# Differential Privacy Over Riemannian Manifolds

arXiv: https://arxiv.org/abs/2111.02516 · HTML: https://mlsanity.com/paper/2111.02516/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 32 references that are in the map.
- First posted on arXiv 3 Nov 2021.

## Builds on (most cited first, 4 of 4)

- [Privately Learning High-Dimensional Distributions](https://mlsanity.com/paper/1805.00216/) (2018)
- [Fingerprinting Codes and the Price of Approximate Differential Privacy](https://mlsanity.com/paper/1311.3158/) (2013)
- [CoinPress: Practical Private Mean and Covariance Estimation](https://mlsanity.com/paper/2006.06618/) (2020)
- [Benefits and Pitfalls of the Exponential Mechanism with Applications to Hilbert Spaces and Functional PCA](https://mlsanity.com/paper/1901.10864/) (2019)

## Built on by (most cited first, 16 of 16)

- [DPZero: Private Fine-Tuning of Language Models without Backpropagation](https://mlsanity.com/paper/2310.09639/) (2023)
- [Differentially private Riemannian optimization](https://mlsanity.com/paper/2205.09494/) (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)
- [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)
- [Federated Learning on Riemannian Manifolds with Differential Privacy](https://mlsanity.com/paper/2404.10029/) (2024)
- [Riemannian coordinate descent algorithms on matrix manifolds](https://mlsanity.com/paper/2406.02225/) (2024)
- [Rieoptax: Riemannian Optimization in JAX](https://mlsanity.com/paper/2210.04840/) (2022)
- [Random matrix theory improved Fr'echet mean of symmetric positive definite matrices](https://mlsanity.com/paper/2405.06558/) (2024)
- [Finite sample bounds for barycenter estimation in geodesic spaces](https://mlsanity.com/paper/2502.14069/) (2025)
- [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)
- [Rolled Gaussian process models for curves on manifolds](https://mlsanity.com/paper/2503.21980/) (2025)
- [Differentially Private Geodesic Regression](https://mlsanity.com/paper/2504.11304/) (2025)
- [Rao Differential Privacy](https://mlsanity.com/paper/2508.17135/) (2025)
- [Locally Private Inference for Riemannian Stochastic Optimization](https://mlsanity.com/paper/2609.22642/) (2026)

## Abstract (arXiv)

In this work we consider the problem of releasing a differentially private statistical summary that resides on a Riemannian manifold. We present an extension of the Laplace or K-norm mechanism that utilizes intrinsic distances and volumes on the manifold. We also consider in detail the specific case where the summary is the Fr'echet mean of data residing on a manifold. We demonstrate that our mechanism is rate optimal and depends only on the dimension of the manifold, not on the dimension of any ambient space, while also showing how ignoring the manifold structure can decrease the utility of the sanitized summary. We illustrate our framework in two examples of particular interest in statistics: the space of symmetric positive definite matrices, which is used for covariance matrices, and the sphere, which can be used as a space for modeling discrete distributions.