We introduce a constrained empirical loss minimization framework for estimating high-dimensional sparse precision matrices and propose a new loss function, called the D-trace loss, for that purpose. A novel sparse precision matrix estimator is defined as the minimizer of the lasso penalized D-trace loss under a positive-definiteness constraint. Under a new irrepresentability condition, the lasso penalized D-trace estimator is shown to have the sparse recovery property. Examples demonstrate that the new condition can hold in situations where the irrepresentability condition for the lasso penalized Gaussian likelihood estimator fails. We establish rates of convergence for the new estimator in the elementwise maximum, Frobenius and operator norms. We develop a very efficient algorithm based on alternating direction methods for computing the proposed estimator. Simulated and real data are used to demonstrate the computational efficiency of our algorithm and the finite-sample performance of the new estimator. The lasso penalized D-trace estimator is found to compare favourably with the lasso penalized Gaussian likelihood estimator.
Bibliographical noteFunding Information:
We thank Dr Shiqian Ma for sharing the Matlab code that implements his alternating direction method for computing the lasso penalized Gaussian likelihood estimator. We thank the editor, associate editor and referees for their suggestions, as well as Professors Peter Bühlmann and Tony Cai for helpful discussions. Zou’s research was supported in part by the U.S. National Science Foundation and the Office of Naval Research.
- Constrained minimization
- D-trace loss
- Graphical lasso
- Graphical model selection
- Precision matrix
- Rate of convergence