  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.
