Differentially Private Fr\'echet Mean on the Manifold of Symmetric Positive Definite (SPD) Matrices with log-Euclidean Metric

TL;DR

This paper introduces a new differentially private mechanism for computing the Fréchet mean on the SPD manifold with log-Euclidean metric, offering improved utility and efficiency over existing methods, with applications in privacy-sensitive fields like medical imaging.

Contribution

It proposes the tangent Gaussian mechanism, a novel, simple, and fast method for differential privacy on SPD manifolds with log-Euclidean metric, outperforming previous approaches.

Findings

The tangent Gaussian mechanism achieves better utility than existing methods.

It is computationally efficient and suitable for real-world applications.

Extensive experiments confirm the advantages of the proposed method.

Abstract

Differential privacy has become crucial in the real-world deployment of statistical and machine learning algorithms with rigorous privacy guarantees. The earliest statistical queries, for which differential privacy mechanisms have been developed, were for the release of the sample mean. In Geometric Statistics, the sample Fr\'echet mean represents one of the most fundamental statistical summaries, as it generalizes the sample mean for data belonging to nonlinear manifolds. In that spirit, the only geometric statistical query for which a differential privacy mechanism has been developed, so far, is for the release of the sample Fr\'echet mean: the \emph{Riemannian Laplace mechanism} was recently proposed to privatize the Fr\'echet mean on complete Riemannian manifolds. In many fields, the manifold of Symmetric Positive Definite (SPD) matrices is used to model data spaces, including in medical imaging where privacy requirements are key. We propose a novel, simple and fast mechanism - the \emph{tangent Gaussian mechanism} - to compute a differentially private Fr\'echet mean on the SPD manifold endowed with the log-Euclidean Riemannian metric. We show that our new mechanism has significantly better utility and is computationally efficient -- as confirmed by extensive experiments.