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

arXiv: https://arxiv.org/abs/2604.20761 · HTML: https://mlsanity.com/paper/2604.20761/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 52 references that are in the map.
- First posted on arXiv 22 Apr 2026.

## Builds on (most cited first, 15 of 15)

- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/1602.06053/) (2016)
- [Fr'echet Regression for Random Objects with Euclidean Predictors](https://mlsanity.com/paper/1608.03012/) (2016)
- [Differential Privacy Over Riemannian Manifolds](https://mlsanity.com/paper/2111.02516/) (2021)
- [Fr'echet random forests for metric space valued regression with non euclidean predictors](https://mlsanity.com/paper/1906.01741/) (2019)
- [Differentially private Riemannian optimization](https://mlsanity.com/paper/2205.09494/) (2022)
- [KNG: The K-Norm Gradient Mechanism](https://mlsanity.com/paper/1905.09436/) (2019)
- [Uniform convergence of local Fr'echet regression, with applications to locating extrema and time warping for metric space valued trajectories](https://mlsanity.com/paper/2006.13548/) (2020)
- [Differentially Private Fr'echet Mean on the Manifold of Symmetric Positive Definite (SPD) Matrices with log-Euclidean Metric](https://mlsanity.com/paper/2208.04245/) (2022)
- [Total Variation Regularized Fr'echet Regression for Metric-Space Valued Data](https://mlsanity.com/paper/1904.09647/) (2019)
- [Shape And Structure Preserving Differential Privacy](https://mlsanity.com/paper/2209.12667/) (2022)
- [Gaussian Differential Privacy on Riemannian Manifolds](https://mlsanity.com/paper/2311.10101/) (2023)
- [Sampling and estimation on manifolds using the Langevin diffusion](https://mlsanity.com/paper/2312.14882/) (2023)
- [Metric Statistics: Exploration and Inference for Random Objects With Distance Profiles](https://mlsanity.com/paper/2202.06117/) (2022)
- [Brownian Noise Reduction: Maximizing Privacy Subject to Accuracy Constraints](https://mlsanity.com/paper/2206.07234/) (2022)

## Built on by (most cited first, 1 of 1)

- [Locally Private Inference for Riemannian Stochastic Optimization](https://mlsanity.com/paper/2609.22642/) (2026)

## Abstract (arXiv)

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.