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

2026-04-22

AI Summary

This work addresses the design of intrinsic mechanisms on Riemannian manifolds that satisfy Rényi differential privacy while preserving data utility. By leveraging heat diffusion and Langevin stochastic processes, the authors construct a continuous privacy mechanism that avoids normalization, and they exploit Ricci curvature to control Rényi divergence—thereby uncovering, for the first time, a deep connection among geometric analysis, heat diffusion models, and differential privacy. Building on dimension-free Harnack inequalities, they establish a privacy framework applicable to manifolds with general Ricci curvature, propose an intrinsic sampling algorithm, and provide a complete analysis of utility and sensitivity. Numerical experiments demonstrate that the proposed method outperforms existing approaches on both non-negatively and generally curved manifolds.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Responsible Web: Data and user privacy-enhancing technologies for the Web

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.

Problem

• Riemannian manifold
• Renyi differential privacy
• heat diffusion
• privacy-utility trade-off
• Frechet mean

Innovation

• Riemannian manifold
• Renyi differential privacy
• heat diffusion
• Ricci curvature
• Langevin process