Statistical and Topological Properties of Sliced Probability Divergences S

We can now prove Theorem 1. Proof of Theorem 1. Now, let us prove the other implication, i.e. Theorem 2. Our result is thus consistent with the existing results in the literature. Next, we show that this result holds for two popular choices of kernels. We conclude that k ˆ k is positive definite, hence (S17) holds for RBF kernels.S1.4 Proof of Theorem 3 Proof of Theorem 3. We start by upper bounding the distance between two regularized measures. The desired result is obtained as a direct application of Theorems 2 and 3.S1.6