The Experts below are selected from a list of 321 Experts worldwide ranked by ideXlab platform
Gherardo Varando - One of the best experts on this subject based on the ideXlab platform.
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Sparse Cholesky Covariance Parametrization for Recovering Latent Structure in Ordered Data
IEEE Access, 2020Co-Authors: Irene Córdoba, Concha Bielza, Pedro Larrañaga, Gherardo VarandoAbstract:The sparse Cholesky parametrization of the inverse covariance matrix is directly related to Gaussian Bayesian networks. Its counterpart, the covariance Cholesky Factorization model, has a natural interpretation as a hidden variable model for ordered signal data. Despite this, it has received little attention so far, with few notable exceptions. To fill this gap, in this paper we focus on arbitrary zero patterns in the Cholesky Factor of a covariance matrix. We discuss how these models can also be extended, in analogy with Gaussian Bayesian networks, to data where no apparent order is available. For the ordered scenario, we propose a novel estimation method that is based on matrix loss penalization, as opposed to the existing regression-based approaches. The performance of this sparse model for the Cholesky Factor, together with our novel estimator,is assessed in a simulation setting, as well as over spatial and temporal real data where a natural ordering arises among the variables. We give guidelines, based on the empirical results, about which of the methods analysed is more appropriate for each setting.
David J. Nott - One of the best experts on this subject based on the ideXlab platform.
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Gaussian variational approximation with sparse precision matrices
Statistics and Computing, 2018Co-Authors: David J. NottAbstract:We consider the problem of learning a Gaussian variational approximation to the posterior distribution for a high-dimensional parameter, where we impose sparsity in the precision matrix to reflect appropriate conditional independence structure in the model. Incorporating sparsity in the precision matrix allows the Gaussian variational distribution to be both flexible and parsimonious, and the sparsity is achieved through parameterization in terms of the Cholesky Factor. Efficient stochastic gradient methods that make appropriate use of gradient information for the target distribution are developed for the optimization. We consider alternative estimators of the stochastic gradients, which have lower variation and are more stable. Our approach is illustrated using generalized linear mixed models and state-space models for time series.
Mohsen Pourahmadi - One of the best experts on this subject based on the ideXlab platform.
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Fused-Lasso Regularized Cholesky Factors of Large Nonstationary Covariance Matrices of Longitudinal Data.
arXiv: Machine Learning, 2020Co-Authors: Aramayis Dallakyan, Mohsen PourahmadiAbstract:Smoothness of the subdiagonals of the Cholesky Factor of large covariance matrices is closely related to the degrees of nonstationarity of autoregressive models for time series and longitudinal data. Heuristically, one expects for a nearly stationary covariance matrix the entries in each subdiagonal of the Cholesky Factor of its inverse to be nearly the same in the sense that sum of absolute values of successive terms is small. Statistically such smoothness is achieved by regularizing each subdiagonal using fused-type lasso penalties. We rely on the standard Cholesky Factor as the new parameters within a regularized normal likelihood setup which guarantees: (1) joint convexity of the likelihood function, (2) strict convexity of the likelihood function restricted to each subdiagonal even when $n
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distribution of random correlation matrices hyperspherical parameterization of the Cholesky Factor
Statistics & Probability Letters, 2015Co-Authors: Mohsen Pourahmadi, Xiao WangAbstract:Abstract We study the distribution of random correlation matrices using the hyperspherical parameterization of their Cholesky Factors and the distributions of the related angles. We highlight the roles of this procedure in generating high-dimensional correlation matrices.
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covariance matrix selection and estimation via penalised normal likelihood
Biometrika, 2006Co-Authors: Jianhua Z Huang, Mohsen Pourahmadi, Naiping Liu, Linxu LiuAbstract:SUMMARY We propose a nonparametric method for identifying parsimony and for producing a statistically efficient estimator of a large covariance matrix. We reparameterise a covariance matrix through the modified Cholesky decomposition of its inverse or the one-step-ahead predictive representation of the vector of responses and reduce the nonintuitive task of modelling covariance matrices to the familiar task of model selection and estimation for a sequence of regression models. The Cholesky Factor containing these regression coefficients is likely to have many off-diagonal elements that are zero or close to zero. Penalised normal likelihoods in this situation with L1 and L2 penalities are shown to be closely related to Tibshirani's (1996) LASSO approach and to ridge regression. Adding either penalty to the likelihood helps to produce more stable estimators by introducing shrinkage to the elements in the Cholesky Factor, while, because of its singularity, the L1 penalty will set some elements to zero and produce interpretable models. An algorithm is developed for computing the estimator and selecting the tuning parameter. The proposed maximum penalised likelihood estimator is illustrated using simulation and a real dataset involving estimation of a 102 x 102 covariance matrix.
