Advances in nonnegative matrix factorization with application on data clustering

Clustering is an important direction in many ﬁelds, e.g., machine learning, data mining and computer vision. It aims to divide data into groups (clusters) for the purposes of summarization or improved understanding. With the rapid development of new technology, high-dimensional data become very common in many real world applications, such as satellite returned large number of images, robot received real-time video streaming, large-scale text database and the mass of information on the social networks (i.e., Facebook, twitter), etc, however, most existing clustering approaches are heavily restricted by the large number of features, and tend to be ineﬃcient and even infeasible. In this thesis, we focus on ﬁnding an optimal low dimensional representation of high-dimensional data, based nonnegative matrix factorization (NMF) framework, for better clustering. Speciﬁcally, there are three methods as follows: - Multiple Components Based Representation Learning 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. To this end, we propose a novel multi-component nonnegative matrix factorization. Instead of seeking for only one representation of data, our approach 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. 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. - Ordered Structure Preserving Representation Learning Real-world applications often process data, such as motion sequences and video clips, are with ordered structure, i.e., consecutive neighbouring data samples are very likely share similar features unless a sudden change occurs. Therefore, traditional NMF assumes the data samples and features to be independently distributed, making it not proper for the analysis of such data. To overcome this limitation, a novel NMF approach is proposed to take full advantage of the ordered nature embedded in the sequential data to improve the accuracy of data representation. With a L2,1-norm based neighbour penalty term, ORNMF enforces the similarity of neighbouring data. ORNMF also adopts the L2,1-norm based loss function to improve its robustness against noises and outliers. Moreover, ORNMF can ﬁnd the cluster boundaries and get the number of clusters without the number of clusters to be given beforehand. A new iterative up- dating optimization algorithm is derived to solve ORNMF’s objective function. The proofs of the convergence and correctness of the scheme are also presented. Experiments on both synthetic and real-world datasets have demonstrated the eﬀectiveness of ORNMF. - Diversity Enhanced Multi-view Representation Learning Multi-view learning aims to explore the correlations of diﬀerent information, such as diﬀerent features or modalities to boost the performance of data analysis. Multi-view data are very common in many real world applications because data is often collected from diverse domains or obtained from diﬀerent feature extractors. For example, color and texture information can be utilized as diﬀerent kinds of features in images and videos. Web pages are also able to be represented using the multi-view features based on text and hyperlinks. Taken alone, these views will often be deﬁcient or incomplete because diﬀerent views describe distinct perspectives of data. Therefore, we propose a Diverse Multi-view NMF approach to explore diverse information among multi-view representations for more comprehensive learning. With a novel diversity regularization term, DiNMF explicitly enforces the orthogonality of diﬀerent data representations. Importantly, DiNMF converges linearly and scales well with large-scale data. By taking into account the manifold structures, we further extend the approach under a graph-based model to preserve the locally geometrical structure of the manifolds for multi-view setting. Compared to other multi-view NMF methods, the enhanced diversity of both approaches reduce the redundancy between the multi-view representations, and improve the accuracy of the clustering results. - Constrained Multi-View Representation Learning To incorporate prior information for learning accurately, we propose a novel semi- supervised multi-view NMF approach, which considers both the label constraints as well as the multi-view consistence simultaneously. In particular, the approach guarantees that data sharing the same label will have the same new representation and be mapped into the same class in the low-dimensional space regardless whether they come from the same view. Moreover, diﬀerent from current NMF- based multi-view clustering methods that require the weight factor of each view to be speciﬁed individually, we introduce a single parameter to control the distribution of weighting factors for NMF-based multi-view clustering. Consequently, the weight factor of each view can be assigned automatically depending on the dissimilarity between each new representation matrix and the consensus matrix. Besides, Using the structured sparsity-inducing, L2,1-norm, our method is robust against noises and hence can achieve more stable clustering results.