  The sparse precision matrix plays an essential role in the Gaussian graphical
model since a zero off-diagonal element indicates conditional independence of
the corresponding two variables given others. In the Gaussian graphical model,
many methods have been proposed, and their theoretical properties are given as
well. Among these, the sparse precision matrix estimation via scaled lasso
(SPMESL) has an attractive feature in which the penalty level is automatically
set to achieve the optimal convergence rate under the sparsity and
invertibility conditions. Conversely, other methods need to be used in
searching for the optimal tuning parameter. Despite such an advantage, the
SPMESL has not been widely used due to its expensive computational cost. In
this paper, we develop a GPU-parallel coordinate descent (CD) algorithm for the
SPMESL and numerically show that the proposed algorithm is much faster than the
least angle regression (LARS) tailored to the SPMESL. Several comprehensive
numerical studies are conducted to investigate the scalability of the proposed
algorithm and the estimation performance of the SPMESL. The results show that
the SPMESL has the lowest false discovery rate for all cases and the best
performance in the case where the level of the sparsity of the columns is high.
