Printed in Great Britain Advance Access publication 12 February 2014 Sparse precision matrix estimation via lasso penalized D-trace loss

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 precisionmatrix estimator is defined as theminimizer 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 con-vergence 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 pro-posed 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