Gaussian Graphical Model exploration and selection in high dimension low sample size setting
CMAP, CNRS, École polytechnique（应用数学中心、法国国家科学研究中心、巴黎综合理工学院）；Aramis project-team, Inria（Aramis项目组、法国国家信息与自动化研究所）；Institut du Cerveau et de la Moelle épinière, Sorbonne University, Inserm U1127, CNRS UMR 7225（脑与脊髓研究所、索邦大学、法国国家健康与医学研究院U1127、法国国家科学研究中心UMR 7225）；Hôpital Européen Georges-Pompidou AP-HP（乔治·蓬皮杜欧洲医院（巴黎公立医院集团））；Centre de Recherche des Cordeliers, Université de Paris, INSERM, Sorbonne Université（科德利耶研究中心、巴黎大学、法国国家健康与医学研究院、索邦大学）
Thomas Lartigue, Simona Bottani, Stephanie Baron, Olivier Colliot, Stanley Durrleman, Stéphanie Allassonnière
Gaussian Graphical Models (GGM) are often used to describe the conditional correlations between the components of a random vector. In this article, we compare two families of GGM inference methods: nodewise edge selection and penalised likelihood maximisation. We demonstrate on synthetic data that, when the sample size is small, the two methods produce graphs with either too few or too many edges when compared to the real one. As a result, we propose a composite procedure that explores a family of graphs with an nodewise numerical scheme and selects a candidate among them with an overall likelihood criterion. We demonstrate that, when the number of observations is small, this selection method yields graphs closer to the truth and corresponding to distributions with better KL divergence with regards to the real distribution than the other two. Finally, we show the interest of our algorithm on two concrete cases: first on brain imaging data, then on biological nephrology data. In both cases our results are more in line with current knowledge in each field.
We characterize the sample size required for accurate graphical model selection from non-stationary samples. The observed data is modeled as a vector-valued zero-mean Gaussian random process whose samples are uncorrelated but have different covariance matrices. This model contains as special cases the standard setting of i.i.d. samples as well as the case of samples forming a stationary or underspread (non-stationary) processes. More generally, our model applies to any process model for which an efficient decorrelation can be obtained. By analyzing a particular model selection method, we deriv…
Accurate model selection is a fundamental requirement for statistical analysis. In many real-world applications of graphical modelling, correct model structure identification is the ultimate objective. Standard model validation procedures such as information theoretic scores and cross validation have demonstrated poor performance in the high dimensional setting. Specialised methods such as EBIC, StARS and RIC have been developed for the explicit purpose of high-dimensional Gaussian graphical model selection. We present a novel model score criterion, Graphical Neighbour Information. This method…
University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
This paper addresses the problem of neighborhood selection for Gaussian graphical models. We present two heuristic algorithms: a forward-backward greedy algorithm for general Gaussian graphical models based on mutual information test, and a threshold-based algorithm for walk summable Gaussian graphical models. Both algorithms are shown to be structurally consistent, and efficient. Numerical results show that both algorithms work very well.
Local approximations are popular methods to scale Gaussian processes (GPs) to big data. Local approximations reduce time complexity by dividing the original dataset into subsets and training a local expert on each subset. Aggregating the experts' prediction is done assuming either conditional dependence or independence between the experts. Imposing the \emph{conditional independence assumption} (CI) between the experts renders the aggregation of different expert predictions time efficient at the cost of poor uncertainty quantification. On the other hand, modeling dependent experts can provide …
Discussion of "Latent variable graphical model selection via convex optimization" by Venkat Chandrasekaran, Pablo A. Parrilo and Alan S. Willsky [arXiv:1008.1290].
Discussion of "Latent variable graphical model selection via convex optimization" by Venkat Chandrasekaran, Pablo A. Parrilo and Alan S. Willsky [arXiv:1008.1290].
Discussion of "Latent variable graphical model selection via convex optimization" by Venkat Chandrasekaran, Pablo A. Parrilo and Alan S. Willsky [arXiv:1008.1290].
Discussion of "Latent variable graphical model selection via convex optimization" by Venkat Chandrasekaran, Pablo A. Parrilo and Alan S. Willsky [arXiv:1008.1290].
