# Gaussian Differential Privacy on Riemannian Manifolds

arXiv: https://arxiv.org/abs/2311.10101 · HTML: https://mlsanity.com/paper/2311.10101/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 52 references that are in the map.
- First posted on arXiv 9 Nov 2023.

## Builds on (most cited first, 11 of 11)

- [Concentrated Differential Privacy: Simplifications, Extensions, and Lower Bounds](https://mlsanity.com/paper/1605.02065/) (2016)
- [The Composition Theorem for Differential Privacy](https://mlsanity.com/paper/1311.0776/) (2013)
- [Privacy Amplification by Subsampling: Tight Analyses via Couplings and Divergences](https://mlsanity.com/paper/1807.01647/) (2018)
- [Improving the Gaussian Mechanism for Differential Privacy: Analytical Calibration and Optimal Denoising](https://mlsanity.com/paper/1805.06530/) (2018)
- [Differential Privacy Over Riemannian Manifolds](https://mlsanity.com/paper/2111.02516/) (2021)
- [Differentially private Riemannian optimization](https://mlsanity.com/paper/2205.09494/) (2022)
- [Wrapped Distributions on homogeneous Riemannian manifolds](https://mlsanity.com/paper/2204.09790/) (2022)
- [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)
- [Identification, Amplification and Measurement: A bridge to Gaussian Differential Privacy](https://mlsanity.com/paper/2210.09269/) (2022)
- [Exponential-wrapped distributions on symmetric spaces](https://mlsanity.com/paper/2009.01983/) (2020)
- [Shape And Structure Preserving Differential Privacy](https://mlsanity.com/paper/2209.12667/) (2022)

## Built on by (most cited first, 6 of 6)

- [A Statistical Viewpoint on Differential Privacy: Hypothesis Testing, Representation and Blackwell's Theorem](https://mlsanity.com/paper/2409.09558/) (2024)
- [Federated Learning on Riemannian Manifolds with Differential Privacy](https://mlsanity.com/paper/2404.10029/) (2024)
- [Geometric Renyi Differential Privacy: Ricci Curvature Characterized by Heat Diffusion Mechanisms](https://mlsanity.com/paper/2604.20761/) (2026)
- [Gaussian Differentially Private Human Faces Under a Face Radial Curve Representation](https://mlsanity.com/paper/2409.08301/) (2024)
- [Differentially Private Geodesic Regression](https://mlsanity.com/paper/2504.11304/) (2025)
- [Locally Private Inference for Riemannian Stochastic Optimization](https://mlsanity.com/paper/2609.22642/) (2026)

## Abstract (arXiv)

We develop an advanced approach for extending Gaussian Differential Privacy (GDP) to general Riemannian manifolds. The concept of GDP stands out as a prominent privacy definition that strongly warrants extension to manifold settings, due to its central limit properties. By harnessing the power of the renowned Bishop-Gromov theorem in geometric analysis, we propose a Riemannian Gaussian distribution that integrates the Riemannian distance, allowing us to achieve GDP in Riemannian manifolds with bounded Ricci curvature. To the best of our knowledge, this work marks the first instance of extending the GDP framework to accommodate general Riemannian manifolds, encompassing curved spaces, and circumventing the reliance on tangent space summaries. We provide a simple algorithm to evaluate the privacy budget μ on any one-dimensional manifold and introduce a versatile Markov Chain Monte Carlo (MCMC)-based algorithm to calculate μ on any Riemannian manifold with constant curvature. Through simulations on one of the most prevalent manifolds in statistics, the unit sphere Sd, we demonstrate the superior utility of our Riemannian Gaussian mechanism in comparison to the previously proposed Riemannian Laplace mechanism for implementing GDP.