Robust Gaussian Graphical Modeling with the Trimmed Graphical Lasso

Gaussian Graphical Models (GGMs) are popular tools for studying network structures. However, many modern applications such as gene network discovery and social interactions analysis often involve high-dimensional noisy data with outliers or heavier tails than the Gaussian distribution. In this paper, we propose the Trimmed Graphical Lasso for robust estimation of sparse GGMs. Our method guards against outliers by an implicit trimming mechanism akin to the popular Least Trimmed Squares method used for linear regression. We provide a rigorous statistical analysis of our estimator in the high-dimensional setting. In contrast, existing approaches for robust sparse GGMs estimation lack statistical guarantees. Our theoretical results are complemented by experiments on simulated and real gene expression data which further demonstrate the value of our approach.

[1]  Susumu Goto,et al.  Data, information, knowledge and principle: back to metabolism in KEGG , 2013, Nucleic Acids Res..

[2]  Hongzhe Li,et al.  High‐Dimensional Heteroscedastic Regression with an Application to eQTL Data Analysis , 2012, Biometrics.

[3]  Y. Nesterov.  Gradient methods for minimizing composite objective function , 2007 .

[4]  Christophe Croux,et al.  Sparse Least Trimmed Squares Regression , 2011 .

[5]  R. Tibshirani,et al.  Sparse inverse covariance estimation with the graphical lasso. , 2008, Biostatistics.

[6]  Po-Ling Loh,et al.  Regularized M-estimators with nonconvexity: statistical and algorithmic theory for local optima , 2013, J. Mach. Learn. Res..

[7]  E. Ising.  Beitrag zur Theorie des Ferromagnetismus , 1925 .

[8]  Martin S. Kochmanski.  NOTE ON THE E. ISING'S PAPER ,,BEITRAG ZUR THEORIE DES FERROMAGNETISMUS" (Zs. Physik, 31, 253 (1925)) , 2008 .

[9]  J. Woods.  Markov image modeling , 1976, 1976 IEEE Conference on Decision and Control including the 15th Symposium on Adaptive Processes.

[10]  Pradeep Ravikumar,et al.  Sparse inverse covariance matrix estimation using quadratic approximation , 2011, MLSLP.

[11]  Hongzhe Li,et al.  Robust Gaussian Graphical Modeling Via l1 Penalization , 2012, Biometrics.

[12]  E. Candès,et al.  Stable signal recovery from incomplete and inaccurate measurements , 2005, math/0503066.

[13]  Trac D. Tran,et al.  Robust Lasso With Missing and Grossly Corrupted Observations , 2011, IEEE Transactions on Information Theory.

[14]  Joseph O. Deasy,et al.  Inference of radio-responsive gene regulatory networks using the graphical lasso algorithm , 2014, BMC Bioinformatics.

[15]  Mathias Drton,et al.  Robust graphical modeling of gene networks using classical and alternative t-distributions , 2010, 1009.3669.

[16]  Steffen L. Lauritzen,et al.  Graphical models in R , 1996 .

[17]  Pradeep Ravikumar,et al.  Elementary Estimators for Graphical Models , 2014, NIPS.

[18]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[19]  M. Yuan,et al.  Model selection and estimation in the Gaussian graphical model , 2007 .

[20]  N. Meinshausen,et al.  High-dimensional graphs and variable selection with the Lasso , 2006, math/0608017.

[21]  Martin J. Wainwright,et al.  A unified framework for high-dimensional analysis of $M$-estimators with decomposable regularizers , 2009, NIPS.

[22]  Bin Yu,et al.  High-dimensional covariance estimation by minimizing ℓ1-penalized log-determinant divergence , 2008, 0811.3628.

[23]  Rachel B. Brem,et al.  The landscape of genetic complexity across 5,700 gene expression traits in yeast. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[24]  Alexandre d'Aspremont,et al.  Model Selection Through Sparse Max Likelihood Estimation Model Selection Through Sparse Maximum Likelihood Estimation for Multivariate Gaussian or Binary Data , 2022 .

[25]  Zhe Jiang,et al.  Spatial Statistics , 2013 .

[26]  Pradeep Ravikumar,et al.  Graphical Models via Generalized Linear Models , 2012, NIPS.

[27]  Anil K. Jain,et al.  Markov Random Field Texture Models , 1983, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[28]  Roman Vershynin,et al.  Introduction to the non-asymptotic analysis of random matrices , 2010, Compressed Sensing.

[29]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .