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Networks are commonly used to model complex systems. The different entities in the system are represented by nodes of the network and their interactions by edges. In most real life systems, the different entities may interact in different ways necessitating the use of multiplex networks where multiple links are used to model the interactions. One of the major tools for inferring network topology is community detection. Although there are numerous works on community detection in single-layer networks, existing community detection methods for multiplex networks mostly learn a common community structure across layers and do not take the heterogeneity across layers into account. In this paper, we introduce a new multiplex community detection method that identifies communities that are common across layers as well as those that are unique to each layer. The proposed method, Multiplex Orthogonal Nonnegative Matrix Tri-Factorization, represents the adjacency matrix of each layer as the sum of two low-rank matrix factorizations corresponding to the common and private communities, respectively. Unlike most of the existing methods which require the number of communities to be pre-determined, the proposed method also introduces a two stage method to determine the number of common and private communities. The proposed algorithm is evaluated on synthetic and real multiplex networks, as well as for multiview clustering applications, and compared to state-of-the-art techniques.

Nonnegative matrix factorization (NMF) decomposes a nonnegative matrix into the product of two lower-rank nonnegative matrices. Since NMF learns parts-based representation, it has been widely used as a feature learning component in many fields. However, standard NMF algorithms ignore the training labels as well as unlabeled data in the test domain. In this paper, we propose a transductive nonnegative matrix tri-factorization method (T-NMTF) to simultaneously exploit the label information of training examples and the statistical structure of features in the test domain. Different from standard NMF, nonnegative matrix tri-factorization (NMTF) decomposes a nonnegative matrix into the product of three lower-rank nonnegative matrices, and thus provides a flexible framework to transduce discriminative information of training examples to test examples. In particular, the proposed T-NMTF projects both training examples and test examples into a unified subspace, and expects the coefficients of training examples close to their label vectors. Since training examples and test examples are assumed to identically distributed, it is reasonable to expect the learned coefficients of test examples approximate their label vectors well. To estimate the T-NMTF parameters, we develop an efficient multiplicative update rule and prove its convergence. In addition, we propose a manifold regularized T-NMTF (MT-NMTF) algorithm that exploits the local geometry structure of the dataset to boost discriminant power. Experimental results on face recognition demonstrate the effectiveness of T-NMTF and MT-NMTF.

Due to the explosive growth of image data, image annotation has been one of the most popular research directions in computer vision. It has been widely used in image retrieval, image analysis and understanding. Because traditional manual image annotation is time consuming, more advanced automatic annotation methods are needed. A major challenge in developing an efficient image annotation method is how to effectively use all available information contained in the data. To this end, this paper proposes a novel image annotation framework that uses multiple information from data. It employs nonnegative matrix tri-factorization (NMTF) to simultaneously factorize image-to-label, image-to-feature, and feature-to-label relation matrices using their intertype relationships and incorporates the intratype information through manifold regularizations. This method can be referred to as the trigraph regularized collective matrix tri-factorization framework (TG-CMTF). TG-CMTF captures the correlations among different labels, different images and different features. By taking advantage of these relations from images, features and labels, TG-CMTF can achieve better annotation performance than most state-of-the-art methods. The promising experimental results on three standard benchmarks have shown the effectiveness of this information. Furthermore, we show the annotation process as a precise optimization problem and solve it by an iterative algorithm, which proves the correctness of the proposed method from the mathematical theory.

With the widespread use of Location-based Social Networks (LBSNs), massive Point-of-Interest (POI) data continuously generated by users. POI clustering is an essential foundation for efficiently processing large amounts of POI data. However, the majority of existing studies only consider artificial labels and geographic information for clustering POIs and rarely take account of the characteristics of user behavior. The main challenge of POI clustering is lack of label information at present. To address the issues above, we propose a method of collaborative clustering based on Nonnegative Matrix Tri-factorization for POI (POI-NMTF), which combines the similarity of users based on time and location by exploiting the user check-in data in our study. Our algorithm provides a co-clustering method that allows clustering users and POIs simultaneously thereby discover the potential preference of users. Moreover, it can also better reflect the multiple interest attributes of users for a single POI, because our algorithm is a soft clustering method. We test our method using real dataset, and the experimental results show the validity and correctness of our algorithm, the clustering result is superior to other compared methods. [ABSTRACT FROM AUTHOR]

