Extensions to the Latent Dirichlet Allocation Topic Model Using Flexible Priors

Intrinsically, topic models have always their likelihood functions fixed to multinomial distributions as they operate on count data instead of Gaussian data. As a result, their performances ultimately depend on the flexibility of the chosen prior distributions when following the Bayesian paradigm compared to classical approaches such as PLSA (probabilistic latent semantic analysis), unigrams and mixture of unigrams that do not use prior information. The standard LDA (latent Dirichlet allocation) topic model operates with symmetric Dirichlet distribution (as a conjugate prior) which has been found to carry some limitations due to its independent structure that tends to hinder performance for instance in topic correlation including positively correlated data processing. Compared to classical ML estimators, the use of priors ultimately presents another unique advantage of smoothing out the multinomials while enhancing predictive topic models. In this thesis, we propose a series of flexible priors such as generalized Dirichlet (GD) and Beta-Liouville (BL) for our topic models within the collapsed representation, leading to much improved CVB (collapsed variational Bayes) update equations compared to ones from the standard LDA. This is because the flexibility of these priors improves significantly the lower bounds in the corresponding CVB algorithms. We also show the robustness of our proposed CVB inferences when using simultaneously the BL and GD in hybrid generative-discriminative models where the generative stage produces good and heterogeneous topic features that are used in the discriminative stage by powerful classifiers such as SVMs (support vector machines) as we propose efficient probabilistic kernels to facilitate processing (classification) of documents based on topic signatures. Doing so, we implicitly cast topic modeling which is an unsupervised learning method into a supervised learning technique. Furthermore, due to the complexity of the CVB algorithm (as it requires second order Taylor expansions) in general, despite its flexibility, we propose a much simpler and tractable update equation using a MAP (maximum a posteriori) framework with the standard EM (expectation-maximization) algorithm. As most Bayesian posteriors are not tractable for complex models, we ultimately propose the MAP-LBLA (latent BL allocation) where we characterize the contributions of asymmetric BL priors over the symmetric Dirichlet (Dir). The proposed MAP technique importantly offers a point estimate (mode) with a much tractable solution. In the MAP, we show that point estimate could be easy to implement than full Bayesian analysis that integrates over the entire parameter space. The MAP implicitly exhibits some equivalent relationship with the CVB especially the zero order approximations CVB0 and its stochastic version SCVB0. The proposed method enhances performances in information retrieval in text document analysis. We show that parametric topic models (as they are finite dimensional methods) have a much smaller hypothesis space and they generally suffer from model selection. We therefore propose a Bayesian nonparametric (BNP) technique that uses the Hierarchical Dirichlet process (HDP) as conjugate prior to the document multinomial distributions where the asymmetric BL serves as a diffuse (probability) base measure that provides the global atoms (topics) that are shared among documents. The heterogeneity in the topic structure helps in providing an alternative to model selection because the nonparametric topic model (which is infinite dimensional with a much bigger hypothesis space) could now prune out irrelevant topics based on the associated probability masses to only retain the most relevant ones. We also show that for large scale applications, stochastic optimizations using natural gradients of the objective functions have demonstrated significant performances when we learn rapidly both data and parameters in online fashion (streaming). We use both predictive likelihood and perplexity as evaluation methods to assess the robustness of our proposed topic models as we ultimately refer to probability as a way to quantify uncertainty in our Bayesian framework. We improve object categorization in terms of inferences through the flexibility of our prior distributions in the collapsed space. We also improve information retrieval technique with the MAP and the HDP-LBLA topic models while extending the standard LDA. These two applications present the ultimate capability of enhancing a search engine based on topic models

Variational-Based Latent Generalized Dirichlet Allocation Model in the Collapsed Space and Applications

In topic modeling framework, many Dirichlet-based models performances have been hindered by the limitations of the conjugate prior. It led to models with more flexible priors, such as the generalized Dirichlet distribution, that tend to capture semantic relationships between topics (topic correlation). Now these extensions also suffer from incomplete generative processes that complicate performances in traditional inferences such as VB (Variational Bayes) and CGS (Collaspsed Gibbs Sampling). As a result, the new approach, the CVB-LGDA (Collapsed Variational Bayesian inference for the Latent Generalized Dirichlet Allocation) presents a scheme that integrates a complete generative process to a robust inference technique for topic correlation and codebook analysis. Its performance in image classification, facial expression recognition, 3D objects categorization, and action recognition in videos shows its merits

Spherical Minimum Description Length

We consider the problem of model selection using the Minimum Description Length (MDL) criterion for distributions with parameters on the hypersphere. Model selection algorithms aim to find a compromise between goodness of fit and model complexity. Variables often considered for complexity penalties involve number of parameters, sample size and shape of the parameter space, with the penalty term often referred to as stochastic complexity. Current model selection criteria either ignore the shape of the parameter space or incorrectly penalize the complexity of the model, largely because typical Laplace approximation techniques yield inaccurate results for curved spaces. We demonstrate how the use of a constrained Laplace approximation on the hypersphere yields a novel complexity measure that more accurately reflects the geometry of these spherical parameters spaces. We refer to this modified model selection criterion as spherical MDL. As proof of concept, spherical MDL is used for bin selection in histogram density estimation, performing favorably against other model selection criteria