Scientific studies increasingly collect multiple modalities of data to investigate a phenomenon from several perspectives. In integrative data analysis it is important to understand how information is heterogeneously spread across these different data sources. To this end, we consider a parametric clustering model for the subjects in a multi-view data set (i.e. multiple sources of data from the same set of subjects) where each view marginally follows a mixture model. In the case of two views, the dependence between them is captured by a cluster membership matrix parameter and we aim to learn the structure of this matrix (e.g. the zero pattern). First, we develop a penalized likelihood approach to estimate the sparsity pattern of the cluster membership matrix. For the specific case of block diagonal structures, we develop a constrained likelihood formulation where this matrix is constrained to be block diagonal up to permutations of the rows and columns. To enforce block diagonal constraints we propose a novel optimization approach based on the symmetric graph Laplacian. We demonstrate the performance of these methods through both simulations and applications to data sets from cancer genetics and neuroscience. Both methods naturally extend to multiple views.
Humans are routinely exposed to mixtures of chemical and other environmental factors, making the quantification of health effects associated with environmental mixtures a critical goal for establishing environmental policy sufficiently protective of human health. The quantification of the effects of exposure to an environmental mixture poses several statistical challenges. It is often the case that exposure to multiple pollutants interact with each other to affect an outcome. Further, the exposure-response relationship between an outcome and some exposures, such as some metals, can exhibit com…
Estimation of the covariance matrix for high-dimensional multivariate datasets is a challenging and important problem in modern statistics. In this paper, we focus on high-dimensional Gaussian DAG models where sparsity is induced on the Cholesky factor L of the inverse covariance matrix. In recent work, ([Cao, Khare, and Ghosh, 2019]), we established high-dimensional sparsity selection consistency for a hierarchical Bayesian DAG model, where an Erdos-Renyi prior is placed on the sparsity pattern in the Cholesky factor L, and a DAG-Wishart prior is placed on the resulting non-zero Cholesky entr…
Choosing the number of mixture components remains an elusive challenge. Model selection criteria can be either overly liberal or conservative and return poorly-separated components of limited practical use. We formalize non-local priors (NLPs) for mixtures and show how they lead to well-separated components with non-negligible weight, interpretable as distinct subpopulations. We also propose an estimator for posterior model probabilities under local and non-local priors, showing that Bayes factors are ratios of posterior to prior empty-cluster probabilities. The estimator is widely applicable …
The sparse structure of the solution for an inverse problem can be modelled using different sparsity enforcing priors when the Bayesian approach is considered. Analytical expression for the unknowns of the model can be obtained by building hierarchical models based on sparsity enforcing distributions expressed via conjugate priors. We consider heavy tailed distributions with this property: the Student-t distribution, which is expressed as a Normal scale mixture, with the mixing distribution the Inverse Gamma distribution, the Laplace distribution, which can also be expressed as a Normal scale …
In probabilistic classification, a discriminative model based on the softmax function has a potential limitation in that it assumes unimodality for each class in the feature space. The mixture model can address this issue, although it leads to an increase in the number of parameters. We propose a sparse classifier based on a discriminative GMM, referred to as a sparse discriminative Gaussian mixture (SDGM). In the SDGM, a GMM-based discriminative model is trained via sparse Bayesian learning. Using this sparse learning framework, we can simultaneously remove redundant Gaussian components and r…
Substantial research on structured sparsity has contributed to analysis of many different applications. However, there have been few Bayesian procedures among this work. Here, we develop a Bayesian model for structured sparsity that uses a Gaussian process (GP) to share parameters of the sparsity-inducing prior in proportion to feature similarity as defined by an arbitrary positive definite kernel. For linear regression, this sparsity-inducing prior on regression coefficients is a relaxation of the canonical spike-and-slab prior that flattens the mixture model into a scale mixture of normals. …
