Supplemental to Shape and Structure Preserving Differential Privacy 1 ProofofLemma1

We first establish that x 7 U(x,D) is strong geodesically convex and derive upper and lower bounds on its Hessian within Br(p0) with r chosen as per Assumption 1. Note that when s < π2 κmax, the function s 7 hmax(s,κmax) > 0, decreasing, and bounded above by 1, while s 7 hmin(s,κmin) is bounded below by 1 and increasing. 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). To derive the lower bound on k U(x,D)kx, note that under our assumption on the radius r, the function U is stronggeodesically convexwithin Br(p0) since thefunction hmax is positive. From the lower bound on the Hessian of U in (2), for any y Br(p0), we hence obtain U(x,D) Γxy U(y,D), exp 1(x,y) x hmax(2r,κmax)ρ(x,y)2, where Γxy is the parallel transport along a geodesic from y to x; applying Cauchy-Schwarz to the inner product results in k U(x,D) Γxy U(y,D)kxρ(x,y) hmax(2r,κmax)ρ(x,y)2. Dividing both sides by ρ(x,y) and taking y = x leads to the desired lower bound, since, within the local coordinates at x, Γx x U( x,D) is the zero gradient vector field under the isometric parallel transport.