The graphical lasso: new insights and alternatives. The graphical lasso [5] is an algorithm for learning the structure in an undirected Gaussian graphical model, using ℓ 1 regularization to control the number of zeros in the precision matrix Θ=Σ -1 [2, 11]. The R package glasso [5] is popular, fast, and allows one to efficiently build a path of models for different values of the tuning parameter. Convergence of glasso can be tricky; the converged precision matrix might not be the inverse of the estimated covariance, and occasionally it fails to converge with warm starts. In this paper we explain this behavior, and propose new algorithms that appear to outperform glasso. By studying the “normal equations” we see that, glasso is solving the dual of the graphical lasso penalized likelihood, by block coordinate ascent; a result which can also be found in [2]. In this dual, the target of estimation is Σ, the covariance matrix, rather than the precision matrix Θ. We propose similar primal algorithms p-glasso and dp-glasso, that also operate by block-coordinate descent, where Θ is the optimization target. We study all of these algorithms, and in particular different approaches to solving their coordinate sub-problems. We conclude that dp-glasso is superior from several points of view.

References in zbMATH (referenced in 346 articles , 1 standard article )

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  1. Liu, Weidong; Luo, Xi: Fast and adaptive sparse precision matrix estimation in high dimensions (2015)
  2. Loh, Po-Ling; Wainwright, Martin J.: Regularized (M)-estimators with nonconvexity: statistical and algorithmic theory for local optima (2015)
  3. Mohammadi, A.; Wit, E. C.: Bayesian structure learning in sparse Gaussian graphical models (2015)
  4. Mohammadi, A.; Wit, E.C.: BDgraph: An R Package for Bayesian Structure Learning in Graphical Models (2015) arXiv
  5. Oates, C. J.; Costa, L.; Nichols, T. E.: Toward a multisubject analysis of neural connectivity (2015)
  6. Pesonen, Maiju; Pesonen, Henri; Nevalainen, Jaakko: Covariance matrix estimation for left-censored data (2015)
  7. Pircalabelu, Eugen; Claeskens, Gerda; Jahfari, Sara; Waldorp, Lourens J.: A focused information criterion for graphical models in fMRI connectivity with high-dimensional data (2015)
  8. Pircalabelu, Eugen; Claeskens, Gerda; Waldorp, Lourens: A focused information criterion for graphical models (2015)
  9. Ren, Zhao; Sun, Tingni; Zhang, Cun-Hui; Zhou, Harrison H.: Asymptotic normality and optimalities in estimation of large Gaussian graphical models (2015)
  10. Rousseeuw, Peter J.; Van den Bossche, Wannes: Comments on: “Robust estimation of multivariate location and scatter in the presence of cellwise and casewise contamination” (2015)
  11. Slawski, Martin; Hein, Matthias: Estimation of positive definite (M)-matrices and structure learning for attractive Gaussian Markov random fields (2015)
  12. Tan, Kean Ming; Witten, Daniela; Shojaie, Ali: The cluster graphical lasso for improved estimation of Gaussian graphical models (2015)
  13. Tibshirani, Ryan J.: A general framework for fast stagewise algorithms (2015)
  14. Ullah, Insha; Jones, Beatrix: Regularised MANOVA for high-dimensional data (2015)
  15. Vujačić, Ivan; Abbruzzo, Antonino; Wit, Ernst: A computationally fast alternative to cross-validation in penalized Gaussian graphical models (2015)
  16. Wainwright, Martin J.: Graphical models and message-passing algorithms: some introductory lectures (2015)
  17. Wang, Hao: Scaling it up: stochastic search structure learning in graphical models (2015)
  18. Wang, Y. X. Rachel; Jiang, Keni; Feldman, Lewis J.; Bickel, Peter J.; Huang, Haiyan: Inferring gene -- gene interactions and functional modules using sparse canonical correlation analysis (2015)
  19. Wright, Stephen J.: Coordinate descent algorithms (2015)
  20. Xiang, Ruoxuan; Khare, Kshitij; Ghosh, Malay: High dimensional posterior convergence rates for decomposable graphical models (2015)

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