In the present work, we consider variable selection and shrinkage for the Gaussian dynamic linear regression within a Bayesian framework. In particular, we propose a novel method that allows for time-varying sparsity, based on an extension of spike-and-slab priors for dynamic models. This is done by assigning appropriate Markov switching priors for the time-varying coefficients' variances, extending the previous work of Ishwaran and Rao (2005). Furthermore, we investigate different priors, including the common Inverted gamma prior for the process variances, and other mixture prior distribution…
This paper provides a framework for estimating the mean and variance of a high-dimensional normal density. The main setting considered is a fixed number of vector following a high-dimensional normal distribution with unknown mean and diagonal covariance matrix. The diagonal covariance matrix can be known or unknown. If the covariance matrix is unknown, the sample size can be as small as $2$. The proposed estimator is based on the idea that the unobserved pairs of mean and variance for each dimension are drawn from an unknown bivariate distribution, which we model as a mixture of normal-inverse…
Biclustering is used for simultaneous clustering of the observations and variables when there is no group structure known \textit{a priori}. It is being increasingly used in bioinformatics, text analytics, etc. Previously, biclustering has been introduced in a model-based clustering framework by utilizing a structure similar to a mixture of factor analyzers. In such models, observed variables $\mathbf{X}$ are modelled using a latent variable $\mathbf{U}$ that is assumed to be from $N(\mathbf{0}, \mathbf{I})$. Clustering of variables is introduced by imposing constraints on the entries of the f…
We consider the problem of detecting sparse heterogeneous mixtures from a nonparametric perspective, and develop distribution-free tests when all effects have the same sign. Specifically, we assume that the null distribution is symmetric about zero, while the true effects have positive median. We evaluate the precise performance of classical tests for the median (t-test, sign test) and classical tests for symmetry (signed-rank, Smirnov, total number of runs, longest run tests) showing that none of them is asymptotically optimal for the normal mixture model in all sparsity regimes. We then sugg…
Gaussian Mixture Model with unknown diagonal covariances via continuous sparse regularization
Centrale Lyon, CNRS UMR 5208, Institut Camille Jordan（里昂中央理工学院，法国国家科学研究中心联合研究实验室5208，卡米尔·乔丹研究所）；Institut Universitaire de France (IUF)（法兰西学院）；Univ. Claude Bernard, CNRS UMR 5208, Institut Camille Jordan（克洛德·贝尔纳大学，法国国家科学研究中心联合研究实验室5208，卡米尔·乔丹研究所）
We develop a mixture model for transition density approximation, together with soft model selection, in the presence of noisy and heterogeneous nonlinear dynamics. Our model builds on the Gaussian mixture transition distribution (MTD) model for continuous state spaces, extending component means with nonlinear functions that are modeled using Gaussian process (GP) priors. The resulting model flexibly captures nonlinear and heterogeneous lag dependence when several mixture components are active, identifies low-order nonlinear dependence while inferring relevant lags when few components are activ…
We consider nonparametric estimation of a mixed discrete-continuous distribution under anisotropic smoothness conditions and possibly increasing number of support points for the discrete part of the distribution. For these settings, we derive lower bounds on the estimation rates in the total variation distance. Next, we consider a nonparametric mixture of normals model that uses continuous latent variables for the discrete part of the observations. We show that the posterior in this model contracts at rates that are equal to the derived lower bounds up to a log factor. Thus, Bayesian mixture o…
This paper introduces a new data-driven methodology for estimating sparse covariance matrices of the random coefficients in logit mixture models. Researchers typically specify covariance matrices in logit mixture models under one of two extreme assumptions: either an unrestricted full covariance matrix (allowing correlations between all random coefficients), or a restricted diagonal matrix (allowing no correlations at all). Our objective is to find optimal subsets of correlated coefficients for which we estimate covariances. We propose a new estimator, called MISC, that uses a mixed-integer op…
We consider Gaussian mixture models in high dimensions and concentrate on the twin tasks of detection and feature selection. Under sparsity assumptions on the difference in means, we derive information bounds and establish the performance of various procedures, including the top sparse eigenvalue of the sample covariance matrix and other projection tests based on moments, such as the skewness and kurtosis tests of Malkovich and Afifi (1973), and other variants which we were better able to control under the null.
