Differential Privacy (DP) is being increasingly adopted for non-Euclidean data that lie on complex, high-dimensional manifolds. Existing DP mechanisms for manifold data consider geometric properties when calibrating privacy perturbations, but they largely fail to capture variations in data density within datasets, leading to biased perturbations and suboptimal privacy-utility trade-offs due to heterogeneous data distributions. In this paper, we propose a novel density-aware differential privacy mechanism on Riemannian manifolds, referred to as Conformal-DP, that leverages conformal transformations to calibrate perturbations based on local densities and to induce a density-balanced geometry. We prove that our mechanism satisfies $ε$-differential privacy on any complete Riemannian manifold under mild regularity assumptions. In addition, we derive a closed-form expected geodesic error bound that depends only on the underlying data density ratio and is independent of global curvature. Our empirical results on synthetic and real-world datasets demonstrate that the proposed Conformal-DP mechanism substantially improves the privacy-utility trade-off in heterogeneous data distribution settings, with worst-case performance comparable to state-of-the-art manifold DP mechanisms that assume uniformly distributed data.

翻译：差分隐私（DP）正越来越多地被应用于位于复杂高维流形上的非欧几里得数据。现有面向流形数据的DP机制在校准隐私扰动时会考虑几何属性，但大多无法捕捉数据集内部的数据密度变化，导致由异构数据分布引起的偏置扰动及欠优的隐私-效用权衡。本文提出一种新颖的黎曼流形上密度感知差分隐私机制，称为Conformal-DP，该机制利用共形变换基于局部密度校准扰动，并诱导出密度平衡的几何结构。我们证明在温和的正则性假设下，该机制在任何完备黎曼流形上均满足ε-差分隐私。此外，我们导出了仅依赖于底层数据密度比且与全局曲率无关的闭合形式期望测地线误差界。在合成数据集和真实数据集上的实验结果表明，所提出的Conformal-DP机制在异构数据分布设置下显著改善了隐私-效用权衡，其最差性能与假设均匀分布数据的最先进流形DP机制相当。