7 research outputs found
On Finding Joint Subspace Boolean Matrix Factorizations
Finding latent factors of the data using matrix factorizations is a tried-and-tested approach in data mining. But finding shared factors over multiple matrices is more novel problem. Specifically, given two matrices, we want to find a set of factors shared by these two matrices and sets of factors specific for the matrices. Not only does such decomposition reveal what is common between the two matrices, it also eliminates the need of explaining that common part twice, thus concentrating the non-shared factors to uniquely specific parts of the data. This paper studies a problem called Joint Subspace Boolean Matrix Factorization asking exactly that: a set of shared factors and sets of specific factors. Furthermore, the matrix factorization is based on the Boolean arithmetic. This restricts the presented approach suitable to only binary matrices. The benefits, however, include much sparser factor matrices and greater interpretability of the results. The paper presents three algorithms for finding the Joint Subspace Boolean Matrix Factorization, an MDLbased method for selecting the subspaces ’ dimensionality, and throughout experimental evaluation of the proposed algorithms.
A mathematical theory of making hard decisions: model selection and robustness of matrix factorization with binary constraints
One of the first and most fundamental tasks in machine learning is to group observations within a dataset. Given a notion of similarity, finding those instances which are outstandingly similar to each other has manifold applications. Recommender systems and topic analysis in text data are examples which are most intuitive to grasp. The interpretation of the groups, called clusters, is facilitated if the assignment of samples is definite. Especially in high-dimensional data, denoting a degree to which an observation belongs to a specified cluster requires a subsequent processing of the model to filter the most important information. We argue that a good summary of the data provides hard decisions on the following question: how many groups are there, and which observations belong to which clusters? In this work, we contribute to the theoretical and practical background of clustering tasks, addressing one or both aspects of this question. Our overview of state-of-the-art clustering approaches details the challenges of our ambition to provide hard decisions. Based on this overview, we develop new methodologies for two branches of clustering: the one concerns the derivation of nonconvex clusters, known as spectral clustering; the other addresses the identification of biclusters, a set of samples together with similarity defining features, via Boolean matrix factorization. One of the main challenges in both considered settings is the robustness to noise. Assuming that the issue of robustness is controllable by means of theoretical insights, we have a closer look at those aspects of established clustering methods which lack a theoretical foundation. In the scope of Boolean matrix factorization, we propose a versatile framework for the optimization of matrix factorizations subject to binary constraints. Especially Boolean factorizations have been computed by intuitive methods so far, implementing greedy heuristics which lack quality guarantees of obtained solutions. In contrast, we propose to build upon recent advances in nonconvex optimization theory. This enables us to provide convergence guarantees to local optima of a relaxed objective, requiring only approximately binary factor matrices. By means of this new optimization scheme PAL-Tiling, we propose two approaches to automatically determine the number of clusters. The one is based on information theory, employing the minimum description length principle, and the other is a novel statistical approach, controlling the false discovery rate. The flexibility of our framework PAL-Tiling enables the optimization of novel factorization schemes. In a different context, where every data point belongs to a pre-defined class, a characterization of the classes may be obtained by Boolean factorizations. However, there are cases where this traditional factorization scheme is not sufficient. Therefore, we propose the integration of another factor matrix, reflecting class-specific differences within a cluster. Our theoretical considerations are complemented by empirical evaluations, showing how our methods combine theoretical soundness with practical advantages
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Analyzing, Mining, and Predicting Networked Behaviors
Network structure exists in various types of data in the real world, such as online and offline social networks, traffic networks, computer networks, brain networks, and countless other cases where there are relationships between different entities in the data. What are the roles of network structures in these data? First, the network captures inherent characteristics of the data themselves. This is clear from the definition of the network, which represents the relationship between entities: e.g., the social links among people in a social network describe how they interact with each other; a road network summarizes how the roads are laid out geographically; a brain network obtained from fMRI images represents pairs of brain regions that are active at the same time; a computer network constrains the paths via which internet packages and thus information or viruses can spread. Second, the network structures affect the evolution of the data over time. For example, new friendship links in an online social network are frequently created between friends of friends. Similarly, the current road network structure is without a doubt taken into consideration when roads are added or temporarily closed. As we grow, our brains also grow, including the additions of useful links or the clean up of unnecessary links between brain regions. Third, the network structures act as guidance for many different processes happening in the data. For instance, the links between users on social network dictate how gossips can spread; the roads influence how traffic flows in a city; the links between brain regions affects the way we think and how effectively we do things; the connections between computers route the transfer of any information on the internet.In this thesis, I studied the network effect in various networked behaviors, including analyzing such effect, finding its patterns, and predicting future networked behaviors. First, I gained insights into the data by analyzing the accompanied network structures as well as its evolution. Second, I proposed algorithms for mining different network patterns that help summarize the effect of the network structures on different networked behaviors. Finally, I proposed models to predict the evolution of networked behaviors over time. Toward these tasks, I explored a wide variety of network data, including protein-protein interaction networks, online social networks, collaboration networks, chemical compounds, and traffic networks. Overall, I tackled these network data in different aspects and developed a number of methods for effectively mining and forecasting networked behaviors in data