We develop necessary and sufficient conditions and a novel provably consistent and efficient algorithm for discovering topics (latent factors) from observations (documents) that are realized from a probabilistic mixture of shared latent factors that have certain properties. Our focus is on the class of topic models in which each shared latent factor contains a novel word that is unique to that factor, a property that has come to be known as separability. Our algorithm is based on the key insight that the novel words correspond to the extreme points of the convex hull formed by the row-vectors of a suitably normalized word co-occurrence matrix. We leverage this geometric insight to establish polynomial computation and sample complexity bounds based on a few isotropic random projections of the rows of the normalized word co-occurrence matrix. Our proposed random-projections-based algorithm is naturally amenable to an efficient distributed implementation and is attractive for modern web-scale distributed data mining applications.
In response to scientific needs for more diverse and structured explanations of statistical data, researchers have discovered how to model individual data points as belonging to multiple groups. Handbook of Mixed Membership Models and Their Applications shows you how to use these flexible modeling tools to uncover hidden patterns in modern high-dimensional multivariate data. It explores the use of the models in various application settings, including survey data, population genetics, text analysis, image processing and annotation, and molecular biology. Through examples using real data sets, youll discover how to characterize complex multivariate data in: Studies involving genetic databases Patterns in the progression of diseases and disabilities Combinations of topics covered by text documents Political ideology or electorate voting patterns Heterogeneous relationships in networks, and much more The handbook spans more than 20 years of the editors and contributors statistical work in the field. Top researchers compare partial and mixed membership models, explain how to interpret mixed membership, delve into factor analysis, and describe nonparametric mixed membership models. They also present extensions of the mixed membership model for text analysis, sequence and rank data, and network data as well as semi-supervised mixed membership models. Probabilistic Latent Semantic Indexing is a novel approach to automated document indexing which is based on a statistical latent class model for factor analysis of count data. Fitted from a training corpus of text documents by a generalization of the Expectation Maximization algorithm, the utilized model is able to deal with domain specific synonymy as well as with polysemous words. In contrast to standard Latent Semantic Indexing (LSI) by Singular Value Decomposition, the probabilistic variant has a solid statistical foundation and defines a proper generative data model. Retrieval experiments on a number of test collections indicate substantial performance gains over direct term matching methods as well as over LSI. In particular, the combination of models with different dimensionalities has proven to be advantageous. Topic models, such as latent Dirichlet allocation (LDA), can be useful tools for the statistical analysis of document collections and other discrete data. The LDA model assumes that the words of each document arise from a mixture of topics, each of which is a distribution over the vocabulary. A limitation of LDA is the inability to model topic correlation even though, for example, a document about genetics is more likely to also be about disease than X-ray astronomy. This limitation stems from the use of the Dirichlet distribution to model the variability among the topic proportions. In this paper we develop the correlated topic model (CTM), where the topic proportions exhibit correlation via the logistic normal distribution [J. Roy. Statist. Soc. Ser. B 44 (1982) 139--177]. We derive a fast variational inference algorithm for approximate posterior inference in this model, which is complicated by the fact that the logistic normal is not conjugate to the multinomial. We apply the CTM to the articles from Science published from 1990--1999, a data set that comprises 57M words. The CTM gives a better fit of the data than LDA, and we demonstrate its use as an exploratory tool of large document collections.  Probabilistic topic modeling provides a suite of tools for the unsupervised analysis of large collections of documents. Topic modeling algorithms can uncover the underlying themes of a collection and decompose its documents according to those themes. This analysis can be used for corpus exploration, document search, and a variety of prediction problems. In this tutorial, I will review the state-of-the-art in probabilistic topic models. I will describe the three components of topic modeling: (1) Topic modeling assumptions (2) Algorithms for computing with topic models (3) Applications of topic models In (1), I will describe latent Dirichlet allocation (LDA), which is one of the simplest topic models, and then describe a variety of ways that we can build on it. These include dynamic topic models, correlated topic models, supervised topic models, author-topic models, bursty topic models, Bayesian nonparametric topic models, and others. I will also discuss some of the fundamental statistical ideas that are used in building topic models, such as distributions on the simplex, hierarchical Bayesian modeling, and models of mixed-membership. In (2), I will review how we compute with topic models. I will describe approximate posterior inference for directed graphical models using both sampling and variational inference, and I will discuss the practical issues and pitfalls in developing these algorithms for topic models. Finally, I will describe some of our most recent work on building algorithms that can scale to millions of documents and documents arriving in a stream. In (3), I will discuss applications of topic models. These include applications to images, music, social networks, and other data in which we hope to uncover hidden patterns. I will describe some of our recent work on adapting topic modeling algorithms to collaborative filtering, legislative modeling, and bibliometrics without citations. Finally, I will discuss some future directions and open research problems in topic models.