This paper studies the topic modeling problem of tagged documents and images. Higher-order relations among tagged documents and images are major and ubiquitous characteristics, and play positive roles in extracting reliable and interpretable topics. In this paper, we propose the tag-topic models (TTM) to depict such higher-order topic structural dependencies within the Markov random field (MRF) framework. First, we use the novel factor graph representation of latent Dirichlet allocation (LDA)-based topic models from the MRF perspective, and present an efficient loopy belief propagation (BP) algorithm for approximate inference and parameter estimation. Second, we propose the factor hypergraph representation of TTM, and focus on both pairwise and higher-order relation modeling among tagged documents and images. Efficient loopy BP algorithm is developed to learn TTM, which encourages the topic labeling smoothness among tagged documents and images. Extensive experimental results confirm the incorporation of higher-order relations to be effective in enhancing the overall topic modeling performance, when compared with current state-of-the-art topic models, in many text and image mining tasks of broad interests such as word and link prediction, document classification, and tag recommendation.

Fitting probabilistic models to data is often difficult, due to the general intractability of the partition function and its derivatives. Here we propose a new parameter estimation technique that does not require computing an intractable normalization factor or sampling from the equilibrium distribution of the model. This is achieved by establishing dynamics that would transform the observed data distribution into the model distribution, and then setting as the objective the minimization of the KL divergence between the data distribution and the distribution produced by running the dynamics for an infinitesimal time. Score matching, minimum velocity learning, and certain forms of contrastive divergence are shown to be special cases of this learning technique. We demonstrate parameter estimation in Ising models, deep belief networks and an independent component analysis model of natural scenes. In the Ising model case, current state of the art techniques are outperformed by at least an order of magnitude in learning time, with lower error in recovered coupling parameters.

Research in reinforcement learning has produced algorithms for optimal decision making under uncertainty that fall within two main types. The first employs a Bayesian framework, where optimality improves with increased computational time. This is because the resulting planning task takes the form of a dynamic programming problem on a belief tree with an infinite number of states. The second type employs relatively simple algorithm which are shown to suffer small regret within a distribution-free framework. This paper presents a lower bound and a high probability upper bound on the optimal value function for the nodes in the Bayesian belief tree, which are analogous to similar bounds in POMDPs. The bounds are then used to create more efficient strategies for exploring the tree. The resulting algorithms are compared with the distribution-free algorithm UCB1, as well as a simpler baseline algorithm on multi-armed bandit problems.

In many real-world networks, nodes have class labels, attributes, or variables that affect the network's topology. If the topology of the network is known but the labels of the nodes are hidden, we would like to select a small subset of nodes such that, if we knew their labels, we could accurately predict the labels of all the other nodes. We develop an active learning algorithm for this problem which uses information-theoretic techniques to choose which nodes to explore. We test our algorithm on networks from three different domains: a social network, a network of English words that appear adjacently in a novel, and a marine food web. Our algorithm makes no initial assumptions about how the groups connect, and performs well even when faced with quite general types of network structure. In particular, we do not assume that nodes of the same class are more likely to be connected to each other---only that they connect to the rest of the network in similar ways.

Recovery of low-rank matrices has recently seen significant activity in many areas of science and engineering, motivated by recent theoretical results for exact reconstruction guarantees and interesting practical applications. A number of methods have been developed for this recovery problem. However, a principled method for choosing the unknown target rank is generally not provided. In this paper, we present novel recovery algorithms for estimating low-rank matrices in matrix completion and robust principal component analysis based on sparse Bayesian learning (SBL) principles. Starting from a matrix factorization formulation and enforcing the low-rank constraint in the estimates as a sparsity constraint, we develop an approach that is very effective in determining the correct rank while providing high recovery performance. We provide connections with existing methods in other similar problems and empirical results and comparisons with current state-of-the-art methods that illustrate the effectiveness of this approach.

We present a new algorithm for exactly solving decision making problems represented as influence diagrams. We do not require the usual assumptions of no forgetting and regularity; this allows us to solve problems with simultaneous decisions and limited information. The algorithm is empirically shown to outperform a state-of-the-art algorithm on randomly generated problems of up to 150 variables and $10^{64}$ solutions. We show that the problem is NP-hard even if the underlying graph structure of the problem has small treewidth and the variables take on a bounded number of states, but that a fully polynomial time approximation scheme exists for these cases. Moreover, we show that the bound on the number of states is a necessary condition for any efficient approximation scheme.

