Differential Privacy Over Riemannian Manifolds
2021
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 Fr\'echet 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.
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