Conformal-DP: Differential Privacy on Riemannian Manifolds via Conformal Transformation

2025-04-29

AI Summary

Existing manifold-aware differential privacy (DP) methods assume uniform data distribution on the manifold, ignoring real-world density heterogeneity—leading to imbalanced noise allocation and degraded privacy-utility trade-offs. This work proposes a density-aware conformal differential privacy mechanism: it estimates local density via kernel density estimation and employs a conformal transformation to reparameterize the Riemannian metric, thereby equalizing sensitivity across regions while preserving intrinsic manifold geometry. We introduce conformal transformations to DP for the first time, establish a theoretical framework for the conformal factor, and prove strict ε-DP guarantees on arbitrary complete Riemannian manifolds, along with a curvature-independent geodesic error bound dependent solely on the maximum density ratio. Empirical evaluation on heterogeneous manifold data—including diffusion tensor MRI and shape analysis—demonstrates significant utility gains while maintaining rigorous privacy guarantees.

Technology Category

• Machine Learning: Privacy
• Data Mining & Knowledge Management: Representing, Reasoning, and Using Provenance, Trust
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Data transparency and provenance
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Responsible Web: Data and user privacy-enhancing technologies for the Web

Abstract

Differential Privacy (DP) has been established as a safeguard for private data sharing by adding perturbations to information release. Prior research on DP has extended beyond data in the flat Euclidean space and addressed data on curved manifolds, e.g., diffusion tensor MRI, social networks, or organ shape analysis, by adding perturbations along geodesic distances. However, existing manifold-aware DP methods rely on the assumption that samples are uniformly distributed across the manifold. In reality, data densities vary, leading to a biased noise imbalance across manifold regions, weakening the privacy-utility trade-offs. To address this gap, we propose a novel mechanism: Conformal-DP, utilizing conformal transformations on the Riemannian manifold to equalize local sample density and to redefine geodesic distances accordingly while preserving the intrinsic geometry of the manifold. Our theoretical analysis yields two main results. First, we prove that the conformal factor computed from local kernel-density estimates is explicitly data-density-aware; Second, under the conformal metric, the mechanism satisfies $ varepsilon $-differential privacy on any complete Riemannian manifold and admits a closed-form upper bound on the expected geodesic error that depends only on the maximal density ratio, not on global curvatureof the manifold. Our experimental results validate that the mechanism achieves high utility while providing the $ varepsilon $-DP guarantee for both homogeneous and especially heterogeneous manifold data.

Problem

• Extends DP to non-uniform data on Riemannian manifolds
• Addresses biased noise in manifold-aware DP methods
• Ensures privacy-utility balance via conformal transformations

Innovation

• Conformal transformations equalize local sample density
• Redefines geodesic distances preserving manifold geometry
• Data-density-aware conformal factor ensures differential privacy