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We introduce a constrained empirical loss minimization framework for estimating highdimensional 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 thatdoi:10.1093/biomet/ast059 fatcat:opynsfak4rd5lov4rqnuusrx3y