Abstract Non-negative Matrix Factorization technique has attracted many interests in overlapping community detection due to its performance and interpretability. However, when adapted to discover community structure the intrinsic geometric information of the network graph is seldom considered. In view of this, we proposed a novel NMF based algorithm called Graph regularized nonnegative matrix tri-factorization (GNMTF) model, which incorporates the intrinsic geometrical properties of the network graph by manifold regularization. Moreover, by using three factor matrices we can not only explicitly obtain the community membership of each node but also learn the interaction among different communities. The experimental results on two well-known real world networks and a benchmark network demonstrate the effectiveness of the algorithm over the representative non-negative matrix factorization based method. Highlights • By consideration of the low-dimensional manifold it could achieve better results. • The number of communities can be determined by analysis of the spectrum properties. • It merges advantages of fuzzy and non-fuzzy overlapping community detection methods. [ABSTRACT FROM AUTHOR]

The goal of document co-clustering is to partition textual data sets into groups by utilizing the duality between documents (i.e., data points) and words (i.e., features). That is, the documents can be grouped based on their distribution on words, while words can be grouped based on their distribution on documents. However, traditional co-clustering methods are usually sensitive to the input affinity matrix since they partition the data based on the fixed data graph. To address this limitation, in this paper, based on nonnegative matrix tri-factorization, we propose a new framework of co-clustering with adaptive local structure learning. The proposed unified learning framework performs intrinsic structure learning and tri-factorization (i.e., 3-factor factorization) simultaneously. The intrinsic structure is adaptively learned from the results of tri-factorization, and the factors are reformulated to preserve the refined local structures of the textual data. In this way, the local structure learning and factorization can be mutually improved. Furthermore, considering the duality between documents and words, the proposed framework explores not only the adaptive local structure of the data space, but also the adaptive local structure of the feature space. In order to solve the optimization problem of our method, an efficient iterative updating algorithm is proposed with guaranteed convergence. Experiments on benchmark textual data sets demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Traditional anomaly detection on microblogging mostly focuses on individual anomalous users or messages. Since anomalous users employ advanced intelligent means, the anomaly detection is greatly poor in performance. In this paper, we propose an innovative framework of anomaly detection based on bipartite graph and co-clustering. A bipartite graph between users and messages is built to model the homogeneous and heterogeneous interactions. The proposed co-clustering algorithm based on nonnegative matrix tri-factorization can detect anomalous users and messages simultaneously. The homogeneous relations modeled by the bipartite graph are used as constraints to improve the accuracy of the co-clustering algorithm. Experimental results show that the proposed scheme can detect individual and group anomalies with high accuracy on a Sina Weibo dataset. [ABSTRACT FROM AUTHOR]

The objective of co-clustering is to simultaneously identify blocks of similarity between the sample set and feature set. Co-clustering has become a widely used technique in data mining, machine learning, and other research areas. The nonnegative matrix tri-factorization (NMTF) algorithm, which aims to decompose an objective matrix into three low-dimensional matrices, is an important tool to achieve co-clustering. However, noise is usually introduced during objective matrix factorization, and the method of square loss is very sensitive to noise, which significantly reduces the performance of the model. To solve this issue, this paper proposes a tri-regularized NMTF (TRNMTF) model for co-clustering, which combines graph regularization, Frobenius norm, and l 1 norm to simultaneously optimize the objective function. TRNMTF can execute feature selection well, enhance the sparseness of the model, adjust the eigenvalues in the low-dimensional matrix, eliminate noise in the model, and obtain cleaner data matrices to approximate the objective matrix, which significantly improves the performance of the model and its generalization ability. Furthermore, to solve the iterative optimization schemes of TRNMTF, this study converts the objective function into elemental form to infer and provide detailed iterative update rules. Experimental results on 8 data sets show that the proposed model displays superior performance. [ABSTRACT FROM AUTHOR]

Nonnegative matrix factorization (NMF) has been a vital data representation technique, and has demonstrated significant potential in the field of machine learning and data mining. Nonnegative matrix tri-factorization (NMTF) is an extension of NMF, and provides more degrees of freedom than NMF. In this paper, we propose the correntropy based orthogonal nonnegative matrix tri-factorization (CNMTF) algorithm, which is robust to noisy data contaminated by non-Gaussian noise and outliers. Different from previous NMF algorithms, CNMTF firstly applies correntropy to NMTF to measure the similarity, and preserves double orthogonality conditions and dual graph regularization. We adopt the half-quadratic technique to solve the optimization problem of CNMTF, and derive the multiplicative update rules. The complexity issue of CNMTF is also presented. Furthermore, the robustness of the proposed algorithm is analyzed, and the relationships between CNMTF and several previous NMF based methods are discussed. Experimental results demonstrate that the proposed CNMTF method has better performance on real world image and text datasets for clustering tasks, compared with several state-of-the-art methods. • The correntropy based orthogonal nonnegative matrix tri-factorization (CNMTF) algorithm is proposed. • The robustness of CNMTF is analyzed by make a comparison with several robust NMF methods. • The relationships between CNMTF and several previous NMF methods are also discussed. [ABSTRACT FROM AUTHOR]