How to Protect 3D Rotations with Differential Privacy?

Differential privacy (DP) is a mathematical tool that ensures published data does not reveal sensitive details. Since 2006 it has been applied to text, images and many other domains. The newer "metric" DP version lets the protection level depend on the distance between data points, perfect for complex spaces. In this paper the authors extend metric DP to the group of three‑dimensional rotations, called SO(3). Imagine sharing a drone's orientation without exposing its exact flight path. They propose two mechanisms: a Laplace one on SO(3) and another based on the Bingham distribution. The first adds noise following a Laplace law adapted to rotations, guaranteeing σ‑privacy with respect to the geodesic distance (the shortest path on the sphere). The second exploits SO(3)'s antipodal symmetry—each rotation is indistinguishable from its opposite—and achieves σ̃‑privacy with σ̃ = π/4. Simulations show the Bingham mechanism preserves more useful accuracy, while the Laplace version is easier to implement. These methods could let us share orientation data in robotics or virtual reality without sacrificing user privacy.

In differential privacy, the "sensitivity" of a function is the maximum change in its output when a single record in the dataset is altered. For data on SO(3) the sensitivity is defined with respect to the geodesic metric, i.e., the length of the shortest arc between two rotations on the three‑dimensional sphere. The Laplace mechanism on SO(3) employs a noise distribution with density proportional to exp(-σ d(g, g')), where d is the geodesic distance and σ is the privacy parameter. This distribution is isotropic, meaning it is invariant under rotations, ensuring the noise does not introduce directional bias. The Bingham distribution, on the other hand, is a probability law on S^3 (the quaternion sphere) characterized by a concentration matrix that controls dispersion around a preferred axis. Transferring privacy from S^3 to SO(3) requires handling antipodal symmetry: each point q and -q represent the same rotation. The Bingham mechanism respects this symmetry by choosing a concentration matrix invariant under the transformation q → -q, thus achieving σ̃‑privacy with σ̃ = π/4. Practically, the choice between the two mechanisms hinges on the privacy‑utility trade‑off: Laplace is more robust to extreme variations but may degrade orientation precision, whereas Bingham preserves better angular fidelity at the cost of a more complex implementation and careful calibration of the concentration matrix.