This paper studies the problem of learning diagnostic policies from training examples. A diagnostic policy is a complete description of the decision-making actions of a diagnostician (i.e., tests followed by a diagnostic decision) for all possible combinations of test results. An optimal diagnostic policy is one that minimizes the expected total cost, which is the sum of measurement costs and misdiagnosis costs. In most diagnostic settings, there is a tradeoff between these two kinds of costs. This paper formalizes diagnostic decision making as a Markov Decision Process (MDP). The paper introduces a new family of systematic search algorithms based on the AO* algorithm to solve this MDP. To make AO* efficient, the paper describes an admissible heuristic that enables AO* to prune large parts of the search space. The paper also introduces several greedy algorithms including some improvements over previously-published methods. The paper then addresses the question of learning diagnostic policies from examples. When the probabilities of diseases and test results are computed from training data, there is a great danger of overfitting. To reduce overfitting, regularizers are integrated into the search algorithms. Finally, the paper compares the proposed methods on five benchmark diagnostic data sets. The studies show that in most cases the systematic search methods produce better diagnostic policies than the greedy methods. In addition, the studies show that for training sets of realistic size, the systematic search algorithms are practical on todays desktop computers.

Stochastic processes that involve the creation of objects and relations over time are widespread, but relatively poorly studied. For example, accurate fault diagnosis in factory assembly processes requires inferring the probabilities of erroneous assembly operations, but doing this efficiently and accurately is difficult. Modeled as dynamic Bayesian networks, these processes have discrete variables with very large domains and extremely high dimensionality. In this paper, we introduce relational dynamic Bayesian networks (RDBNs), which are an extension of dynamic Bayesian networks (DBNs) to first-order logic. RDBNs are a generalization of dynamic probabilistic relational models (DPRMs), which we had proposed in our previous work to model dynamic uncertain domains. We first extend the Rao-Blackwellised particle filtering described in our earlier work to RDBNs. Next, we lift the assumptions associated with Rao-Blackwellization in RDBNs and propose two new forms of particle filtering. The first one uses abstraction hierarchies over the predicates to smooth the particle filters estimates. The second employs kernel density estimation with a kernel function specifically designed for relational domains. Experiments show these two methods greatly outperform standard particle filtering on the task of assembly plan execution monitoring.

We study linear models under heavy-tailed priors from a probabilistic viewpoint. Instead of computing a single sparse most probable (MAP) solution as in standard deterministic approaches, the focus in the Bayesian compressed sensing framework shifts towards capturing the full posterior distribution on the latent variables, which allows quantifying the estimation uncertainty and learning model parameters using maximum likelihood. The exact posterior distribution under the sparse linear model is intractable and we concentrate on variational Bayesian techniques to approximate it. Repeatedly computing Gaussian variances turns out to be a key requisite and constitutes the main computational bottleneck in applying variational techniques in large-scale problems. We leverage on the recently proposed Perturb-and-MAP algorithm for drawing exact samples from Gaussian Markov random fields (GMRF). The main technical contribution of our paper is to show that estimating Gaussian variances using a relatively small number of such efficiently drawn random samples is much more effective than alternative general-purpose variance estimation techniques. By reducing the problem of variance estimation to standard optimization primitives, the resulting variational algorithms are fully scalable and parallelizable, allowing Bayesian computations in extremely large-scale problems with the same memory and time complexity requirements as conventional point estimation techniques. We illustrate these ideas with experiments in image deblurring.

In many applications, such as economics, operations research and reinforcement learning, one often needs to estimate a multivariate regression function f subject to a convexity constraint. For example, in sequential decision processes the value of a state under optimal subsequent decisions may be known to be convex or concave. We propose a new Bayesian nonparametric multivariate approach based on characterizing the unknown regression function as the max of a random collection of unknown hyperplanes. This specification induces a prior with large support in a Kullback-Leibler sense on the space of convex functions, while also leading to strong posterior consistency. Although we assume that f is defined over R^p, we show that this model has a convergence rate of log(n)^{-1} n^{-1/(d+2)} under the empirical L2 norm when f actually maps a d dimensional linear subspace to R. We design an efficient reversible jump MCMC algorithm for posterior computation and demonstrate the methods through application to value function approximation.