Booth School of Business University of Chicago（芝加哥大学商学院）；Volgenau School of Engineering, George Mason University（乔治·马歇尔大学工程学院）；Department of Systems Engineering and Operations Research George Mason University（乔治·马歇尔大学系统工程与运营管理系）
University of California, San Diego（加州大学圣地亚哥分校）；University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）
Mixture models are widely used to fit complex and multimodal datasets. In this paper we study mixtures with high dimensional sparse latent parameter vectors and consider the problem of support recovery of those vectors. While parameter learning in mixture models is well-studied, the sparsity constraint remains relatively unexplored. Sparsity of parameter vectors is a natural constraint in variety of settings, and support recovery is a major step towards parameter estimation. We provide efficient algorithms for support recovery that have a logarithmic sample complexity dependence on the dimensi…
A Novel Multi-view Mixture Model Framework for Longitudinal Clustering with Application to ANCA-Associated Vasculitis
School of Computer Science and Statistics, Trinity College Dublin（都柏林三一学院计算机科学与统计学院）；ADAPT Center（ADAPT中心）；German Research Centre for Artificial Intelligence (DFKI)（德国人工智能研究中心（DFKI））；Trinity Kidney Centre, Trinity College Dublin（都柏林三一学院肾脏病中心）
Riemannian Gradient Descent for Gaussian Mixture Models with unknown diagonal covariances
Centrale Lyon, INSA Lyon, Université Lyon 1, Université Jean Monnet, CNRS, ICJ UMR5208（里昂中央理工学院、里昂国立应用科学学院、里昂第一大学、让·莫内大学、法国国家科学研究中心、ICJ UMR5208）；Institut Universitaire de France (IUF)（法兰西学院）
Learning a Gaussian Mixture Model (GMM) is hard when the number of parameters is too large given the amount of available data. As a remedy, we propose restricting the GMM to a Gaussian Markov Random Field Mixture Model (GMRF-MM), as well as a new method for estimating the latter's sparse precision (i.e., inverse covariance) matrices. When the sparsity pattern of each matrix is known, we propose an efficient optimization method for the Maximum Likelihood Estimate (MLE) of that matrix. When it is unknown, we utilize the popular Graphical Least Absolute Shrinkage and Selection Operator (GLASSO) t…
In this paper, we propose a generalized scale mixture family of distributions, namely the Power Exponential Scale Mixture (PESM) family, to model the sparsity inducing priors currently in use for sparse signal recovery (SSR). We show that the successful and popular methods such as LASSO, Reweighted $\ell_1$ and Reweighted $\ell_2$ methods can be formulated in an unified manner in a maximum a posteriori (MAP) or Type I Bayesian framework using an appropriate member of the PESM family as the sparsity inducing prior. In addition, exploiting the natural hierarchical framework induced by the PESM f…
Deep neural networks with dependent weights: Gaussian Process mixture limit, heavy tails, sparsity and compressibility
KAIST（韩国科学技术院）；Huawei Technologies（华为技术有限公司）；Fordham University（福特汉姆大学）；Kim Jaechul Graduate School of AI, KAIST（韩国科学技术院金在哲人工智能研究生院）；Institute for Basic Science (IBS)（基础科学研究院）；University of Oxford（牛津大学）
Japan Advanced Institute of Science and Technology（北陆先端科学技术大学院大学）
Inference is an integral part of probabilistic topic models, but is often non-trivial to derive an efficient algorithm for a specific model. It is even much more challenging when we want to find a fast inference algorithm which always yields sparse latent representations of documents. In this article, we introduce a simple framework for inference in probabilistic topic models, denoted by FW. This framework is general and flexible enough to be easily adapted to mixture models. It has a linear convergence rate, offers an easy way to incorporate prior knowledge, and provides us an easy way to dir…
Bi-clustering is a technique that allows for the simultaneous clustering of observations and features in a dataset. This technique is often used in bioinformatics, text mining, and time series analysis. An important advantage of biclustering algorithm is the ability to uncover multiple ``views'' (i.e., through rows and column groupings) in the data. Several Gaussian mixture model based biclustering approach currently exist in the literature. However, they impose severe restrictions on the structure of the covariance matrix. Here, we propose a Gaussian mixture model-based bi-clustering approach…
Duke University, Fuqua School of Business（杜克大学富卡商学院）；Institute for Data, Systems, and Society, MIT（麻省理工学院数据、系统与社会研究所）；Operations Research Center, MIT（麻省理工学院运筹学中心）
