Differential Privacy Over Riemannian Manifolds
This work addresses privacy-preserving data analysis for statistical summaries on manifolds, such as covariance matrices and discrete distributions, with incremental advancements in differential privacy methods.
The authors tackled the problem of releasing differentially private statistical summaries on Riemannian manifolds by extending the Laplace or K-norm mechanism using intrinsic distances and volumes, demonstrating rate optimality and improved utility compared to ignoring manifold structure.
In this work we consider the problem of releasing a differentially private statistical summary that resides on a Riemannian manifold. We present an extension of the Laplace or K-norm mechanism that utilizes intrinsic distances and volumes on the manifold. We also consider in detail the specific case where the summary is the Fréchet mean of data residing on a manifold. We demonstrate that our mechanism is rate optimal and depends only on the dimension of the manifold, not on the dimension of any ambient space, while also showing how ignoring the manifold structure can decrease the utility of the sanitized summary. We illustrate our framework in two examples of particular interest in statistics: the space of symmetric positive definite matrices, which is used for covariance matrices, and the sphere, which can be used as a space for modeling discrete distributions.