Università Cattolica del Sacro Cuore（圣心天主教大学）；Università degli Studi di Torino（都灵大学）
This paper proposes a new algorithm for an automatic variable selection procedure in High Dimensional Graphical Models. The algorithm selects the relevant variables for the node of interest on the basis of mutual information. Several contributions in literature have investigated the use of mutual information in selecting the appropriate number of relevant features in a large data-set, but most of them have focused on binary outcomes or required high computational effort. The algorithm here proposed overcomes these drawbacks as it is an extension of Chow and Liu's algorithm. Once, the probabili…
California Institute of Technology（加州理工学院）；Massachusetts Institute of Technology（麻省理工学院）
Rejoinder to "Latent variable graphical model selection via convex optimization" by Venkat Chandrasekaran, Pablo A. Parrilo and Alan S. Willsky [arXiv:1008.1290].
We consider the problem of covariance matrix estimation in the presence of latent variables. Under suitable conditions, it is possible to learn the marginal covariance matrix of the observed variables via a tractable convex program, where the concentration matrix of the observed variables is decomposed into a sparse matrix (representing the graphical structure of the observed variables) and a low rank matrix (representing the marginalization effect of latent variables). We present an efficient first-order method based on split Bregman to solve the convex problem. The algorithm is guaranteed to…
John A. Paulson School of Engineering and Applied Sciences, Harvard University（哈佛大学约翰·A·保尔森工程与应用科学学院）；Department of Electrical and Computer Engineering, Duke University（杜克大学电气与计算机工程系）
We consider the problem of model selection in Gaussian Markov fields in the sample deficient scenario. In many practically important cases, the underlying networks are embedded into Euclidean spaces. Using the natural geometric structure, we introduce the notion of spatially stationary distributions over geometric graphs. This directly generalizes the notion of stationary time series to the multidimensional setting lacking time axis. We show that the idea of spatial stationarity leads to a dramatic decrease in the sample complexity of the model selection compared to abstract graphs with the sa…
This thesis studies two problems in modern statistics. First, we study selective inference, or inference for hypothesis that are chosen after looking at the data. The motiving application is inference for regression coefficients selected by the lasso. We present the Condition-on-Selection method that allows for valid selective inference, and study its application to the lasso, and several other selection algorithms. In the second part, we consider the problem of learning the structure of a pairwise graphical model over continuous and discrete variables. We present a new pairwise model for grap…
Rice University（莱斯大学）；Baylor College of Medicine（贝勒医学院）
In neuroscience, researchers seek to uncover the connectivity of neurons from large-scale neural recordings or imaging; often people employ graphical model selection and estimation techniques for this purpose. But, existing technologies can only record from a small subset of neurons leading to a challenging problem of graph selection in the presence of extensive latent variables. Chandrasekaran et al. (2012) proposed a convex program to address this problem that poses challenges from both a computational and statistical perspective. To solve this problem, we propose an incredibly simple soluti…
Yale School of Public Health（耶鲁大学公共卫生学院）；University of Washington（华盛顿大学）；Ohio State University（俄亥俄州立大学）
In this article, we propose a new class of priors for Bayesian inference with multiple Gaussian graphical models. We introduce fully Bayesian treatments of two popular procedures, the group graphical lasso and the fused graphical lasso, and extend them to a continuous spike-and-slab framework to allow self-adaptive shrinkage and model selection simultaneously. We develop an EM algorithm that performs fast and dynamic explorations of posterior modes. Our approach selects sparse models efficiently with substantially smaller bias than would be induced by alternative regularization procedures. The…
We examine a variety of graphical models to construct optimal portfolios. Graphical models such as PCA-KMeans, autoencoders, dynamic clustering, and structural learning can capture the time varying patterns in the covariance matrix and allow the creation of an optimal and robust portfolio. We compared the resulting portfolios from the different models with baseline methods. In many cases our graphical strategies generated steadily increasing returns with low risk and outgrew the S&P 500 index. This work suggests that graphical models can effectively learn the temporal dependencies in time seri…
University of Hawaii（夏威夷大学）；John A. Burns School of Medicine（约翰·A·伯恩斯医学院）
A classic application of description length is for model selection with the minimum description length (MDL) principle. The focus of this paper is to extend description length for data analysis beyond simple model selection and sequences of scalars. More specifically, we extend the description length for data analysis in Gaussian graphical models. These are powerful tools to model interactions among variables in a sequence of i.i.d Gaussian data in the form of a graph. Our method uses universal graph coding methods to accurately account for model complexity, and therefore provide a more rigoro…