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covariance matrix selection and estimation via penalised normal likelihood
Biometrika, 2006Co-Authors: Jianhua Z Huang, Mohsen Pourahmadi, Naiping Liu, Linxu LiuAbstract:We propose a nonparametric method for identifying parsimony and for producing a statistically efficient estimator of a large covariance matrix. We reparameterise a covariance matrix through the modified Cholesky decomposition of its inverse or the one-step-ahead predictive representation of the vector of responses and reduce the nonintuitive task of modelling covariance matrices to the familiar task of model selection and estimation for a sequence of regression models. The Cholesky Factor containing these regression coefficients is likely to have many off-diagonal elements that are zero or close to zero. Penalised normal likelihoods in this situation with L-sub-1 and L-sub-2 penalities are shown to be closely related to Tibshirani's (1996) LASSO approach and to ridge regression. Adding either penalty to the likelihood helps to produce more stable estimators by introducing shrinkage to the elements in the Cholesky Factor, while, because of its singularity, the L-sub-1 penalty will set some elements to zero and produce interpretable models. An algorithm is developed for computing the estimator and selecting the tuning parameter. The proposed maximum penalised likelihood estimator is illustrated using simulation and a real dataset involving estimation of a 102 × 102 covariance matrix. Copyright 2006, Oxford University Press.
Martin H Bucker - One of the best experts on this subject based on the ideXlab platform.
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interactively exploring elimination orderings in symbolic sparse Cholesky Factorization
International Conference on Conceptual Structures, 2010Co-Authors: Michael Lulfesmann, Simon Robert Lesenich, Martin H BuckerAbstract:When large sparse symmetric systems of linear equations are solved by the Cholesky Factorization, nonzero elements can be generated at positions where the original matrix contains zero elements. This phenomenon is called fill-in and it is often crucial in large-scale problems. The symbolic Cholesky Factorization solely takes into account the nonzero structure of a sparse matrix to determine the nonzero structure of its Cholesky Factor. Sequences of elimination graphs are typically used to model this combinatorial problem. We propose an interactive educational module to visualize and explore the symbolic Cholesky Factorization in terms of both elimination graphs and matrix representation. We describe the design and implementation of this interactive module that is intended to be used in a face-to-face learning environment.
James F. O'brien - One of the best experts on this subject based on the ideXlab platform.
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Updated sparse Cholesky Factors for corotational elastodynamics
ACM Transactions on Graphics, 2012Co-Authors: Florian Hecht, Yeon J. Lee, Jonathan Richard Shewchuk, James F. O'brienAbstract:We present warp-canceling corotation, a nonlinear finite element formulation for elastodynamic simulation that achieves fast performance by making only partial or delayed changes to the simulation's linearized system matrices. Coupled with an algorithm for incremental updates to a sparse Cholesky Factorization, the method realizes the stability and scalability of a sparse direct method without the need for expensive reFactorization at each time step. This finite element formulation combines the widely used corotational method with stiffness warping so that changes in the per-element rotations are initially approximated by inexpensive per-node rotations. When the errors of this approximation grow too large, the per-element rotations are selectively corrected by updating parts of the matrix chosen according to locally measured errors. These changes to the system matrix are propagated to its Cholesky Factor by incremental updates that are much faster than reFactoring the matrix from scratch. A nested dissection ordering of the system matrix gives rise to a hierarchical Factorization in which changes to the system matrix cause limited, well-structured changes to the Cholesky Factor. We show examples of simulations that demonstrate that the proposed formulation produces results that are visually comparable to those produced by a standard corotational formulation. Because our method requires computing only partial updates of the Cholesky Factor, it is substantially faster than full reFactorization and outperforms widely used iterative methods such as preconditioned conjugate gradients. Our method supports a controlled trade-off between accuracy and speed, and unlike most iterative methods its performance does not slow for stiffer materials but rather it actually improves.