1. Sparse Inverse Covariance Estimation with Graphical LASSO J. Friedman, T. Hastie, R. Tibshirani Biostatistics, 2008 Presented by Minhua Chen 1

2. Motivation Mathematical Model Mathematical Tools Graphical LASSO Related papers 2 Outline

3. Motivation (M. Choi, V. Chandrasekaran and A.S. Willsky, 2009) (O. Banerjee, L. Ghaoui, and A. d’Aspremont, 2008) 3

4. The optimization problem is concave (M. Yuan and Y. Lin, 2007). Various optimization algorithms have been proposed (M. Yuan and Y. Lin, 2007; O. Banerjee, L. Ghaoui, and A. d’Aspremont, 2008; N. Meinshausen and P. Buhlmann, 2006). The Graphical LASSO algorithm, built on a previous paper (O. Banerjee, L. Ghaoui, and A. d’Aspremont, 2008), is widely used due to its computational efficiency. It transforms the above optimization to LASSO regressions. Mathematical Model 4

5. Subgradient (J. Tropp, 2006) Mathematical Tools (1) Example 1:Example 2: 5

6. Mathematical Tools (2) Matrix inversion identity: The above equations reveal the relationship between the inverse covariance matrix and the covariance matrix. 6

7. Graphical LASSO (1) 7

8. Graphical LASSO (2) 8

9. Graphical LASSO (3) 9

10. Graphical LASSO (4) Ground TruthInferred 10

11. Related papers: N. Stadler and P. Buhlmann, Missing Values: Sparse Inverse Covariance Estimation and an Extension to Sparse Regression Proposed a MissGLasso algorithm to impute the missing data and infer the inverse covariance matrix simultaneously. O. Banerjee, L. El Ghaoui and A. d’Aspremont, Model Selection Through Sparse Maximum Likelihood Estimation for Multivariate Gaussian or Binary Data Used a constrained quadratic programming algorithm (COVSEL) to solve the same optimization problem as Graphical LASSO. N. Meinshausen and P. Buhlmann, High-Dimensional Graphs and Variable Selection with the Lasso Proposed a neighborhood selection method to approximate the Gaussian Graph. 11