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 167 articles , 1 standard article )

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  1. Champion, Magali; Picheny, Victor; Vignes, Matthieu: Inferring large graphs using $\ell_1$-penalized likelihood (2018)
  2. Popovic, Gordana C.; Hui, Francis K.C.; Warton, David I.: A general algorithm for covariance modeling of discrete data (2018)
  3. Aste, Tomaso; Di Matteo, T.: Sparse causality network retrieval from short time series (2017)
  4. 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)
  5. Cheng, Lulu; Shan, Liang; Kim, Inyoung: Multilevel Gaussian graphical model for multilevel networks (2017)
  6. Hirose, Kei; Fujisawa, Hironori; Sese, Jun: Robust sparse Gaussian graphical modeling (2017)
  7. Janková, Jana; van de Geer, Sara: Honest confidence regions and optimality in high-dimensional precision matrix estimation (2017)
  8. Jonatan Kallus, Jose Sanchez, Alexandra Jauhiainen, Sven Nelander, Rebecka Jornsten: ROPE: high-dimensional network modeling with robust control of edge FDR (2017) arXiv
  9. Kühnel, Line; Sommer, Stefan; Pai, Akshay; Raket, Lars Lau: Most likely separation of intensity and warping effects in image registration (2017)
  10. Kwon, Sunghoon; Ahn, Jeongyoun; Jang, Woncheol; Lee, Sangin; Kim, Yongdai: A doubly sparse approach for group variable selection (2017)
  11. Leppä-aho, Janne; Pensar, Johan; Roos, Teemu; Corander, Jukka: Learning Gaussian graphical models with fractional marginal pseudo-likelihood (2017)
  12. Li, Fan-qun; Zhang, Xin-sheng: Bayesian Lasso with neighborhood regression method for Gaussian graphical model (2017)
  13. Lu, Zhaosong: Randomized block proximal damped Newton method for composite self-concordant minimization (2017)
  14. Nanty, Simon; Helbert, Céline; Marrel, Amandine; Pérot, Nadia; Prieur, Clémentine: Uncertainty quantification for functional dependent random variables (2017)
  15. Peyhardi, Jean; Fernique, Pierre: Characterization of convolution splitting graphical models (2017)
  16. van Wieringen, Wessel N.: On the mean squared error of the ridge estimator of the covariance and precision matrix (2017)
  17. Xie, Jichun; Kang, Jian: High-dimensional tests for functional networks of brain anatomic regions (2017)
  18. Xudong, Huang; Mengmeng, Li: Confidence intervals for sparse precision matrix estimation via Lasso penalized D-trace loss (2017)
  19. Zhang, Hai; Wang, Puyu; Dong, Qing; Wang, Pu: Sparse Bayesian linear regression using generalized normal priors (2017)
  20. Zhang, Qiang; Ip, Edward H.; Pan, Junhao; Plemmons, Robert: Individual-specific, sparse inverse covariance estimation in generalized estimating equations (2017)

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