Precision matrix estimation is a fundamental topic in multivariate statistics and modern machine learning. This paper proposes an adversarially perturbed precision matrix estimation framework, motivated by recent developments in adversarial training. The proposed framework is versatile for the precision matrix problem since, by adapting to different perturbation geometries, the proposed framework can not only recover the existing distributionally robust method but also inspire a novel moment-adaptive approach to precision matrix estimation, proven capable of sparsity recovery and adversarial robustness. Notably, the proposed perturbed precision matrix framework is proven to be asymptotically equivalent to regularized precision matrix estimation, and the asymptotic normality can be established accordingly. The resulting asymptotic distribution highlights the asymptotic bias introduced by perturbation and identifies conditions under which the perturbed estimation can be unbiased in the asymptotic sense. Numerical experiments on both synthetic and real data demonstrate the desirable performance of the proposed adversarially perturbed approach in practice.

翻译：精度矩阵估计是多元统计学与现代机器学习中的一个基础课题。本文受对抗训练近期发展的启发，提出了一种对抗性扰动精度矩阵估计框架。该框架在精度矩阵问题上具有广泛适用性：通过适配不同的扰动几何结构，该框架不仅能复原现有的分布鲁棒方法，还能启发一种新颖的矩自适应精度矩阵估计方法，并被证明具备稀疏性恢复与对抗鲁棒性。值得注意的是，所提出的扰动精度矩阵框架被证明在渐近意义上等价于正则化精度矩阵估计，并可据此建立渐近正态性。所得的渐近分布揭示了扰动引入的渐近偏差，并识别出在何种条件下扰动估计能在渐近意义上保持无偏。在合成数据与真实数据上的数值实验均表明，所提出的对抗性扰动方法在实践中具有优越性能。