Principal Component Analysis in the local differential privacy model

In this paper, we study the Principal Component Analysis (PCA) problem under the (distributed) non-interactive local differential privacy model. For the low dimensional case (i.e., p≪n), we show the optimal rate of Θ(kpnϵ2) (omitting the eigenvalue terms) for the private minimax risk of the k-dimensional PCA using the squared subspace distance as the measurement, where n is the sample size and ϵ is the privacy parameter. For the high dimensional (i.e., p≫n) row sparse case, we first give a lower bound of [Formula presented] on the private minimax risk, where s is the underlying sparsity parameter. Then we provide an efficient algorithm to achieve the upper bound of [Formula presented]. Experiments on both synthetic and real world datasets confirm our theoretical guarantees.