Exponential-Wrapped Mechanisms: Differential Privacy on Hadamard Manifolds Made Practical
Abstract
We propose a general and computationally efficient framework for achieving differential privacy (DP) on Hadamard manifolds, which are complete and simply connected Riemannian manifolds with non-positive curvature. Leveraging the Cartan-Hadamard theorem, we introduce Exponential-Wrapped Laplace and Gaussian mechanisms that achieve $\epsilon$-DP, $(\epsilon, \delta)$-DP, Gaussian DP (GDP), and Rényi DP (RDP) without relying on computationally intensive MCMC sampling. Our methods operate entirely within the intrinsic geometry of the manifold, ensuring both theoretical soundness and practical scalability. We derive utility bounds for privatized Fréchet means and demonstrate superior utility and runtime performances on both synthetic data and real-world data in the space of symmetric positive definite matrices (SPDM) equipped with three different metrics. To our knowledge, this work constitutes the first unified extension of multiple DP notions to general Hadamard manifolds with practical and scalable implementations.
Cite
Text
Jiang et al. "Exponential-Wrapped Mechanisms: Differential Privacy on Hadamard Manifolds Made Practical." International Conference on Learning Representations, 2026.Markdown
[Jiang et al. "Exponential-Wrapped Mechanisms: Differential Privacy on Hadamard Manifolds Made Practical." International Conference on Learning Representations, 2026.](https://mlanthology.org/iclr/2026/jiang2026iclr-exponentialwrapped/)BibTeX
@inproceedings{jiang2026iclr-exponentialwrapped,
title = {{Exponential-Wrapped Mechanisms: Differential Privacy on Hadamard Manifolds Made Practical}},
author = {Jiang, Yangdi and Chang, Xiaotian and Ding, Lei and Kong, Linglong and Jiang, Bei},
booktitle = {International Conference on Learning Representations},
year = {2026},
url = {https://mlanthology.org/iclr/2026/jiang2026iclr-exponentialwrapped/}
}