We propose a novel and systematic differentially private (DP) inference framework for non-Euclidean data. First, we design two types of DP mechanisms for the Fréchet mean and variance with i.i.d. Riemannian manifold-valued data, tailored to different geometric structures and accompanied by analytic privacy budgets calibrated to the geometry of the underlying manifold. Second, we establish the consistency and central limit theorems (CLTs) of the proposed DP estimators, enabling a suite of statistical inference procedures under privacy protection. Furthermore, we provide comprehensive implementation guidelines and feasible procedures, including consistent DP estimators of the asymptotic variance in the CLTs. Extensive numerical experiments support the proposed methodologies. Finally, we demonstrate the effectiveness of our approach on real-world medical image and sociological datasets lying on two representative manifolds.

翻译：我们提出了一种新颖且系统化的面向非欧几里得数据的差分隐私（DP）推断框架。首先，针对独立同分布的黎曼流形值数据，我们设计了两种适用于弗里切均值与方差的DP机制，这两种机制分别适配不同的几何结构，并附有依据底层流形几何特性校准的解析隐私预算。其次，我们建立了所提出DP估计量的一致性与中心极限定理（CLT），从而在隐私保护下实现一系列统计推断程序。此外，我们提供了全面的实施指南与可行流程，包括CLT中渐近方差的一致DP估计量。大量数值实验支持了所提方法。最后，我们在两个代表性流形上的真实医学图像与社会学数据集上验证了该方法的效果。