Differential Privacy (DP) enables privacy-preserving data analysis by adding calibrated noise. While recent works extend DP to curved manifolds such as diffusion-tensor MRI or social networks by adding geodesic noise, these assume uniform data distribution and are not always practical. Hence, these approaches may introduce biased noise and suboptimal privacy-utility tradeoffs for non-uniform data. To address these shortcomings, we develop a density-aware differential privacy mechanism based on conformal transformations over Riemannian manifolds, which calibrates perturbations according to local density while preserving intrinsic geometric structure. We construct the conformal factor based on local kernel density estimates and establish that it inherently adapts to variations in data density. Our mechanism achieves a local balance of sample density and redefines geodesic distances while faithfully preserving the intrinsic geometry of the underlying manifold. We demonstrate that, through conformal transformation, our mechanism satisfies epsilon-differential privacy on any complete Riemannian manifold and derives a closed-form expected geodesic error bound that is contingent solely on the maximal density ratio, independent of global curvature. Empirical results on synthetic and real-world datasets demonstrate that our mechanism substantially improves the privacy-utility tradeoff in heterogeneous manifold settings and remains on par with state-of-the-art approaches when data are uniformly distributed.

翻译：差分隐私（DP）通过添加校准噪声实现隐私保护的数据分析。虽然近期研究通过添加测地噪声将DP扩展到弯曲流形（如扩散张量MRI或社交网络），但这些方法假设数据均匀分布且并不总是实用。因此，对于非均匀数据，这些方法可能引入有偏噪声并导致次优的隐私-效用权衡。为解决这些缺陷，我们基于黎曼流形上的共形变换开发了一种密度感知差分隐私机制，该机制根据局部密度校准扰动，同时保持内在几何结构。我们基于局部核密度估计构建共形因子，并证明其能自适应数据密度的变化。我们的机制实现了样本密度的局部平衡，重新定义了测地距离，同时忠实保持了底层流形的内在几何结构。我们证明，通过共形变换，该机制在任何完备黎曼流形上满足ε-差分隐私，并推导出仅取决于最大密度比、与全局曲率无关的闭式期望测地误差界。在合成和真实数据集上的实证结果表明，该机制在异构流形设置中显著改善了隐私-效用权衡，且在数据均匀分布时与最先进方法性能相当。