Differentially private high dimensional sparse covariance matrix estimation

Di Wang, Jinhui Xu

Abstract: Highlights • We give the first study on estimating the sparse covariance matrix under DP model. • We propose a simple but nontrivial DP method, and show that it could achieve non-trivial error bound. • The method can also easily extend to local differential privacy model. • Experiments on synthetic datasets confirm our theoretical results. Abstract In this paper, we study the problem of estimating the covariance matrix under differential privacy, where the underlying covariance matrix is assumed to be sparse and of high dimensions. We propose a new method, called DP-Thresholding, to achieve a non-trivial ℓ 2 -norm based error bound whose dependence on the dimension drops to logarithmic instead of polynomial, it is significantly better than the existing ones, which add noise directly to the empirical covariance matrix. We also extend the ℓ 2 -norm based error bound to a general ℓ w -norm based one for any 1 ≤ w ≤ ∞ , and show that they share the same upper bound asymptotically. Our approach can be easily extended to local differential privacy. Experiments on the synthetic datasets show results that are consistent with theoretical claims.