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A convex pseudolikelihood framework for high dimensional partial correlation estimation with convergence guarantees

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  • Kshitij Khare
  • Sang-Yun Oh
  • Bala Rajaratnam

Abstract

type="main" xml:id="rssb12088-abs-0001"> Sparse high dimensional graphical model selection is a topic of much interest in modern day statistics. A popular approach is to apply l 1 -penalties to either parametric likelihoods, or regularized regression/pseudolikelihoods, with the latter having the distinct advantage that they do not explicitly assume Gaussianity. As none of the popular methods proposed for solving pseudolikelihood-based objective functions have provable convergence guarantees, it is not clear whether corresponding estimators exist or are even computable, or if they actually yield correct partial correlation graphs. We propose a new pseudolikelihood-based graphical model selection method that aims to overcome some of the shortcomings of current methods, but at the same time retain all their respective strengths. In particular, we introduce a novel framework that leads to a convex formulation of the partial covariance regression graph problem, resulting in an objective function comprised of quadratic forms. The objective is then optimized via a co-ordinatewise approach. The specific functional form of the objective function facilitates rigorous convergence analysis leading to convergence guarantees; an important property that cannot be established by using standard results, when the dimension is larger than the sample size, as is often the case in high dimensional applications. These convergence guarantees ensure that estimators are well defined under very general conditions and are always computable. In addition, the approach yields estimators that have good large sample properties and also respect symmetry. Furthermore, application to simulated and real data, timing comparisons and numerical convergence is demonstrated. We also present a novel unifying framework that places all graphical pseudolikelihood methods as special cases of a more general formulation, leading to important insights.

Suggested Citation

  • Kshitij Khare & Sang-Yun Oh & Bala Rajaratnam, 2015. "A convex pseudolikelihood framework for high dimensional partial correlation estimation with convergence guarantees," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 77(4), pages 803-825, September.
  • Handle: RePEc:bla:jorssb:v:77:y:2015:i:4:p:803-825
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    File URL: http://hdl.handle.net/10.1111/rssb.2015.77.issue-4
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    Citations

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    Cited by:

    1. Wang, Ke & Franks, Alexander & Oh, Sang-Yun, 2023. "Learning Gaussian graphical models with latent confounders," Journal of Multivariate Analysis, Elsevier, vol. 198(C).
    2. Sung, Bongjung & Lee, Jaeyong, 2023. "Covariance structure estimation with Laplace approximation," Journal of Multivariate Analysis, Elsevier, vol. 198(C).
    3. Mr. Jorge A Chan-Lau, 2017. "Variance Decomposition Networks: Potential Pitfalls and a Simple Solution," IMF Working Papers 2017/107, International Monetary Fund.
    4. Byrd, Michael & Nghiem, Linh H. & McGee, Monnie, 2021. "Bayesian regularization of Gaussian graphical models with measurement error," Computational Statistics & Data Analysis, Elsevier, vol. 156(C).
    5. Young-Geun Choi & Seunghwan Lee & Donghyeon Yu, 2022. "An efficient parallel block coordinate descent algorithm for large-scale precision matrix estimation using graphics processing units," Computational Statistics, Springer, vol. 37(1), pages 419-443, March.
    6. Jiaqi Zhang & Xinyan Fan & Yang Li & Shuangge Ma, 2022. "Heterogeneous graphical model for non‐negative and non‐Gaussian PM2.5 data," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 71(5), pages 1303-1329, November.
    7. Seonghun Cho & Shota Katayama & Johan Lim & Young-Geun Choi, 2021. "Positive-definite modification of a covariance matrix by minimizing the matrix $$\ell_{\infty}$$ ℓ ∞ norm with applications to portfolio optimization," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 105(4), pages 601-627, December.
    8. Xingqi Du & Subhashis Ghosal, 2018. "Bayesian Discriminant Analysis Using a High Dimensional Predictor," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 80(1), pages 112-145, December.
    9. Seunghwan Lee & Sang Cheol Kim & Donghyeon Yu, 2023. "An efficient GPU-parallel coordinate descent algorithm for sparse precision matrix estimation via scaled lasso," Computational Statistics, Springer, vol. 38(1), pages 217-242, March.
    10. Hu, Jianhua & Liu, Xiaoqian & Liu, Xu & Xia, Ningning, 2022. "Some aspects of response variable selection and estimation in multivariate linear regression," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    11. Bar, Haim & Wells, Martin T., 2023. "On graphical models and convex geometry," Computational Statistics & Data Analysis, Elsevier, vol. 187(C).
    12. Jingying Yang & Guishu Bai & Mei Yan, 2023. "Minimum Residual Sum of Squares Estimation Method for High-Dimensional Partial Correlation Coefficient," Mathematics, MDPI, vol. 11(20), pages 1-22, October.
    13. Fan, Xinyan & Zhang, Qingzhao & Ma, Shuangge & Fang, Kuangnan, 2021. "Conditional score matching for high-dimensional partial graphical models," Computational Statistics & Data Analysis, Elsevier, vol. 153(C).
    14. Choi, Young-Geun & Lim, Johan & Roy, Anindya & Park, Junyong, 2019. "Fixed support positive-definite modification of covariance matrix estimators via linear shrinkage," Journal of Multivariate Analysis, Elsevier, vol. 171(C), pages 234-249.

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