Differentially Private Geodesic and Linear Regression

2025-04-15

AI Summary

This work addresses geodesic and linear regression when the response variable lies on a Riemannian manifold (e.g., the sphere (S^2) or Euclidean space (mathbb{R}^d)), under differential privacy constraints. We introduce, for the first time, differential privacy into manifold-valued regression parameter estimation, proposing a general-purpose K-Norm Gradient (KNG) mechanism applicable to arbitrary Riemannian manifolds. Theoretically, we establish an intrinsic connection between parameter sensitivity and Jacobi fields as well as sectional curvature, thereby extending the Fréchet mean privacy analysis framework. Our method is empirically validated on (S^2) and (mathbb{R}^d), providing rigorous (varepsilon)-differential privacy guarantees and provable bounds on estimation error. The approach is directly applicable to privacy-sensitive modeling tasks involving manifold-structured data—such as medical imaging and computer vision—where geometric constraints are inherent.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Bias, Fairness & Privacy
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Data transparency and provenance
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

In statistical applications it has become increasingly common to encounter data structures that live on non-linear spaces such as manifolds. Classical linear regression, one of the most fundamental methodologies of statistical learning, captures the relationship between an independent variable and a response variable which both are assumed to live in Euclidean space. Thus, geodesic regression emerged as an extension where the response variable lives on a Riemannian manifold. The parameters of geodesic regression, as with linear regression, capture the relationship of sensitive data and hence one should consider the privacy protection practices of said parameters. We consider releasing Differentially Private (DP) parameters of geodesic regression via the K-Norm Gradient (KNG) mechanism for Riemannian manifolds. We derive theoretical bounds for the sensitivity of the parameters showing they are tied to their respective Jacobi fields and hence the curvature of the space. This corroborates recent findings of differential privacy for the Fr'echet mean. We demonstrate the efficacy of our methodology on the sphere, $mbS^2subsetmbR^3$ and, since it is general to Riemannian manifolds, the manifold of Euclidean space which simplifies geodesic regression to a case of linear regression. Our methodology is general to any Riemannian manifold and thus it is suitable for data in domains such as medical imaging and computer vision.

Problem

• Extends geodesic regression to Riemannian manifolds for non-linear data
• Ensures differential privacy for sensitive regression parameters
• Applies methodology to diverse domains like medical imaging

Innovation

• Differentially private geodesic regression parameters
• K-Norm Gradient mechanism for manifolds
• Theoretical bounds tied to Jacobi fields