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Large Scale Distributed Sparse Precision Estimation
"... We consider the problem of sparse precision matrix estimation in high dimensions using the CLIME estimator, which has several desirable theoretical properties. We present an inexact alternating direction method of multiplier (ADMM) algorithm for CLIME, and establish rates of convergence for both the ..."
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We consider the problem of sparse precision matrix estimation in high dimensions using the CLIME estimator, which has several desirable theoretical properties. We present an inexact alternating direction method of multiplier (ADMM) algorithm for CLIME, and establish rates of convergence for both the objective and optimality conditions. Further, we develop a large scale distributed framework for the computations, which scales to millions of dimensions and trillions of parameters, using hundreds of cores. The proposed framework solves CLIME in columnblocks and only involves elementwise operations and parallel matrix multiplications. We evaluate our algorithm on both sharedmemory and distributedmemory architectures, which can use block cyclic distribution of data and parameters to achieve load balance and improve the efficiency in the use of memory hierarchies. Experimental results show that our algorithm is substantially more scalable than stateoftheart methods and scales almost linearly with the number of cores. 1
Gaussian Copula Precision Estimation with Missing
 Values, International Conference on Artificial Intelligence and Statistics
, 2014
"... We consider the problem of estimating sparse precision matrix of Gaussian copula distributions using samples with missing values in high dimensions. Existing approaches, primarily designed for Gaussian distributions, suggest using plugin estimators by disregarding the missing values. In this paper ..."
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We consider the problem of estimating sparse precision matrix of Gaussian copula distributions using samples with missing values in high dimensions. Existing approaches, primarily designed for Gaussian distributions, suggest using plugin estimators by disregarding the missing values. In this paper, we propose double plugin Gaussian (DoPinG) copula estimators to estimate the sparse precision matrix corresponding to nonparanormal distributions. DoPinG uses two plugin procedures and consists of three steps: (1) estimate nonparametric correlations based on observed values, including Kendall’s tau and Spearman’s rho; (2) estimate the nonparanormal correlation matrix; (3) plug into existing sparse precision estimators. We prove that DoPinG copula estimators consistently estimate the nonparanormal correlation matrix at a rate of O ( 1(1−δ) log p n), where δ is the probability of missing values. We provide experimental results to illustrate the effect of sample size and percentage of missing data on the model performance. Experimental results show that DoPinG is significantly better than estimators like mGlasso, which are primarily designed for Gaussian data.
Nonconvex Statistical Optimization for Sparse Tensor Graphical Model
, 2015
"... We consider the estimation of sparse graphical models that characterize the dependency structure of highdimensional tensorvalued data. To facilitate the estimation of the precision matrix corresponding to each way of the tensor, we assume the data follow a tensor normal distribution whose covarian ..."
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We consider the estimation of sparse graphical models that characterize the dependency structure of highdimensional tensorvalued data. To facilitate the estimation of the precision matrix corresponding to each way of the tensor, we assume the data follow a tensor normal distribution whose covariance has a Kronecker product structure. The penalized maximum likelihood estimation of this model involves minimizing a nonconvex objective function. In spite of the nonconvexity of this estimation problem, we prove that an alternating minimization algorithm, which iteratively estimates each sparse precision matrix while fixing the others, attains an estimator with the optimal statistical rate of convergence as well as consistent graph recovery. Notably, such an estimator achieves estimation consistency with only one tensor sample, which is unobserved in previous work. Our theoretical results are backed by thorough numerical studies.
Geometrizing Local Rates of Convergence for HighDimensional Linear Inverse Problems
"... This paper presents a unified theoretical framework for the analysis of a general illposed linear inverse model which includes as special cases noisy compressed sensing, sign vector recovery, trace regression, orthogonal matrix estimation, and noisy matrix completion. We propose a computationally f ..."
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This paper presents a unified theoretical framework for the analysis of a general illposed linear inverse model which includes as special cases noisy compressed sensing, sign vector recovery, trace regression, orthogonal matrix estimation, and noisy matrix completion. We propose a computationally feasible convex program for the linear inverse problem and develop a theoretical framework to characterize the local rate of convergence. The unified theory is built based on the local conic geometry and duality. The difficulty of estimation is captured by the geometric characterization of the local tangent cone through the complexity measures – the Gaussian width and covering entropy. 1
Joint Estimation of Multiple Highdimensional Precision Matrices *
"... Abstract Motivated by the analysis of gene expression data measured in different tissues or disease states, we consider joint estimation of multiple precision matrices to effectively utilize the partially shared graphical structures of the corresponding graphs. The procedure is based on a weighted ..."
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Abstract Motivated by the analysis of gene expression data measured in different tissues or disease states, we consider joint estimation of multiple precision matrices to effectively utilize the partially shared graphical structures of the corresponding graphs. The procedure is based on a weighted constrained ∞ / 1 minimization approach, which can be effectively implemented by a secondorder cone programming. Both theoretical and numerical properties of the procedure are investigated. It is shown that the proposed joint estimation procedure leads to a faster convergence rate than estimating the precision matrices individually under various losses. The supports of the precision matrices can also be recovered after an additional thresholding step. Under regularity conditions, the proposed procedure leads to an exact graph structure recovery with probability tending to 1. The method is illustrated through an analysis of an ovarian cancer gene expression data. The results indicate that the patients of the poor prognostic subtype lack some important links between the genes of the apoptosis pathway.
Sure Screening for Gaussian Graphical Models
, 2014
"... We propose graphical sure screening, or GRASS, a very simple and computationallyefficient screening procedure for recovering the structure of a Gaussian graphical model in the highdimensional setting. The GRASS estimate of the conditional dependence graph is obtained by thresholding the elements ..."
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We propose graphical sure screening, or GRASS, a very simple and computationallyefficient screening procedure for recovering the structure of a Gaussian graphical model in the highdimensional setting. The GRASS estimate of the conditional dependence graph is obtained by thresholding the elements of the sample covariance matrix. The proposed approach possesses the sure screening property: with very high probability, the GRASS estimated edge set contains the true edge set. Furthermore, with high probability, the size of the estimated edge set is controlled. We provide a choice of threshold for GRASS that can control the expected false positive rate. We illustrate the performance of GRASS in a simulation study and on a gene expression data set, and show that in practice it performs quite competitively with more complex and computationallydemanding techniques for graph estimation. 1
3 Volume Ratio, Sparsity, and Minimaxity under Unitarily Invariant Norms
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