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arXiv · 2604.20761

Geometric Renyi Differential Privacy: Ricci Curvature Characterized by Heat Diffusion Mechanisms

Xiaotian Chang·Yangdi Jiang·Cyrus Mostajeran·Qirui Hu
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Abstract

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.

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Xiaotian Chang, Yangdi Jiang, Cyrus Mostajeran, Qirui Hu. 2026-04-22. Geometric Renyi Differential Privacy: Ricci Curvature Characterized by Heat Diffusion Mechanisms. https://arxiv.org/abs/2604.20761

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