We consider a seller offering a large network of $N$ products over a time horizon of $T$ periods. The seller does not know the parameters of the products' linear demand model, and can dynamically adjust product prices to learn the demand model based on sales observations. The seller aims to minimize its pseudo-regret, i.e., the expected revenue loss relative to a clairvoyant who knows the underlying demand model. We consider a sparse set of demand relationships between products to characterize various connectivity properties of the product network. In particular, we study three different spars…
Structured sparsity is an important part of the modern statistical toolkit. We say a set of model parameters has block diagonal sparsity up to permutations if its elements can be viewed as the edges of a graph that has multiple connected components. For example, a block diagonal correlation matrix with K blocks of variables corresponds to a graph with K connected components whose nodes are the variables and whose edges are the correlations. This type of sparsity captures clusters of model parameters. To learn block diagonal sparsity patterns we develop the folded concave Laplacian spectral pen…
We consider the problem of clustering data points in high dimensions, i.e. when the number of data points may be much smaller than the number of dimensions. Specifically, we consider a Gaussian mixture model (GMM) with non-spherical Gaussian components, where the clusters are distinguished by only a few relevant dimensions. The method we propose is a combination of a recent approach for learning parameters of a Gaussian mixture model and sparse linear discriminant analysis (LDA). In addition to cluster assignments, the method returns an estimate of the set of features relevant for clustering. …
This paper is about variable selection, clustering and estimation in an unsupervised high-dimensional setting. Our approach is based on fitting constrained Gaussian mixture models, where we learn the number of clusters $K$ and the set of relevant variables $S$ using a generalized Bayesian posterior with a sparsity inducing prior. We prove a sparsity oracle inequality which shows that this procedure selects the optimal parameters $K$ and $S$. This procedure is implemented using a Metropolis-Hastings algorithm, based on a clustering-oriented greedy proposal, which makes the convergence to the po…
Penalized regression methods, such as $L_1$ regularization, are routinely used in high-dimensional applications, and there is a rich literature on optimality properties under sparsity assumptions. In the Bayesian paradigm, sparsity is routinely induced through two-component mixture priors having a probability mass at zero, but such priors encounter daunting computational problems in high dimensions. This has motivated an amazing variety of continuous shrinkage priors, which can be expressed as global-local scale mixtures of Gaussians, facilitating computation. In sharp contrast to the frequent…
This paper studies sparse density estimation via $\ell_1$ penalization (SPADES). We focus on estimation in high-dimensional mixture models and nonparametric adaptive density estimation. We show, respectively, that SPADES can recover, with high probability, the unknown components of a mixture of probability densities and that it yields minimax adaptive density estimates. These results are based on a general sparsity oracle inequality that the SPADES estimates satisfy. We offer a data driven method for the choice of the tuning parameter used in the construction of SPADES. The method uses the gen…
Penalized regression methods, such as $L_1$ regularization, are routinely used in high-dimensional applications, and there is a rich literature on optimality properties under sparsity assumptions. In the Bayesian paradigm, sparsity is routinely induced through two-component mixture priors having a probability mass at zero, but such priors encounter daunting computational problems in high dimensions. This has motivated an amazing variety of continuous shrinkage priors, which can be expressed as global-local scale mixtures of Gaussians, facilitating computation. In sharp contrast to the correspo…
In this paper we propose a novel framework for the construction of sparsity-inducing priors. In particular, we define such priors as a mixture of exponential power distributions with a generalized inverse Gaussian density (EP-GIG). EP-GIG is a variant of generalized hyperbolic distributions, and the special cases include Gaussian scale mixtures and Laplace scale mixtures. Furthermore, Laplace scale mixtures can subserve a Bayesian framework for sparse learning with nonconvex penalization. The densities of EP-GIG can be explicitly expressed. Moreover, the corresponding posterior distribution al…
