Dimension-free Private Mean Estimation for Anisotropic Distributions

This rate is unavoidable when the distribution is isotropic, namely, when the covariance is a multiple of the identity matrix. Yet, real-world data is often highly anisotropic, with signals concentrated on a small number of principal components. We develop estimators that are appropriate for such signals--our estimators are (ε, δ)-differentially private and have sample complexity that is dimension-independent for anisotropic subgaussian distributions.