In this paper, we develop a novel privacy mechanism for Riemannian manifold-valued data. Our key contribution lies in uncovering unexpected connections among geometric analysis, heat diffusion models, and differential privacy (DP). We characterize the Renyi divergence via dimension-free Harnack inequalities on Riemannian manifolds and establish Renyi differential privacy guarantees governed by Ricci curvature. For manifolds with nonnegative Ricci curvature, we propose a mechanism based on heat diffusion. In contrast, for general manifolds we introduce a Langevin-process-based approach that yields intrinsic mechanisms supporting normalization-free sampling and continuous privacy-utility trade-offs. We derive detailed utility analyses for both mechanisms. As a statistical application, we develop privacy-preserving estimation of the generalized Frechet mean, including nontrivial sensitivity analysis and phase transition characterizations. Numerical experiments further demonstrate the advantages of the proposed DP mechanisms over existing approaches.

翻译：本文针对黎曼流形上的数据，提出了一种新颖的隐私机制。我们的核心贡献在于揭示了几何分析、热扩散模型与差分隐私之间意想不到的联系。我们通过黎曼流形上的无维数 Harnack 不等式刻画出 Renyi 散度，并建立了由里奇曲率控制的 Renyi 差分隐私保证。对于非负里奇曲率的流形，我们提出了一种基于热扩散的机制；相反，对于一般流形，我们引入了一种基于 Langevin 过程的方法，该方法支持免归一化采样和连续的隐私-效用权衡，从而得到内在的机制。我们为这两种机制推导了详细的效用分析。作为一种统计应用，我们开发了广义 Frechet 均值的隐私保护估计，包括非平凡的灵敏度分析和相变刻画。数值实验进一步证明了所提出的差分隐私机制相较于现有方法的优势。