Research ArticleOpen AccessGoogle Scholar indexed
Positive-Definite Sparse Precision Matrix Estimation
School of Mathematics and Computer Science, Anhui Normal University, Wuhu, China
School of Mathematics and Computer Science, Anhui Normal University, Wuhu, China
School of Mathematics and Computer Science, Anhui Normal University, Wuhu, China
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, China
- 1 School of Mathematics and Computer Science, Anhui Normal University, Wuhu, China
- 2 School of Mathematics and Computer Science, Anhui Normal University, Wuhu, China
- 3 School of Mathematics and Computer Science, Anhui Normal University, Wuhu, China
- 4 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, China
Advances in Pure Mathematics·Volume 07 (2017)·Pages 21–30·Published 23 January 2017·DOI10.4236/apm.2017.71002
Copy link · social · email
Abstract
The positive-definiteness and sparsity are the most important property of high-dimensional precision matrices. To better achieve those property, this paper uses a sparse lasso penalized D-trace loss under the positive-definiteness constraint to estimate high-dimensional precision matrices. This paper derives an efficient accelerated gradient method to solve the challenging optimization problem and establish its converges rate as . The numerical simulations illustrated our method have competitive advantage than other methods.
KeywordsPositive-DefinitenessSparsityD-Trace LossAccelerated Gradient Method
- Huang, J., Liu, N., Pourahmadi, M. and Liu, L. (2006) Covariance Matrix Selection and Estimation via Penalised Normal Likelihood. Biometrika, 93, 85-98. https://doi.org/10.1093/biomet/93.1.85
- Meinshausen, N. and Bühlmann, P. (2006) High-Dimensional Graphs and Variable Selection with the Lasso. Annals of Statist, 34, 1436-1462. https://doi.org/10.1214/009053606000000281
- Peng, J., Wang, P., Zhou, N. and Zhu, J. (2009) Partial Correlation Estimation by Joint Sparse Regression Models. Journal of the American Statistical Association, 104, 735-746. https://doi.org/10.1198/jasa.2009.0126
- Yuan, M. (2010) High Dimensional Inverse Covariance Matrix Estimation via Linear Programming. Journal of Machine Learning Research, 11, 2261-2286.
- Cai, T., Liu, W. and Luo, X. (2011) A Constrained Minimization Approach to Sparse Precision Matrix Estimation. Journal of the American Statistical Association, 106, 594-607. https://doi.org/10.1198/jasa.2011.tm10155
- Yuan, M. and Lin, Y. (2007) Model Selection and Estimation in the Gaussian Graphical Model. Biometrika, 94, 19-35. https://doi.org/10.1093/biomet/asm018
- Friedman, J.H., Hastie, T.J. and Tibshirani, R.J. (2008) Sparse Inverse Covariance Estimation with the Graphical Lasso. Biostatistics, 9, 432-441. https://doi.org/10.1093/biostatistics/kxm045
- Witten, D., Frienman, J.H. and Simon, N. (2011) New Insights and Faster Computations for the Graphical Lasso. Journal of Computational and Graphical Statistics, 20, 892-900. https://doi.org/10.1198/jcgs.2011.11051a
- Zhang, T. and Zou, H. (2014) Sparse Precision Matrix Estimation via Lasso Penalized D-Trace Loss. Biometrika, 101, 103-120. https://doi.org/10.1093/biomet/ast059
- Ji, S. and Ye, J. (2009) An Accelerated Gradient Method for Trace Norm Minimization. International Conference on Machine Learning, 58, 457-464. https://doi.org/10.1145/1553374.1553434
- Nesterov, Y. (1983) A Method for Solving a Convex Programming Problem with Convergence Rate O(1/K 2 ) . Soviet Mathematics Doklady, 27, 372-367.
- Nesterov, Y. (2003) Introductory Lectures on Convex Optimization: A Basic Course. Kluwer Academic.
- Toh, K.C. and Yun, S. (2010) An Accelerated Proximal Gradient Algorithm for Nuclear Norm Regularized Linear Least Squares Problems. Pacific Journal of Optimization, 6, 615-640.