Bayesian DAG Structure Learning with Simultaneous Shrinkage Covariance Estimation under Scale-Mixture Error Distributions in the Proportional High-Dimensional Regime
Ferdowsi University of Mashhad（马什哈德 Ferdowsi 大学）；North Carolina Agricultural and Technical State University（北卡罗来纳农业与技术州立大学）
Model selection, via penalized likelihood type criteria, is a standard task in many statistical inference and machine learning problems. Progress has led to deriving criteria with asymptotic consistency results and an increasing emphasis on introducing non-asymptotic criteria. We focus on the problem of modeling non-linear relationships in regression data with potential hidden graph-structured interactions between the high-dimensional predictors, within the mixture of experts modeling framework. In order to deal with such a complex situation, we investigate a block-diagonal localized mixture o…
Multivariate categorical data are routinely collected in many application areas. As the number of cells in the table grows exponentially with the number of variables, many or even most cells will contain zero observations. This severe sparsity motivates appropriate statistical methodologies that effectively reduce the number of free parameters, with penalized log-linear models and latent structure analysis being popular options. This article proposes a fundamentally new class of methods, which we refer to as Mixture of Log Linear models (mills). Combining latent class analysis and log-linear m…
Statistical inference for sparse covariance matrices is crucial to reveal dependence structure of large multivariate data sets, but lacks scalable and theoretically supported Bayesian methods. In this paper, we propose beta-mixture shrinkage prior, computationally more efficient than the spike and slab prior, for sparse covariance matrices and establish its minimax optimality in high-dimensional settings. The proposed prior consists of beta-mixture shrinkage and gamma priors for off-diagonal and diagonal entries, respectively. To ensure positive definiteness of the resulting covariance matrix,…
Given two sets of variables, derived from a common set of samples, sparse Canonical Correlation Analysis (CCA) seeks linear combinations of a small number of variables in each set, such that the induced canonical variables are maximally correlated. Sparse CCA is NP-hard. We propose a novel combinatorial algorithm for sparse diagonal CCA, i.e., sparse CCA under the additional assumption that variables within each set are standardized and uncorrelated. Our algorithm operates on a low rank approximation of the input data and its computational complexity scales linearly with the number of input va…
In this paper, we present several estimators of the diagonal elements of the inverse of the covariance matrix, called precision matrix, of a sample of iid random vectors. The focus is on high dimensional vectors having a sparse precision matrix. It is now well understood that when the underlying distribution is Gaussian, the columns of the precision matrix can be estimated independently form one another by solving linear regression problems under sparsity constraints. This approach leads to a computationally efficient strategy for estimating the precision matrix that starts by estimating the r…
Feature aware covariance estimation, with application to mixtures of chemical exposures
Laboratory for Information and Decision Systems, Massachusetts Institute of Technology（麻省理工学院信息与决策系统实验室）；Nicholas School of the Environment, Duke University（杜克大学尼古拉环境学院）；Department of Statistical Science, Duke University（杜克大学统计科学系）
Normandie Univ（诺曼底大学）；UNICAEN（卡昂大学）；CNRS（法国国家科学研究中心）；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；Inria（法国国家信息与自动化研究所）；Grenoble INP（格勒诺布尔国立理工学院）；La Trobe University（乐卓博大学）；University of Queensland（昆士兰大学）
Mixture of experts (MoE) are a popular class of statistical and machine learning models that have gained attention over the years due to their flexibility and efficiency. In this work, we consider Gaussian-gated localized MoE (GLoME) and block-diagonal covariance localized MoE (BLoME) regression models to present nonlinear relationships in heterogeneous data with potential hidden graph-structured interactions between high-dimensional predictors. These models pose difficult statistical estimation and model selection questions, both from a computational and theoretical perspective. This paper is…
