The Annals of Statistics

Innovated scalable efficient estimation in ultra-large Gaussian graphical models

Yingying Fan and Jinchi Lv

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Large-scale precision matrix estimation is of fundamental importance yet challenging in many contemporary applications for recovering Gaussian graphical models. In this paper, we suggest a new approach of innovated scalable efficient estimation (ISEE) for estimating large precision matrix. Motivated by the innovated transformation, we convert the original problem into that of large covariance matrix estimation. The suggested method combines the strengths of recent advances in high-dimensional sparse modeling and large covariance matrix estimation. Compared to existing approaches, our method is scalable and can deal with much larger precision matrices with simple tuning. Under mild regularity conditions, we establish that this procedure can recover the underlying graphical structure with significant probability and provide efficient estimation of link strengths. Both computational and theoretical advantages of the procedure are evidenced through simulation and real data examples.

Article information

Ann. Statist., Volume 44, Number 5 (2016), 2098-2126.

Received: May 2015
Revised: November 2015
First available in Project Euclid: 12 September 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H12: Estimation 62F12: Asymptotic properties of estimators
Secondary: 62J05: Linear regression

Gaussian graphical model precision matrix big data scalability efficiency sparsity


Fan, Yingying; Lv, Jinchi. Innovated scalable efficient estimation in ultra-large Gaussian graphical models. Ann. Statist. 44 (2016), no. 5, 2098--2126. doi:10.1214/15-AOS1416.

