Metric Differential Privacy on the Special Orthogonal Group SO(3)

Differential privacy (DP) is an important framework to provide strong theoretical guarantees on the privacy and utility of released data. Since its introduction in 2006, DP has been applied to various data types and domains. More recently, the introduction of metric differential privacy has improved the applicability and interpretability of DP in cases where the data resides in more general metric spaces. In metric DP, indistinguishability of data points is modulated by their distance. In this work, we demonstrate how to extend metric differential privacy to datasets representing three-dimensional rotations in SO(3) through two mechanisms: a Laplace mechanism on SO(3), and a novel privacy mechanism based on the Bingham distribution. In contrast to other applications of metric DP to directional data, we demonstrate how to handle the antipodal symmetry inherent in SO(3) while transferring privacy from S3 to SO(3). We show that the Laplace mechanism fulfills ϵϕ-privacy, where ϕ is the geodesic metric on SO(3), and that the Bingham mechanism fulfills ϵ˜ϕ-privacy with ϵ˜=π4ϵ. Through a simulation study, we compare the distribution of samples from both mechanisms and argue about their respective privacy–utility tradeoffs.