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

TL;DR

This paper introduces new differential privacy mechanisms for Riemannian manifold data, leveraging heat diffusion and Ricci curvature, with theoretical guarantees and practical utility analyses.

Contribution

It uncovers connections between geometric analysis, heat diffusion, and differential privacy, proposing novel mechanisms tailored to manifold curvature properties.

Findings

Mechanisms support normalization-free sampling.

Detailed utility analyses for both mechanisms.

Numerical experiments show advantages over existing methods.

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.