Supplemental to Shape and Structure Preserving Differential Privacy 1 Proof of Lemma 1

The equality is due to the fact that the parallel transport map is an isometry between tangent spaces and fixes the origin; the first inequality follows from the reverse triangle inequality, while the last follows from the upper bound on the Hessian of U in (2). U ( x,D) is the zero gradient vector field under the isometric parallel transport. Simulations pertaining to the sphere and Kendall shape space are done on a desktop computer with an Intel Xeon processor at 3.60GHz with 31.9 Simulations pertaining to symmetric positive-definite matrices were performed on the Pennsylvania State University's Institute for Computational and Data Sciences' Roar supercomputer. All simulations are done in Matlab.