Conformal-DP: Data Density Aware Privacy on Riemannian Manifolds via Conformal Transformation

Abstract: Differential Privacy (DP) enables privacy-preserving data analysis by adding calibrated noise. While recent works extend DP to curved manifolds (e.g., diffusion-tensor MRI, social networks) by adding geodesic noise, these assume uniform data distribution. This assumption is not always practical, hence these approaches may introduce biased noise and suboptimal privacy-utility trade-offs for non-uniform data. To address this issue, we propose \emph{Conformal}-DP that utilizes conformal transformations on Riemannian manifolds. This approach locally equalizes sample density and redefines geodesic distances while preserving intrinsic manifold geometry. Our theoretical analysis demonstrates that the conformal factor, which is derived from local kernel density estimates, is data density-aware. We show that under these conformal metrics, \emph{Conformal}-DP satisfies $\varepsilon$-differential privacy on any complete Riemannian manifold and offers a closed-form expected geodesic error bound dependent only on the maximal density ratio, and not global curvature. We show through experiments on synthetic and real-world datasets that our mechanism achieves superior privacy-utility trade-offs, particularly for heterogeneous manifold data, and also is beneficial for homogeneous datasets.