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Non-negative matrix factorization (NMF) provides a lower rank approximation of a matrix. Due to nonnegativity imposed on the factors, it gives a latent structure that is often more physically meaningful than other lower rank approximations such as singular value decomposition (SVD). Most of the algorithms proposed in literature for NMF have been based on minimizing the Frobenius norm. This is partly due to the fact that the minimization problem based on the Frobenius norm provides much more flexibility in algebraic manipulation than other divergences. In this paper we propose a fast NMF algorithm that is applicable to general Bregman divergences. Through Taylor series expansion of the Bregman divergences, we reveal a relationship between Bregman divergences and Euclidean distance. This key relationship provides a new direction for NMF algorithms with general Bregman divergences when combined with the scalar block coordinate descent method. The proposed algorithm generalizes several recently proposed methods for computation of NMF with Bregman divergences and is computationally faster than existing alternatives. We demonstrate the effectiveness of our approach with experiments conducted on artificial as well as real world data.

We consider a search task as a set of queries that serve the same user information need. Analyzing search tasks from user query streams plays an important role in building a set of modern tools to improve search engine performance. In this paper, we propose a probabilistic method for identifying and labeling search tasks based on the following intuitive observations: queries that are issued temporally close by users in many sequences of queries are likely to belong to the same search task, meanwhile, different users having the same information needs tend to submit topically coherent search queries. To capture the above intuitions, we directly model query temporal patterns using a special class of point processes called Hawkes processes, and combine topic models with Hawkes processes for simultaneously identifying and labeling search tasks. Essentially, Hawkes processes utilize their self-exciting properties to identify search tasks if influence exists among a sequence of queries for individual users, while the topic model exploits query co-occurrence across different users to discover the latent information needed for labeling search tasks. More importantly, there is mutual reinforcement between Hawkes processes and the topic model in the unified model that enhances the performance of both. We evaluate our method based on both synthetic data and real-world query log data. In addition, we also apply our model to query clustering and search task identification. By comparing with state-of-the-art methods, the results demonstrate that the improvement in our proposed approach is consistent and promising.

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In many applications in social network analysis, it is important to model the interactions and infer the influence between pairs of actors, leading to the problem of dyadic event modeling which has attracted increasing interests recently. In this paper we focus on the problem of dyadic event attribution, an important missing data problem in dyadic event modeling where one needs to infer the missing actor-pairs of a subset of dyadic events based on their observed timestamps. Existing works either use fixed model parameters and heuristic rules for event attribution, or assume the dyadic events across actor-pairs are independent. To address those shortcomings we propose a probabilistic model based on mixtures of Hawkes processes that simultaneously tackles event attribution and network parameter inference, taking into consideration the dependency among dyadic events that share at least one actor. We also investigate using additive models to incorporate regularization to avoid overfitting. Our experiments on both synthetic and real-world data sets on international armed conflicts suggest that the proposed new method is capable of significantly improve accuracy when compared with the state-of-the-art for dyadic event attribution.