Differentially Private Inference for Geodesic Regression on Riemannian Manifolds

In this paper, we study inference for geodesic regression with Euclidean predictors and a manifold-valued response under Gaussian differential privacy (GDP). We develop a one-shot GDP inference procedure via sample splitting. The released outputs include a private parameter estimator, a private Wald region for the footpoint and shooting-vector parameters, and pointwise private confidence regions for the fitted geodesic. We derive a range of theoretical results, including sensitivity bounds and GDP guarantee, consistency, a central limit theorem, and a consistent private covariance estimator. We also illustrate our methodology on several important manifolds and explain how the main theorems specialize in these settings, including Euclidean spaces, spheres, Hadamard manifolds, affine-invariant or log-Euclidean SPD manifolds, and Grassmann manifolds. Extensive numerical experiments provide positive support for the proposed GDP inference framework. We further apply the methodology to the tropical-cyclone track data.