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.
The prevailing approaches for estimation and inference problems in topic modeling are based on MAP or ML estimation @cite_17 . However, the computation of posterior distributions conditioned on observations @math is intractable @cite_12 . Moreover, the MAP estimation objective is non-convex and has been shown to be @math -hard @cite_47 @cite_21 . Therefore various approximation and heuristic strategies have been employed. These approaches fall into two major categories -- sampling approaches and optimization approaches. Most sampling approaches are based on Markov Chain Monte Carlo (MCMC) algorithms that seek to generate (approximately) independent samples from a Markov Chain that is carefully designed to ensure that the sample distribution converges to the true posterior @cite_27 @cite_43 . Optimization approaches are typically based on the so-called Variational-Bayes methods. These methods optimize the parameters of a simpler parametric distribution so that it is close to the true posterior in terms of KL divergence @cite_12 @cite_2 . Expectation- -Maximization- -type algorithms are typically used in these methods. In practice, while both Variational-Bayes and MCMC algorithms have similar performance, Variational-Bayes is typically faster than MCMC @cite_15 @cite_17 .
@cite_15 @cite_21 @cite_43 @cite_27 @cite_2 @cite_47 @cite_12 @cite_17
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We describe latent Dirichlet allocation (LDA), a generative probabilistic model for collections of discrete data such as text corpora. LDA is a three-level hierarchical Bayesian model, in which each item of a collection is modeled as a finite mixture over an underlying set of topics. Each topic is, in turn, modeled as an infinite mixture over an underlying set of topic probabilities. In the context of text modeling, the topic probabilities provide an explicit representation of a document. We present efficient approximate inference techniques based on variational methods and an EM algorithm for empirical Bayes parameter estimation. We report results in document modeling, text classification, and collaborative filtering, comparing to a mixture of unigrams model and the probabilistic LSI model. A natural evaluation metric for statistical topic models is the probability of held-out documents given a trained model. While exact computation of this probability is intractable, several estimators for this probability have been used in the topic modeling literature, including the harmonic mean method and empirical likelihood method. In this paper, we demonstrate experimentally that commonly-used methods are unlikely to accurately estimate the probability of held-out documents, and propose two alternative methods that are both accurate and efficient. The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building large-scale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fields, including bioinformatics, communication theory, statistical physics, combinatorial optimization, signal and image processing, information retrieval and statistical machine learning. Many problems that arise in specific instances — including the key problems of computing marginals and modes of probability distributions — are best studied in the general setting. Working with exponential family representations, and exploiting the conjugate duality between the cumulant function and the entropy for exponential families, we develop general variational representations of the problems of computing likelihoods, marginal probabilities and most probable configurations. We describe how a wide variety of algorithms — among them sum-product, cluster variational methods, expectation-propagation, mean field methods, max-product and linear programming relaxation, as well as conic programming relaxations — can all be understood in terms of exact or approximate forms of these variational representations. The variational approach provides a complementary alternative to Markov chain Monte Carlo as a general source of approximation methods for inference in large-scale statistical models. We consider the computational complexity of probabilistic inference in Latent Dirichlet Allocation (LDA). First, we study the problem of finding the maximum a posteriori (MAP) assignment of topics to words, where the document's topic distribution is integrated out. We show that, when the effective number of topics per document is small, exact inference takes polynomial time. In contrast, we show that, when a document has a large number of topics, finding the MAP assignment of topics to words in LDA is NP-hard. Next, we consider the problem of finding the MAP topic distribution for a document, where the topic-word assignments are integrated out. We show that this problem is also NP-hard. Finally, we briefly discuss the problem of sampling from the posterior, showing that this is NP-hard in one restricted setting, but leaving open the general question. 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.