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

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