B0743
Title: Differential privacy over Riemannian manifolds
Authors: Carlos Soto - University of Massachusetts Amherst (United States) [presenting]
Matthew Reimherr - Pennsylvania State University (United States)
Karthik Bharath - University of Nottingham (United Kingdom)
Abstract: The problem of releasing a differentially private statistical summary that resides on a Riemannian manifold is considered. It presents an extension of the Laplace, or K-norm, a mechanism that utilizes intrinsic distances and volumes while specifically considering the case where the summary is the Fr\'echet mean. The mechanism is shown to be rate optimal and depends only on the dimension of the manifold, not on the dimension of an ambient space, while also showing that ignoring the manifold structure can decrease the utility of the privatized summary. The proposed framework is illustrated 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.