Differentially Private Upper Bounds for Covariance and Inverse Covariance Matrices

ABSTRACT Prior work established upper bounds for spiked covariance matrices under Gaussian and sub‐Gaussian distributions in the Schatten‐q norm, a specific class of unitarily invariant norms. In contrast, our work investigates differentially private estimation for general covariance matrices under sub‐Gaussian distributions, and we analyse the estimation error under any unitarily invariant norm. Furthermore, we establish the conditions for the invertibility of the covariance matrix and provide an upper bound on its spectral norm under differential privacy. To demonstrate the optimality of the upper bound, we derive provide a lower bound estimation. We also use numerical experiments to illustrate the effectiveness and tightness of the upper bound estimation.