High-dimensional Functional Graphical Model Structure Learning via Neighborhood Selection Approach
Booth School of Business, The University of Chicago（芝加哥大学布斯商学院）；Department of Statistics and Data Science, Cornell University（康奈尔大学统计与数据科学系）；Department of Data Sciences and Operations, Marshall School of Business, University of Southern California（南加州大学马歇尔商学院数据科学与运营系）
We introduce the model selection problem in pure exploration linear bandits, where the learner needs to adapt to the instance-dependent complexity measure of the smallest hypothesis class containing the true model. We design algorithms in both fixed confidence and fixed budget settings with near instance optimal guarantees. The core of our algorithms is a new optimization problem based on experimental design that leverages the geometry of the action set to identify a near-optimal hypothesis class. Our fixed budget algorithm is developed based on a novel selection-validation procedure, which pr…
Northeastern University（东北大学）；Mitsubishi Electric Research Laboratories (MERL)（三菱电机研究实验室（MERL））
Learning data representations that capture task-related features, but are invariant to nuisance variations remains a key challenge in machine learning. We introduce an automated Bayesian inference framework, called AutoBayes, that explores different graphical models linking classifier, encoder, decoder, estimator and adversarial network blocks to optimize nuisance-invariant machine learning pipelines. AutoBayes also enables learning disentangled representations, where the latent variable is split into multiple pieces to impose various relationships with the nuisance variation and task labels. …
KU Leuven（鲁汶大学）；Leuven Statistics Research Center (LStat)（鲁汶统计研究中心）；Université Paris Descartes（巴黎笛卡尔大学）；CNRS（法国国家科学研究中心）；Université Paris-Sud（巴黎南大学）；INRA（法国国家农业研究院）
Gaussian graphical models are widely utilized to infer and visualize networks of dependencies between continuous variables. However, inferring the graph is difficult when the sample size is small compared to the number of variables. To reduce the number of parameters to estimate in the model, we propose a non-asymptotic model selection procedure supported by strong theoretical guarantees based on an oracle inequality and a minimax lower bound. The covariance matrix of the model is approximated by a block-diagonal matrix. The structure of this matrix is detected by thresholding the sample covar…
Center for Pervasive Communications and Computing（普适通信与计算中心）；University of California, Irvine（加利福尼亚大学欧文分校）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Laboratory for Information and Decision Systems（信息与决策系统实验室）；Massachusetts Institute of Technology（麻省理工学院）
We consider the problem of high-dimensional Gaussian graphical model selection. We identify a set of graphs for which an efficient estimation algorithm exists, and this algorithm is based on thresholding of empirical conditional covariances. Under a set of transparent conditions, we establish structural consistency (or sparsistency) for the proposed algorithm, when the number of samples n=omega(J_{min}^{-2} log p), where p is the number of variables and J_{min} is the minimum (absolute) edge potential of the graphical model. The sufficient conditions for sparsistency are based on the notion of…
Lawrence Berkeley National Lab（劳伦斯伯克利国家实验室）；Stanford University（斯坦福大学）；University of Florida（佛罗里达大学）
Sparse high dimensional graphical model selection is a popular topic in contemporary machine learning. To this end, various useful approaches have been proposed in the context of $\ell_1$-penalized estimation in the Gaussian framework. Though many of these inverse covariance estimation approaches are demonstrably scalable and have leveraged recent advances in convex optimization, they still depend on the Gaussian functional form. To address this gap, a convex pseudo-likelihood based partial correlation graph estimation method (CONCORD) has been recently proposed. This method uses coordinate-wi…
University of California, Santa Barbara（加州大学圣巴巴拉分校）
Building on a recent framework for distributionally robust optimization, we consider estimation of the inverse covariance matrix for multivariate data. We provide a novel notion of a Wasserstein ambiguity set specifically tailored to this estimation problem, leading to a tractable class of regularized estimators. Special cases include penalized likelihood estimators for Gaussian data, specifically the graphical lasso estimator. As a consequence of this formulation, the radius of the Wasserstein ambiguity set is directly related to the regularization parameter in the estimation problem. Using t…
The problem of learning the structure of a high dimensional graphical model from data has received considerable attention in recent years. In many applications such as sensor networks and proteomics it is often expensive to obtain samples from all the variables involved simultaneously. For instance, this might involve the synchronization of a large number of sensors or the tagging of a large number of proteins. To address this important issue, we initiate the study of a novel graphical model selection problem, where the goal is to optimize the total number of scalar samples obtained by allowin…
Graphical Gaussian models have proven to be useful tools for exploring network structures based on multivariate data. Applications to studies of gene expression have generated substantial interest in these models, and resulting recent progress includes the development of fitting methodology involving penalization of the likelihood function. In this paper we advocate the use of the multivariate t and related distributions for more robust inference of graphs. In particular, we demonstrate that penalized likelihood inference combined with an application of the EM algorithm provides a simple and c…