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  • [1] Benjamini, Y. and Hochberg, Y. (1995). Controlling the false discovery rate: A practical and powerful approach to multiple testing. J. Roy. Statist. Soc. Ser. B 57 289–300.
  • [2] Bickel, P. J. and Levina, E. (2008). Regularized estimation of large covariance matrices. Ann. Statist. 36 199–227.
  • [3] Bickel, P. J. and Levina, E. (2008). Covariance regularization by thresholding. Ann. Statist. 36 2577–2604.
  • [4] Cai, T. and Liu, W. (2011). Adaptive thresholding for sparse covariance matrix estimation. J. Amer. Statist. Assoc. 106 672–684.
  • [5] Cai, T., Liu, W. and Luo, X. (2011). A constrained $\ell_{1}$ minimization approach to sparse precision matrix estimation. J. Amer. Statist. Assoc. 106 594–607.
  • [6] Cai, T. T., Liu, W. and Zhou, H. H. (2014). Estimating sparse precision matrix: Optimal rates of convergence and adaptive estimation. Ann. Statist. To appear.
  • [7] Cai, T. T. and Yuan, M. (2012). Adaptive covariance matrix estimation through block thresholding. Ann. Statist. 40 2014–2042.
  • [8] Candes, E. and Tao, T. (2007). The Dantzig selector: Statistical estimation when $p$ is much larger than $n$. Ann. Statist. 35 2313–2351.
  • [9] Dempster, A. (1972). Covariance selection. Biometrics 28 157–175.
  • [10] Efron, B. (1979). Bootstrap methods: Another look at the jackknife. Ann. Statist. 7 1–26.
  • [11] Fan, J. and Fan, Y. (2008). High-dimensional classification using features annealed independence rules. Ann. Statist. 36 2605–2637.
  • [12] Fan, J., Fan, Y. and Lv, J. (2008). High dimensional covariance matrix estimation using a factor model. J. Econometrics 147 186–197.
  • [13] Fan, J., Feng, Y. and Wu, Y. (2009). Network exploration via the adaptive lasso and SCAD penalties. Ann. Appl. Stat. 3 521–541.
  • [14] Fan, J., Han, X. and Gu, W. (2012). Estimating false discovery proportion under arbitrary covariance dependence. J. Amer. Statist. Assoc. 107 1019–1035.
  • [15] Fan, J. and Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. J. Amer. Statist. Assoc. 96 1348–1360.
  • [16] Fan, J. and Lv, J. (2008). Sure independence screening for ultrahigh dimensional feature space. J. R. Stat. Soc. Ser. B. Stat. Methodol. 70 849–911.
  • [17] Fan, J. and Peng, H. (2004). Nonconcave penalized likelihood with a diverging number of parameters. Ann. Statist. 32 928–961.
  • [18] Fan, Y., Jin, J. and Yao, Z. (2013). Optimal classification in sparse Gaussian graphic model. Ann. Statist. 41 2537–2571.
  • [19] Fan, Y., Kong, Y., Li, D. and Zheng, Z. (2015). Innovated interaction screening for high-dimensional nonlinear classification. Ann. Statist. 43 1243–1272.
  • [20] Fan, Y. and Lv, J. (2013). Asymptotic equivalence of regularization methods in thresholded parameter space. J. Amer. Statist. Assoc. 108 1044–1061.
  • [21] Fan, Y. and Lv, J. (2016). Supplement to “Innovated scalable efficient estimation in ultra-large Gaussian graphical models.” DOI:10.1214/15-AOS1416SUPP.
  • [22] Friedman, J., Hastie, T. and Tibshirani, R. (2008). Sparse inverse covariance estimation with the graphical lasso. Biostatistics 9 432–441.
  • [23] Hall, P. and Jin, J. (2010). Innovated higher criticism for detecting sparse signals in correlated noise. Ann. Statist. 38 1686–1732.
  • [24] Hall, P. and Li, K.-C. (1993). On almost linearity of low-dimensional projections from high-dimensional data. Ann. Statist. 21 867–889.
  • [25] Hall, P., Titterington, D. M. and Xue, J.-H. (2009). Tilting methods for assessing the influence of components in a classifier. J. R. Stat. Soc. Ser. B. Stat. Methodol. 71 783–803.
  • [26] Hess, K. R., Anderson, K., Symmans, W. F., Valero, V., Ibrahim, N., Mejia, J. A., Booser, D., Theriault, R. L., Buzdar, A. U., Dempsey, P. J., Rouzier, R., Sneige, N., Ross, J. S., Vidaurre, T., Gómez, H. L., Hortobagyi, G. N. and Pusztai, L. (2006). Pharmacogenomic predictor of sensitivity to preoperative chemotherapy with paclitaxel and fluorouracil, doxorubicin, and cyclophosphamide in breast cancer. J. Clin. Oncol. 24 4236–4244.
  • [27] Jin, J. (2012). Comment: “Estimating false discovery proportion under arbitrary covariance dependence” [MR3010887]. J. Amer. Statist. Assoc. 107 1042–1045.
  • [28] Lam, C. and Fan, J. (2009). Sparsistency and rates of convergence in large covariance matrix estimation. Ann. Statist. 37 4254–4278.
  • [29] Lauritzen, S. L. (1996). Graphical Models. Oxford Univ. Press, New York.
  • [30] Li, K.-C. (1991). Sliced inverse regression for dimension reduction. J. Amer. Statist. Assoc. 86 316–342.
  • [31] Liu, W. (2013). Gaussian graphical model estimation with false discovery rate control. Ann. Statist. 41 2948–2978.
  • [32] Lv, J. (2013). Impacts of high dimensionality in finite samples. Ann. Statist. 41 2236–2262.
  • [33] Lv, J. and Fan, Y. (2009). A unified approach to model selection and sparse recovery using regularized least squares. Ann. Statist. 37 3498–3528.
  • [34] Markowitz, H. M. (1952). Portfolio selection. J. Finance 7 77–91.
  • [35] Meinshausen, N. and Bühlmann, P. (2006). High-dimensional graphs and variable selection with the lasso. Ann. Statist. 34 1436–1462.
  • [36] Peng, J., Wang, P., Zhou, N. and Zhu, J. (2009). Partial correlation estimation by joint sparse regression models. J. Amer. Statist. Assoc. 104 735–746.
  • [37] Ravikumar, P., Wainwright, M. J., Raskutti, G. and Yu, B. (2011). High-dimensional covariance estimation by minimizing $\ell_{1}$-penalized log-determinant divergence. Electron. J. Stat. 5 935–980.
  • [38] Ren, Z., Sun, T., Zhang, C.-H. and Zhou, H. H. (2015). Asymptotic normality and optimalities in estimation of large Gaussian graphical models. Ann. Statist. 43 991–1026.
  • [39] Rothman, A. J., Bickel, P. J., Levina, E. and Zhu, J. (2008). Sparse permutation invariant covariance estimation. Electron. J. Stat. 2 494–515.
  • [40] Sun, T. and Zhang, C.-H. (2012). Scaled sparse linear regression. Biometrika 99 879–898.
  • [41] Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. J. Roy. Statist. Soc. Ser. B 58 267–288.
  • [42] Wainwright, M. J. and Jordan, M. I. (2008). Graphical models, exponential families, and variational inference. Faund. Trends Mach. Learn. 1 1–305.
  • [43] Ye, F. and Zhang, C.-H. (2010). Rate minimaxity of the Lasso and Dantzig selector for the $\ell_{q}$ loss in $\ell_{r}$ balls. J. Mach. Learn. Res. 11 3519–3540.
  • [44] Yuan, M. (2010). High dimensional inverse covariance matrix estimation via linear programming. J. Mach. Learn. Res. 11 2261–2286.
  • [45] Yuan, M. and Lin, Y. (2007). Model selection and estimation in the Gaussian graphical model. Biometrika 94 19–35.
  • [46] Zhang, C.-H. (2010). Nearly unbiased variable selection under minimax concave penalty. Ann. Statist. 38 894–942.
  • [47] Zhang, T. and Zou, H. (2014). Sparse precision matrix estimation via lasso penalized D-trace loss. Biometrika 101 103–120.
  • [48] Zou, H. (2006). The adaptive lasso and its oracle properties. J. Amer. Statist. Assoc. 101 1418–1429.

Supplemental materials

  • Supplement to “Innovated scalable efficient estimation in ultra-large Gaussian graphical models”. Due to space constraints, the proofs of Theorem 3 and Proposition 1 and additional technical details are provided in the Supplementary Material [21].