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.

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  1. Ayyıldız, Ezgi; Purutçuoğlu, Vilda; Weber, Gerhard Wilhelm: Loop-based conic multivariate adaptive regression splines is a novel method for advanced construction of complex biological networks (2018)
  2. Champion, Magali; Picheny, Victor; Vignes, Matthieu: Inferring large graphs using $\ell_1$-penalized likelihood (2018)
  3. Karl Sjöstrand; Line Clemmensen; Rasmus Larsen; Gudmundur Einarsson; Bjarne Ersbøll: SpaSM: A MATLAB Toolbox for Sparse Statistical Modeling (2018)
  4. Li, Peili; Xiao, Yunhai: An efficient algorithm for sparse inverse covariance matrix estimation based on dual formulation (2018)
  5. Loh, Po-Ling; Tan, Xin Lu: High-dimensional robust precision matrix estimation: cellwise corruption under $\epsilon $-contamination (2018)
  6. Mair, Patrick: Modern psychometrics with R (2018)
  7. Perrot-Dockès, Marie; Lévy-Leduc, Céline; Sansonnet, Laure; Chiquet, Julien: Variable selection in multivariate linear models with high-dimensional covariance matrix estimation (2018)
  8. Popovic, Gordana C.; Hui, Francis K. C.; Warton, David I.: A general algorithm for covariance modeling of discrete data (2018)
  9. Torri, Gabriele; Giacometti, Rosella; Paterlini, Sandra: Robust and sparse banking network estimation (2018)
  10. Zheng, Yuchen; Lee, Ilbin; Serban, Nicoleta: Regularized optimization with spatial coupling for robust decision making (2018)
  11. Al-Najjar, Elias; Adragni, Kofi P.: Sufficient dimension reduction constrained through sub-populations (2017)
  12. Aste, Tomaso; Di Matteo, T.: Sparse causality network retrieval from short time series (2017)
  13. Baek, Changryong; Davis, Richard A.; Pipiras, Vladas: Sparse seasonal and periodic vector autoregressive modeling (2017)
  14. Boutsidis, Christos; Drineas, Petros; Kambadur, Prabhanjan; Kontopoulou, Eugenia-Maria; Zouzias, Anastasios: A randomized algorithm for approximating the log determinant of a symmetric positive definite matrix (2017)
  15. Cheng, Lulu; Shan, Liang; Kim, Inyoung: Multilevel Gaussian graphical model for multilevel networks (2017)
  16. Hirose, Kei; Fujisawa, Hironori; Sese, Jun: Robust sparse Gaussian graphical modeling (2017)
  17. Janková, Jana; Van de Geer, Sara: Honest confidence regions and optimality in high-dimensional precision matrix estimation (2017)
  18. Jonatan Kallus, Jose Sanchez, Alexandra Jauhiainen, Sven Nelander, Rebecka Jornsten: ROPE: high-dimensional network modeling with robust control of edge FDR (2017) arXiv
  19. Kühnel, Line; Sommer, Stefan; Pai, Akshay; Raket, Lars Lau: Most likely separation of intensity and warping effects in image registration (2017)
  20. Kwon, Sunghoon; Ahn, Jeongyoun; Jang, Woncheol; Lee, Sangin; Kim, Yongdai: A doubly sparse approach for group variable selection (2017)

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