Learning Gaussian Graphical Models with Observed or Latent FVSs

Gaussian Graphical Models (GGMs) or Gauss Markov random fields are widely used in many applications, and the trade-off between the modeling capacity and the efficiency of learning and inference has been an important research problem. In this paper, we study the family of GGMs with small feedback vertex sets (FVSs), where an FVS is a set of nodes whose removal breaks all the cycles. Exact inference such as computing the marginal distributions and the partition function has complexity $O(k^{2}n)$ using message-passing algorithms, where k is the size of the FVS, and n is the total number of nodes. We propose efficient structure learning algorithms for two cases: 1) All nodes are observed, which is useful in modeling social or flight networks where the FVS nodes often correspond to a small number of high-degree nodes, or hubs, while the rest of the networks is modeled by a tree. Regardless of the maximum degree, without knowing the full graph structure, we can exactly compute the maximum likelihood estimate in $O(kn^2+n^2\log n)$ if the FVS is known or in polynomial time if the FVS is unknown but has bounded size. 2) The FVS nodes are latent variables, where structure learning is equivalent to decomposing a inverse covariance matrix (exactly or approximately) into the sum of a tree-structured matrix and a low-rank matrix. By incorporating efficient inference into the learning steps, we can obtain a learning algorithm using alternating low-rank correction with complexity $O(kn^{2}+n^{2}\log n)$ per iteration. We also perform experiments using both synthetic data as well as real data of flight delays to demonstrate the modeling capacity with FVSs of various sizes

Learning Sparse Gaussian Graphical Model with l0-regularization

For the problem of learning sparse Gaussian graphical models, it is desirable to obtain both sparse structures as well as good parameter estimates. Classical techniques, such as optimizing the l1-regularized maximum likelihood or Chow-Liu algorithm, either focus on parameter estimation or constrain to speci c structure. This paper proposes an alternative that is based on l0-regularized maximum likelihood and employs a greedy algorithm to solve the optimization problem. We show that, when the graph is acyclic, the greedy solution finds the optimal acyclic graph. We also show it can update the parameters in constant time when connecting two sub-components, thus work efficiently on sparse graphs. Empirical results are provided to demonstrate this new algorithm can learn sparse structures with cycles efficiently and that it dominates l1-regularized approach on graph likelihood.ARO MURI grant W911NF-11-1-0391

Quantifying Nonlocal Informativeness in High-Dimensional, Loopy Gaussian Graphical Models

We consider the problem of selecting informative observations in Gaussian graphical models containing both cycles and nuisances. More specifically, we consider the subproblem of quantifying conditional mutual information measures that are nonlocal on such graphs. The ability to efficiently quantify the information content of observations is crucial for resource-constrained data acquisition (adaptive sampling) and data processing (active learning) systems. While closed-form expressions for Gaussian mutual information exist, standard linear algebraic techniques, with complexity cubic in the network size, are intractable for high-dimensional distributions. We investigate the use of embedded trees for computing nonlocal pairwise mutual information and demonstrate through numerical simulations that the presented approach achieves a significant reduction in computational cost over inversion-based methods.United States. Defense Advanced Research Projects Agency (Mathematics of Sensing, Exploitation and Execution

Low Rank Directed Acyclic Graphs and Causal Structure Learning

Despite several important advances in recent years, learning causal structures represented by directed acyclic graphs (DAGs) remains a challenging task in high dimensional settings when the graphs to be learned are not sparse. In particular, the recent formulation of structure learning as a continuous optimization problem proved to have considerable advantages over the traditional combinatorial formulation, but the performance of the resulting algorithms is still wanting when the target graph is relatively large and dense. In this paper we propose a novel approach to mitigate this problem, by exploiting a low rank assumption regarding the (weighted) adjacency matrix of a DAG causal model. We establish several useful results relating interpretable graphical conditions to the low rank assumption, and show how to adapt existing methods for causal structure learning to take advantage of this assumption. We also provide empirical evidence for the utility of our low rank algorithms, especially on graphs that are not sparse. Not only do they outperform state-of-the-art algorithms when the low rank condition is satisfied, the performance on randomly generated scale-free graphs is also very competitive even though the true ranks may not be as low as is assumed

The Econometrics of Bayesian Graphical Models: A Review With Financial Application

Recent advances in empirical finance has shown that the adoption of network theory is critical to understand contagion and systemic vulnerabilities. While interdependencies among financial markets have been widely examined, only few studies review networks, however, they do not focus on the econometrics aspects. This paper presents a state-of-the-art review on the interface between statistics and econometrics in the inference and application of Bayesian graphical models. We specifically highlight the connections and possible applications of network models in financial econometrics, in the context of systemic risk

Distributed Learning, Prediction and Detection in Probabilistic Graphs.

Critical to high-dimensional statistical estimation is to exploit the structure in the data distribution. Probabilistic graphical models provide an efficient framework for representing complex joint distributions of random variables through their conditional dependency graph, and can be adapted to many high-dimensional machine learning applications. This dissertation develops the probabilistic graphical modeling technique for three statistical estimation problems arising in real-world applications: distributed and parallel learning in networks, missing-value prediction in recommender systems, and emerging topic detection in text corpora. The common theme behind all proposed methods is a combination of parsimonious representation of uncertainties in the data, optimization surrogate that leads to computationally efficient algorithms, and fundamental limits of estimation performance in high dimension. More specifically, the dissertation makes the following theoretical contributions: (1) We propose a distributed and parallel framework for learning the parameters in Gaussian graphical models that is free of iterative global message passing. The proposed distributed estimator is shown to be asymptotically consistent, improve with increasing local neighborhood sizes, and have a high-dimensional error rate comparable to that of the centralized maximum likelihood estimator. (2) We present a family of latent variable Gaussian graphical models whose marginal precision matrix has a “low-rank plus sparse” structure. Under mild conditions, we analyze the high-dimensional parameter error bounds for learning this family of models using regularized maximum likelihood estimation. (3) We consider a hypothesis testing framework for detecting emerging topics in topic models, and propose a novel surrogate test statistic for the standard likelihood ratio. By leveraging the theory of empirical processes, we prove asymptotic consistency for the proposed test and provide guarantees of the detection performance.PhDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/110499/1/mengzs_1.pd