Differentially Private Geodesic and Linear Regression
This work addresses privacy concerns in statistical applications like medical imaging and computer vision where data lives on manifolds, though it is incremental as it extends existing differential privacy techniques to a new context.
The authors tackled the problem of protecting sensitive data in geodesic regression on Riemannian manifolds by developing a differentially private method using the K-Norm Gradient mechanism, achieving theoretical bounds tied to curvature and demonstrating efficacy on the sphere and Euclidean space.
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 Fréchet mean. We demonstrate the efficacy of our methodology on the sphere, $\mbS^2\subset\mbR^3$ 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.