Differentially private inference framework of Riemannian manifold data

TL;DR

This paper introduces a new differentially private inference framework for Riemannian manifold data, including mechanisms for mean and variance estimation, with theoretical guarantees and real-world applications.

Contribution

It develops tailored DP mechanisms for manifold-valued data, establishing their statistical properties and demonstrating practical effectiveness on real datasets.

Findings

Proposed DP mechanisms for Fréchet mean and variance with geometric calibration.

Proved consistency and CLTs for the DP estimators.

Validated methods on medical and sociological datasets.

Abstract

We propose a novel and systematic differentially private (DP) inference framework for non-Euclidean data. First, we design two types of DP mechanisms for the Fr\'echet mean and variance with i.i.d. Riemannian manifold-valued data, tailored to different geometric structures and accompanied by analytic privacy budgets calibrated to the geometry of the underlying manifold. Second, we establish the consistency and central limit theorems (CLTs) of the proposed DP estimators, enabling a suite of statistical inference procedures under privacy protection. Furthermore, we provide comprehensive implementation guidelines and feasible procedures, including consistent DP estimators of the asymptotic variance in the CLTs. Extensive numerical experiments support the proposed methodologies. Finally, we demonstrate the effectiveness of our approach on real-world medical image and sociological datasets lying on two representative manifolds.