Exponential-Wrapped Mechanisms for Differential Privacy on Hadamard Manifolds

Yangdi Jiang, Xiaotian Chang, Lei Ding, Linglong Kong, Bei Jiang

Keywords: Differential Privacy, Riemannian Manifold, Gaussian Differential Privacy, Renyi Differential Privacy, Hadamard Manifold, Frechet Mean, Symmetric Positive Definite Matrices

TL;DR: Extending differential privacy framework to Hadamard manifolds using exponential-wrapped mechanisms.

Abstract: We extend the Differential Privacy (DP) framework to Hadamard manifolds, the class of complete and simply connected Riemannian manifolds with non-positive sectional curvature. Inspired by the Cartan–Hadamard theorem, we introduce Exponential-Wrapped Laplace and Gaussian mechanisms to achieve $\varepsilon$-DP, $(\varepsilon, \delta)$-DP, Gaussian DP (GDP), and R\'enyi DP (RDP) on these manifolds. Our approach employs efficient, straightforward algorithms that circumvent the computationally intensity Monte Carlo Markov Chain (MCMC) methods. This work is the first to extend $(\varepsilon, \delta)$-DP, GDP, and RDP to Hadamard manifolds. We further demonstrate the effectiveness of our methodology through simulations on the space of Symmetric Positive Definite Matrices, a frequently used Hadamard manifold in statistics. Our findings reveal that our Exponential-Wrapped mechanisms surpass traditional MCMC-based approaches, which require careful tuning and extensive diagnostics, in both performance and ease of use. Additionally, our methods achieve comparable utility to the Riemannian Laplace mechanism with enhanced utility for smaller privacy budgets ($\varepsilon$) and operate orders of magnitude faster computationally.

Primary Area: alignment, fairness, safety, privacy, and societal considerations

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Submission Number: 4853