  In this paper, we develop a general framework to design differentially
private expectation-maximization (EM) algorithms in high-dimensional latent
variable models, based on the noisy iterative hard-thresholding. We derive the
statistical guarantees of the proposed framework and apply it to three specific
models: Gaussian mixture, mixture of regression, and regression with missing
covariates. In each model, we establish the near-optimal rate of convergence
with differential privacy constraints, and show the proposed algorithm is
minimax rate optimal up to logarithm factors. The technical tools developed for
the high-dimensional setting are then extended to the classic low-dimensional
latent variable models, and we propose a near rate-optimal EM algorithm with
differential privacy guarantees in this setting. Simulation studies and real
data analysis are conducted to support our results.
