Gradient-based Riemannian Gaussian Differential Privacy

Taking into account the non-linear structure of the Riemannian manifold, this paper proposes a Riemannian Gaussian differential privacy method based on gradient optimization. Specifically, the proposed method defines a Riemannian Gaussian distribution on the manifold by using the geodesic metric, and constructs a new sensitivity calculation method by using an exponential mapping. Furthermore, this method is the first to extend the gradient-based Euclidean Gaussian differential privacy technique to the Riemannian manifold, broadening the scope of differential privacy techniques. Simulation experiments were conducted on the unit sphere and the symmetric positive definite matrix manifold (SPD matrices), and the results show that the proposed method satisfies the differential privacy conditions, thereby demonstrating the effectiveness of the method.