Necessary and Sufficient Conditions for Novel Word Detection in Separable Topic Models

The simplicial condition and other stronger conditions that imply it have recently played a central role in developing polynomial time algorithms with provable asymptotic consistency and sample complexity guarantees for topic estimation in separable topic models. Of these algorithms, those that rely solely on the simplicial condition are impractical while the practical ones need stronger conditions. In this paper, we demonstrate, for the first time, that the simplicial condition is a fundamental, algorithm-independent, information-theoretic necessary condition for consistent separable topic estimation. Furthermore, under solely the simplicial condition, we present a practical quadratic-complexity algorithm based on random projections which consistently detects all novel words of all topics using only up to second-order empirical word moments. This algorithm is amenable to distributed implementation making it attractive for 'big-data' scenarios involving a network of large distributed databases

A Topic Modeling Approach to Ranking

We propose a topic modeling approach to the prediction of preferences in pairwise comparisons. We develop a new generative model for pairwise comparisons that accounts for multiple shared latent rankings that are prevalent in a population of users. This new model also captures inconsistent user behavior in a natural way. We show how the estimation of latent rankings in the new generative model can be formally reduced to the estimation of topics in a statistically equivalent topic modeling problem. We leverage recent advances in the topic modeling literature to develop an algorithm that can learn shared latent rankings with provable consistency as well as sample and computational complexity guarantees. We demonstrate that the new approach is empirically competitive with the current state-of-the-art approaches in predicting preferences on some semi-synthetic and real world datasets

Learning mixed membership models with a separable latent structure: theory, provably efficient algorithms, and applications

In a wide spectrum of problems in science and engineering that includes hyperspectral imaging, gene expression analysis, and machine learning tasks such as topic modeling, the observed data is high-dimensional and can be modeled as arising from a data-specific probabilistic mixture of a small collection of latent factors. Being able to successfully learn the latent factors from the observed data is important for efficient data representation, inference, and prediction. Popular approaches such as variational Bayesian and MCMC methods exhibit good empirical performance on some real-world datasets, but make heavy use of approximations and heuristics for dealing with the highly non-convex and computationally intractable optimization objectives that accompany them. As a consequence, consistency or efficiency guarantees for these algorithms are rather weak. This thesis develops a suite of algorithms with provable polynomial statistical and computational efficiency guarantees for learning a wide class of high-dimensional Mixed Membership Latent Variable Models (MMLVMs). Our approach is based on a natural separability property of the shared latent factors that is known to be either exactly or approximately satisfied by the estimates produced by variational Bayesian and MCMC methods. Latent factors are called separable when each factor contains a novel part that is predominantly unique to that factor. For a broad class of problems, we establish that separability is not only an algorithmically convenient structural condition, but is in fact an inevitable consequence of a having a relatively small number of latent factors in a high-dimensional observation space. The key insight underlying our algorithms is the identification of novel parts of each latent factor as extreme points of certain convex polytopes in a suitable representation space. We show that this can be done efficiently through appropriately defined random projections in the representation space. We establish statistical and computational efficiency bounds that are both polynomial in all the model parameters. Furthermore, the proposed random-projections-based algorithm turns out to be naturally amenable to a low-communication-cost distributed implementation which is attractive for modern web-scale distributed data mining applications. We explore in detail two distinct classes of MMLVMs in this thesis: learning topic models for text documents based on their empirical word frequencies and learning mixed membership ranking models based on pairwise comparison data. For each problem, we demonstrate that separability is inevitable when the data dimension scales up and then establish consistency and efficiency guarantees for identifying all novel parts and estimating the latent factors. As a by-product of this analysis, we obtain the first asymptotic consistency and polynomial sample and computational complexity results for learning permutation-mixture and Mallows-mixture models for rankings based on pairwise comparison data. We demonstrate empirically that the performance of our approach is competitive with the current state-of-the-art on a number of real-world datasets