibler Divergence ◽  
Update Rules ◽  
Quadratic Distance

This letter presents a general parametric divergence measure. The metric includes as special cases quadratic error and Kullback-Leibler divergence. A parametric generalization of the two different multiplicative update rules for nonnegative matrix factorization by Lee and Seung (2001) is shown to lead to locally optimal solutions of the nonnegative matrix factorization problem with this new cost function. Numeric simulations demonstrate that the new update rule may improve the quadratic distance convergence speed. A proof of convergence is given that, as in Lee and Seung, uses an auxiliary function known from the expectation-maximization theoretical framework.

scholarly journals Multi-Component Nonnegative Matrix Factorization

2017 ◽  
Author(s):  
Jing Wang ◽  
Feng Tian ◽  
Xiao Wang ◽  
Hongchuan Yu ◽  
Chang Hong Liu ◽  
...  
Keyword(s):  
Matrix Factorization ◽  
Nonnegative Matrix Factorization ◽  
Multiple Representations ◽  
Nonnegative Matrix ◽  
Real Data ◽  
Multiple Components ◽  
Face Images ◽  
The Arts ◽  
Accurate Performance ◽  
Real World Datasets

Real data are usually complex and contain various components. For example, face images have expressions and genders. Each component mainly reflects one aspect of data and provides information others do not have. Therefore, exploring the semantic information of multiple components as well as the diversity among them is of great benefit to understand data comprehensively and in-depth. However, this cannot be achieved by current nonnegative matrix factorization (NMF)-based methods, despite that NMF has shown remarkable competitiveness in learning parts-based representation of data. To overcome this limitation, we propose a novel multi-component nonnegative matrix factorization (MCNMF). Instead of seeking for only one representation of data, MCNMF learns multiple representations simultaneously, with the help of the Hilbert Schmidt Independence Criterion (HSIC) as a diversity term. HSIC explores the diverse information among the representations, where each representation corresponds to a component. By integrating the multiple representations, a more comprehensive representation is then established. A new iterative updating optimization scheme is derived to solve the objective function of MCNMF, along with its correctness and convergence guarantees. Extensive experimental results on real-world datasets have shown that MCNMF not only achieves more accurate performance over the state-of-the-arts using the aggregated representation, but also interprets data from different aspects with the multiple representations, which is beyond what current NMFs can offer.

scholarly journals Multiview Clustering via Robust Neighboring Constraint Nonnegative Matrix Factorization

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Feiqiong Chen ◽  
Guopeng Li ◽  
Shuaihui Wang ◽  
Zhisong Pan
Keyword(s):  
Real World ◽  
Matrix Factorization ◽  
Nonnegative Matrix Factorization ◽  
Nonnegative Matrix ◽  
Data Representation ◽  
Structure Representation ◽  
Cut Method ◽  
Accurate Performance ◽  
Real World Datasets ◽  
Multiview Clustering

Many real-world datasets are described by multiple views, which can provide complementary information to each other. Synthesizing multiview features for data representation can lead to more comprehensive data description for clustering task. However, it is often difficult to preserve the locally real structure in each view and reconcile the noises and outliers among views. In this paper, instead of seeking for the common representation among views, a novel robust neighboring constraint nonnegative matrix factorization (rNNMF) is proposed to learn the neighbor structure representation in each view, and L2,1-norm-based loss function is designed to improve its robustness against noises and outliers. Then, a final comprehensive representation of data was integrated with those representations of multiviews. Finally, a neighboring similarity graph was learned and the graph cut method was used to partition data into its underlying clusters. Experimental results on several real-world datasets have shown that our model achieves more accurate performance in multiview clustering compared to existing state-of-the-art methods.

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