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. Bollhöfer, Matthias; Eftekhari, Aryan; Scheidegger, Simon; Schenk, Olaf: Large-scale sparse inverse covariance matrix estimation (2019)
  2. Pun, Chi Seng; Wong, Hoi Ying: A linear programming model for selection of sparse high-dimensional multiperiod portfolios (2019)
  3. Yue, Mu; Li, Jialiang; Cheng, Ming-Yen: Two-step sparse boosting for high-dimensional longitudinal data with varying coefficients (2019)
  4. Avanesov, Valeriy; Buzun, Nazar: Change-point detection in high-dimensional covariance structure (2018)
  5. 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)
  6. Azose, Jonathan J.; Raftery, Adrian E.: Estimating large correlation matrices for international migration (2018)
  7. Barber, Rina Foygel; Kolar, Mladen: ROCKET: robust confidence intervals via Kendall’s tau for transelliptical graphical models (2018)
  8. Bilgrau, Anders Ellern; Brøndum, Rasmus Froberg; Eriksen, Poul Svante; Dybkær, Karen; Bøgsted, Martin: Estimating a common covariance matrix for network meta-analysis of gene expression datasets in diffuse large B-cell lymphoma (2018)
  9. Castelletti, Federico; Consonni, Guido; Della Vedova, Marco L.; Peluso, Stefano: Learning Markov equivalence classes of directed acyclic graphs: an objective Bayes approach (2018)
  10. Champion, Magali; Picheny, Victor; Vignes, Matthieu: Inferring large graphs using $\ell_1$-penalized likelihood (2018)
  11. Chang, Jinyuan; Qiu, Yumou; Yao, Qiwei; Zou, Tao: Confidence regions for entries of a large precision matrix (2018)
  12. Chen, Shuo; Kang, Jian; Xing, Yishi; Zhao, Yunpeng; Milton, Donald K.: Estimating large covariance matrix with network topology for high-dimensional biomedical data (2018)
  13. Devijver, Emilie; Gallopin, Mélina: Block-diagonal covariance selection for high-dimensional Gaussian graphical models (2018)
  14. Fan, Jianqing; Liu, Han; Wang, Weichen: Large covariance estimation through elliptical factor models (2018)
  15. Fan, Jianqing; Liu, Han; Wang, Weichen; Zhu, Ziwei: Heterogeneity adjustment with applications to graphical model inference (2018)
  16. Hao, Botao; Sun, Will Wei; Liu, Yufeng; Cheng, Guang: Simultaneous clustering and estimation of heterogeneous graphical models (2018)
  17. Hong, Younghee; Kim, Choongrak: Recent developments in high dimensional covariance estimation and its related issues, a review (2018)
  18. Janková, Jana; van de Geer, Sara: Semiparametric efficiency bounds for high-dimensional models (2018)
  19. Kakourou, Alexia; Mertens, Bart: Bayesian variable selection logistic regression with paired proteomic measurements (2018)
  20. Karl Sjöstrand; Line Clemmensen; Rasmus Larsen; Gudmundur Einarsson; Bjarne Ersbøll: SpaSM: A MATLAB Toolbox for Sparse Statistical Modeling (2018) not zbMATH

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