Adversarially Perturbed Precision Matrix Estimation

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 achieve high-dimensional model selection consistency under the scale-adaptive incoherence condition, which can be viewed as a relaxation of the classic incoherence condition in the heteroscedastic settings. Additionally, the proposed perturbed precision matrix estimation framework is asymptotically equivalent to the regularized precision matrix estimation, and the asymptotic normality can be established accordingly, where the asymptotic bias introduced by perturbation is highlighted. Numerical experiments demonstrate the desirable practical performance of the proposed adversarially perturbed approach.