Supplemental to Differential Privacy Over Riemannian Manifolds 1 Simulation details

We use a gradient descent algorithm to compute the Fr echet mean of a sample D ={x1,x2,...,xn}. We initialize the mean ˆµ0 at any data point, take a small step in the average direction of the gradient of energy functional F2:M R, and iterate. Then, the estimate of the Fr echet mean at iterate k is ˆµk = expˆµk 1(tkvk) where tk (0,1] is the step size. The algorithm is assumed to have converged once the change in the mean across subsequent steps is no longer significant, measured using the intrinsic distance ρ on M; that is, the algorithm terminates if ρ(µk,µk 1)<λ for some pre-specifiedλ>0. Wechoosethestepsizetk =0.5andλ=10 5. Inaddition, one could set a maximum number of iterations for situations when the mean oscillates between local optima, and we set this at 500 but note that in our settings the algorithm typically converges in fewer than 200 iterations.