We propose Nodewise Loreg, a nodewise $L_0$-penalized regression method for estimating high-dimensional sparse precision matrices. We establish its asymptotic properties, including convergence rates, support recovery, and asymptotic normality under high-dimensional sub-Gaussian settings. Notably, the Nodewise Loreg estimator is asymptotically unbiased and normally distributed, eliminating the need for debiasing required by Nodewise Lasso. We also develop a desparsified version of Nodewise Loreg, similar to the desparsified Nodewise Lasso estimator. The asymptotic variances of the undesparsified Nodewise Loreg estimator are upper bounded by those of both desparsified Nodewise Loreg and Lasso estimators for Gaussian data, potentially offering more powerful statistical inference. Extensive simulations show that the undesparsified Nodewise Loreg estimator generally outperforms the two desparsified estimators in asymptotic normal behavior. Moreover, Nodewise Loreg surpasses Nodewise Lasso, CLIME, and GLasso in most simulations in terms of matrix norm losses, support recovery, and timing performance. Application to a breast cancer gene expression dataset further demonstrates Nodewise Loreg's superiority over the three $L_1$-norm based methods.

翻译：我们提出了一种节点式Loreg方法，即基于节点式$L_0$惩罚回归的高维稀疏精度矩阵估计方法。我们建立了其渐近性质，包括在高维次高斯设定下的收敛速率、支持恢复和渐近正态性。值得注意的是，节点式Loreg估计量是渐近无偏且服从正态分布的，无需节点式Lasso所需的去偏处理。我们还开发了节点式Loreg的去稀疏化版本，类似于去稀疏化节点式Lasso估计量。对于高斯数据，未去稀疏化的节点式Loreg估计量的渐近方差均不超过去稀疏化节点式Loreg和Lasso估计量的渐近方差，从而可能提供更强大的统计推断能力。大量模拟实验表明，未去稀疏化的节点式Loreg估计量在渐近正态行为上通常优于两种去稀疏化估计量。此外，在大多数模拟中，节点式Loreg在矩阵范数损失、支持恢复和运行时间性能方面均优于节点式Lasso、CLIME和GLasso。应用于乳腺癌基因表达数据集的结果进一步证明了节点式Loreg相较于三种基于$L_1$范数的方法的优越性。