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

• URL: http://arxiv.org/abs/2604.20761v1
• Date: Wed, 22 Apr 2026 16:48:55 GMT
• Title: Geometric Renyi Differential Privacy: Ricci Curvature Characterized by Heat Diffusion Mechanisms
• Abstract summary: Key contribution lies in uncovering unexpected connections among geometric analysis, heat diffusion models, and differential privacy.<n>We establish Renyi differential privacy guarantees governed by Ricci curvature.<n>For manifold with nonnegative Ricci curvature, we propose a mechanism based on heat diffusion.<n>As a statistical application, we develop privacy-preserving estimation of the generalized Frechet mean.
• Score: 4.997037172860666
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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