Differentially Private Geodesic Regression

In statistical applications it has become increasingly common to encounter data structures that live on non-linear spaces such as manifolds. For data living on such non-linear spaces geodesic regression emerged as a natural extension of linear regression where the response variable lives on a Riemannian manifold. The parameters of geodesic 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. We demonstrate the efficacy of our methodology on the sphere, $S_2\subset\mathbb{R}^3$, the space of symmetric positive definite matrices, and Kendall's planar shape space. Our methodology is general to any Riemannian manifold, and thus it is suitable for data in domains such as medical imaging and computer vision.

Many important types of data, such as medical images, brain scans, and body or object shapes, contain strong structural constraints. This makes it more challenging to study patterns in the data using standard methods. Our work develops a method to discover patterns while protecting the privacy of the people which contributed to the data. This is especially important when working with sensitive information such as patient data. We also show that the shape of the space where the data lives affects how privacy protection should be implemented. We test our approach on examples involving spheres, medical-style measurements, and shape data. This work can help researchers use complex human data more safely and responsibly.