Trees and beyond : exploiting and improving tree-structured graphical models

Abstract

Probabilistic models commonly assume that variables are independent of each other conditioned on a subset of other variables. Graphical models provide a powerful framework for encoding such conditional independence structure of a large collection of random variables. A special class of graphical models with significant theoretical and practical importance is the class of tree-structured graphical models. Tree models have several advantages: they can be easily learned given data, their structures are often intuitive, and inference in tree models is highly efficient. However, tree models make strong conditional independence assumptions, which limit their modeling power substantially. This thesis exploits the advantages of tree-structured graphical models and considers modifications to overcome their limitations. To improve the modeling accuracy of tree models, we consider latent trees in which variables at some nodes represent the original (observed) variables of interest while others represent the latent variables added during the learning procedure. The appeal of such models is clear: the additional latent variables significantly increase the modeling power, and inference on trees is scalable with or without latent variables. We propose two computationally efficient and statistically consistent algorithms for learning latent trees, and compare the proposed algorithms to other methods by performing extensive numerical experiments on various latent tree models. We exploit the advantages of tree models in the application of modeling contextual information of an image. Object co-occurrences and spatial relationships can be important cues in recognizing and localizing object instances. We develop tree-based context models and demonstrate that its simplicity enables us to integrate many sources of contextual information efficiently. In addition to object recognition, we are interested in using context models to detect objects that are out of their normal context. This task requires precise and careful modeling of object relationships, so we use a latent tree for object co-occurrences. Many of the latent variables can be interpreted as scene categories, capturing higher-order dependencies among object categories. Tree-structured graphical models have been widely used in multi-resolution (MR) modeling. In the last part of the thesis, we move beyond trees, and propose a new modeling framework that allows additional dependency structure at each scale of an MR tree model. We mainly focus on MR models with jointly Gaussian variables, and assume that variables at each scale have sparse covariance structure (as opposed to fully-uncorrelated structure in MR trees) conditioned on variables at other scales. We develop efficient inference algorithms that are partly based on inference on the embedded MR tree and partly based on local filtering at each scale. In addition, we present methods for learning such models given data at the finest scale by formulating a convex optimization problem.