Positive Definite Generalized Lasso to Large Sparse Precision Matrix Estimation

Estimating large sparse inverse covariance matrices (precision matrices) is an interesting and challenging problem in many fields of sciences, engineering, and humanities, thanks to advances in computing technologies. Recent applications often encounter high dimensionality with a limited number of data points leading to a number of covariance parameters that greatly exceeds the number of observations. Several methods have been proposed to deal with this problem, but there is no guarantee that the obtained estimator is positive-definite. Furthermore, in many cases, one needs to capture some additional information on the setting of the problem. We propose HPDGLasso (Positive definite Generalized Lasso) approach to fix these problems. To obtain the precision matrix estimator, we suggest to estimate the precision matrix by minimizing a criterion combining a relaxation of the gradient-log likelihood and an adaptive lasso penalty of the form |D ◦ Ω|1, where D is an arbitrary matrix with non-negative elements and ◦ denotes the Hadamard product (elementwise multiplication). We derive an efficient Alternating Direction Method of multipliers (ADMM) algorithm to obtain the optimal solution. 1