양은호 교수

Undirected graphical models, also known as Markov networks, enjoy popularity in a variety of applications. The popular instances of these models such as Gaus-sian Markov Random Fields (GMRFs), Ising models, and multinomial discrete models, however do not capture the characteristics of data in many settings. We introduce a new class of graphical models based on generalized linear models (GLMs) by assuming that node-wise conditional distributions arise from expo-nential families. Our models allow

exponential family distributions, such as the Poisson, negative binomial, and exponential distributions. Our key contributions include a class of M-estimators to fit these graphical model distributions; and rigorous statistical analysis showing that these M-estimators recover the true graphical model structure exactly, with high probability. We provide examples of genomic and proteomic networks learned via instances of our class of graphical models derived from Poisson and exponential distributi

Markov Random Fields, or undirected graphical models are widely used to model highdimensional multivariate data. Classical instances of these models, such as Gaussian Graphical and Ising Models, as well as recent extensions (Yang et al., 2012) to graphical models specified by univariate exponential families, assume all variables arise from the same distribution. Complex data from high-throughput genomics and social networking for example, often contain discrete, count, and continuous variables m

Undirected graphical models, or Markov networks, are a popular class of statistical models, used in a wide variety of applications. Popular instances of this class include Gaussian graphical models and Ising models. In many settings, however, it might not be clear which subclass of graphical models to use, particularly for non-Gaussian and non-categorical data. In this paper, we consider a general sub-class of graphical models where the node-wise conditional distributions arise from exponential

Undirected graphical models, such as Gaussian graphical models, Ising, and multinomial/categorical graphical models, are widely used in a variety of applications for modeling distributions over a large number of variables. These standard instances, however, are ill-suited to modeling count data, which are increasingly ubiquitous in big-data settings such as genomic sequencing data, user-ratings data, spatial incidence data, climate studies, and site visits. Existing classes of Poisson graphical

We provide a unified framework for the high-dimensional analysis of superposition-structured or dirty statistical models: where the model parameters are a superposition of structurally constrained parameters. We allow for any number and types of structures, and any statistical model. We consider the general class of M-estimators that minimize the sum of any loss function, and an instance of what we call a hybrid regularization, that is the infimal convolution of weighted regularization functions

We consider the problem of robustifying high-dimensional structured estimation. Robust techniques are key in real-world applications which often involve outliers and data corruption. We focus on trimmed versions of structurally regularized M-estimators in the high-dimensional setting, including the popular Least Trimmed Squares estimator, as well as analogous estimators for generalized linear models and graphical models, using convex and non-convex loss functions. We present a general analysis o

We propose a class of closed-form estimators for sparsity-structured graphical models, expressed as exponential family distributions, under high-dimensional settings. Our approach builds on observing the precise manner in which the classi-cal graphical model MLE “breaks down ” under high-dimensional settings. Our es-timator uses a carefully constructed, well-defined and closed-form backward map, and then performs thresholding operations to ensure the desired sparsity structure. We provide a rigo

We consider the problem of structurally con-strained high-dimensional linear regression. This has attracted considerable attention over the last decade, with state of the art statistical estimators based on solving regularized convex programs. While these typically non-smooth convex pro-grams can be solved by the state of the art op-timization methods in polynomial time, scaling them to very large-scale problems is an ongoing and rich area of research. In this paper, we at-tempt to address this

We consider the problem of estimating expecta-tions of vector-valued feature functions; a spe-cial case of which includes estimating the co-variance matrix of a random vector. We are in-terested in recovery under high-dimensional set-tings, where the number of features p is poten-tially larger than the number of samples n, and where we need to impose structural constraints. In a natural distributional setting for this prob-lem, the feature functions comprise the sufficient statistics of an expon

Gaussian Graphical Models (GGMs) are popular tools for studying network structures. However, many modern applications such as gene network discovery and social interactions analysis often involve high-dimensional noisy data with outliers or heavier tails than the Gaussian distribution. In this paper, we propose the Trimmed Graphical Lasso for robust estimation of sparse GGMs. Our method guards against outliers by an implicit trimming mechanism akin to the popular Least Trimmed Squares method use

"Mixed Data" comprising a large number of heterogeneous variables (e.g. count, binary, continuous, skewed continuous, among other data types) are prevalent in varied areas such as genomics and proteomics, imaging genetics, national security, social networking, and Internet advertising. There have been limited efforts at statistically modeling such mixed data jointly, in part because of the lack of computationally amenable multivariate distributions that can capture direct dependencies between su

We study the general class of estimators for graphical model structure based on optimizing l1-regularized approximate log-likelihood, where the approximate likelihood uses tractable variational approximations of the partition function. We provide a message-passing algorithm that directly computes the l1 regularized approximate MLE. Further, in the case of certain reweighted entropy approximations to the partition function, we show that surprisingly the l1 regularized approximate MLE estimator ha

Conditional random fields, which model the distribution of a multivariate response conditioned on a set of covariates using undirected graphs, are widely used in a variety of multivariate prediction applications. Popular instances of this class of models, such as categorical-discrete CRFs, Ising CRFs, and conditional Gaussian based CRFs, are not well suited to the varied types of response variables in many applications, including count-valued responses. We thus introduce a novel subclass of CRFs

We study robust high-dimensional estimation of generalized linear models (GLMs); where a small number k of the n observations can be arbitrarily corrupted, and where the true parameter is high di-mensional in the “p n ” regime, but only has a small number s of non-zero entries. There has been some recent work connecting robustness and sparsity, in the context of linear regression with cor-rupted observations, by using an explicitly mod-eled outlier response vector that is assumed to be sparse. I