We introduce normalized nonnegative models (NNM) for explorative data analysis. NNMs are partial convexifications of models from probability theory. We demonstrate their value at the example of item recommendation. We show that NNM-based recommender systems satisfy three criteria that all recommender systems should ideally satisfy: high predictive power, computational tractability, and expressive representations of users and items. Expressive user and item representations are important in practice to succinctly summarize the pool of customers and the pool of items. In NNMs, user representations are expressive because each user's preference can be regarded as normalized mixture of preferences of stereotypical users. The interpretability of item and user representations allow us to arrange properties of items (e.g., genres of movies or topics of documents) or users (e.g., personality traits) hierarchically.
These new interpretations of @math and @math allow us to deal with situations where @math is a categorical random variable and we can extract hierarchical structures from NNMs in a straightforward manner. Aspect models can deal with both of these tasks too. However, we think that dealing with these tasks in terms of aspect models is less natural then dealing with these tasks with NNMs. For instance, to model multiple outcomes like ( @math ) we need to first decide on a particular graphical model (see section 2.3 in @cite_3 ). Moreover, to extract hierarchical classifications of topics, we need to imagine generative processes like the nested Chinese restaurant process @cite_5 , or we need to employ nested hierarchical Dirichlet processes @cite_8 .
@cite_5 @cite_3 @cite_8
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We develop a nested hierarchical Dirichlet process (nHDP) for hierarchical topic modeling. The nHDP generalizes the nested Chinese restaurant process (nCRP) to allow each word to follow its own path to a topic node according to a per-document distribution over the paths on a shared tree. This alleviates the rigid, single-path formulation assumed by the nCRP, allowing documents to easily express complex thematic borrowings. We derive a stochastic variational inference algorithm for the model, which enables efficient inference for massive collections of text documents. We demonstrate our algorithm on 1.8 million documents from The New York Times and 2.7 million documents from Wikipedia . Inspired by a two-level theory from political science that unifies agenda setting and ideological framing, we propose supervised hierarchical latent Dirichlet allocation (SHLDA), which jointly captures documents' multi-level topic structure and their polar response variables. Our model extends the nested Chinese restaurant processes to discover tree-structured topic hierarchies and uses both per-topic hierarchical and per-word lexical regression parameters to model response variables. SHLDA improves prediction on political affiliation and sentiment tasks in addition to providing insight into how topics under discussion are framed. We address the problem of learning topic hierarchies from data. The model selection problem in this domain is dauntingwhich of the large collection of possible trees to use? We take a Bayesian approach, generating an appropriate prior via a distribution on partitions that we refer to as the nested Chinese restaurant process. This nonparametric prior allows arbitrarily large branching factors and readily accommodates growing data collections. We build a hierarchical topic model by combining this prior with a likelihood that is based on a hierarchical variant of latent Dirichlet allocation. We illustrate our approach on simulated data and with an application to the modeling of NIPS abstracts.