Differential privacy over Riemannian manifolds

Reimherr, Matthew; Bharath, Karthik; Soto, Carlos

Abstract

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 Frechet 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.

• Conference Name: Thirty-fifth Conference on Neural Information Processing Systems (NeurIPS 2021)
• Conference Location: Online
• Start Date: Dec 7, 2021
• End Date: Dec 10, 2021
• Acceptance Date: Sep 28, 2021
• Online Publication Date: Dec 10, 2021
• Publication Date: Dec 10, 2021
• Deposit Date: Oct 26, 2021
• Publicly Available Date: Dec 10, 2021
• Public URL: https://nottingham-repository.worktribe.com/output/6538427
• Publisher URL: https://proceedings.neurips.cc/paper/2021/hash/6600e06fe9350b62c1e343504d4a7b86-Abstract.html
• Related Public URLs: https://nips.cc/