This paper considers the problem of networks reconstruction from heterogeneous data using a Gaussian Graphical Mixture Model (GGMM). It is well known that parameter estimation in this context is challenging due to large numbers of variables coupled with the degeneracy of the likelihood. We propose as a solution a penalized maximum likelihood technique by imposing an $l_{1}$ penalty on the precision matrix. Our approach shrinks the parameters thereby resulting in better identifiability and variable selection. We use the Expectation Maximization (EM) algorithm which involves the graphical LASSO …
We propose an AdaPtive Noise Augmentation (PANDA) technique to regularize the estimation and construction of undirected graphical models. PANDA iteratively optimizes the objective function given the noise augmented data until convergence to achieve regularization on model parameters. The augmented noises can be designed to achieve various regularization effects on graph estimation, such as the bridge (including lasso and ridge), elastic net, adaptive lasso, and SCAD penalization; it also realizes the group lasso and fused ridge. We examine the tail bound of the noise-augmented loss function an…
This paper presents a new algorithm for automatic variables selection. In particular, using the Graphical Models properties it is possible to develop a method that can be used in the contest of large dataset. The advantage of this algorithm is that can be combined with different forecasting models. In this research we have used the OLS method and we have compared the result with the LASSO method.
École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院 (EPFL)）
We consider the problem of estimating the underlying graph associated with a Markov random field, with the added twist that the decoding algorithm can iteratively choose which subsets of nodes to sample based on the previous samples, resulting in an active learning setting. Considering both Ising and Gaussian models, we provide algorithm-independent lower bounds for high-probability recovery within the class of degree-bounded graphs. Our main results are minimax lower bounds for the active setting that match the best known lower bounds for the passive setting, which in turn are known to be tig…
We consider a scenario where an agent has multiple available strategies to explore an unknown environment. For each new interaction with the environment, the agent must select which exploration strategy to use. We provide a new strategy-agnostic method that treat the situation as a Multi-Armed Bandits problem where the reward signal is the diversity of effects that each strategy produces. We test the method empirically on a simulated planar robotic arm, and establish that the method is both able discriminate between strategies of dissimilar quality, even when the differences are tenuous, and t…
Arizona State University（亚利桑那州立大学）；ByteDance（字节跳动）
Feature selection has been an essential step in developing industry-scale deep Click-Through Rate (CTR) prediction systems. The goal of neural feature selection (NFS) is to choose a relatively small subset of features with the best explanatory power as a means to remove redundant features and reduce computational cost. Inspired by gradient-based neural architecture search (NAS) and network pruning methods, people have tackled the NFS problem with Gating approach that inserts a set of differentiable binary gates to drop less informative features. The binary gates are optimized along with the ne…
We propose estimation methods for change points in high-dimensional covariance structures with an emphasis on challenging scenarios with missing values. We advocate three imputation like methods and investigate their implications on common losses used for change point detection. We also discuss how model selection methods have to be adapted to the setting of incomplete data. The methods are compared in a simulation study and applied to a time series from an environmental monitoring system. An implementation of our proposals within the R-package hdcd is available via the Supplementary materials…
Neural networks are suggested for learning a map from $d$-dimensional samples with any underlying dependence structure to multivariate uniformity in $d'$ dimensions. This map, termed DecoupleNet, is used for dependence model assessment and selection. If the data-generating dependence model was known, and if it was among the few analytically tractable ones, one such transformation for $d'=d$ is Rosenblatt's transform. DecoupleNets have multiple advantages. For example, they only require an available sample and are applicable to $d'<d$, in particular $d'=2$. This allows for simpler model assessm…
University of Hawaii（夏威夷大学）；UC Berkeley（加州大学伯克利分校）
The problem of graphical model selection is to correctly estimate the graph structure of a Markov random field given samples from the underlying distribution. We analyze the information-theoretic limitations of the problem of graph selection for binary Markov random fields under high-dimensional scaling, in which the graph size $p$ and the number of edges $k$, and/or the maximal node degree $d$ are allowed to increase to infinity as a function of the sample size $n$. For pairwise binary Markov random fields, we derive both necessary and sufficient conditions for correct graph selection over th…
