Lower Bounds for Private Estimation of Gaussian Covariance Matrices under All Reasonable Parameter Regimes

Victor S. Portella, Nick Harvey

Abstract: We prove lower bounds on the number of samples needed to privately estimate the covariance matrix of a Gaussian distribution. Our bounds match existing upper bounds in the widest known setting of parameters. Our analysis relies on the Stein-Haff identity, an extension of the classical Stein's identity used in previous fingerprinting lemma arguments.