Distributional stability of sparse inverse covariance matrix estimators

Finding an approximation of the inverse of the covariance matrix, also known as precision matrix, of a random vector with empirical data is widely discussed in finance and engineering. In data-driven problems, empirical data may be contaminated''. This raises the question as to whether the approximate precision matrix is reliable from a statistical point of view. In this paper, we concentrate on a much-noticed sparse estimator of the precision matrix and investigate the issue from the perspective of distributional stability. Specifically, we derive an explicit local Lipschitz bound for the distance between the distributions of the sparse estimator under two different distributions (regarded as the true data distribution and the distribution of contaminated’’ data). The distance is measured by the Kantorovich metric on the set of all probability measures on a matrix space. We also present analogous results for the standard estimators of the covariance matrix and its eigenvalues. Furthermore, we discuss several applications and conduct some numerical experiments.

💡 Research Summary

The paper addresses a fundamental problem in multivariate statistics and machine learning: estimating the inverse covariance (precision) matrix of a random vector from empirical data, especially when the data may be contaminated by outliers, measurement errors, or distributional shifts. Classical estimators such as the sample covariance matrix (\hat\Sigma_N) and its inverse often fail in high‑dimensional settings—(\hat\Sigma_N) may be singular, and even when invertible its inverse is typically dense, contradicting the sparsity assumptions that underlie many modern applications (graphical model selection, portfolio optimization, etc.).

To overcome these issues, the authors focus on the well‑known ℓ₁‑regularized sparse precision estimator introduced by Banerjee et al.:

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