Abstract: Kernel mean embeddings and maximum mean discrepancies (MMD) associated with positive semi-definite kernels are important tools in machine learning that allow to compare probability measures and sample distributions. Two kernels are said equivalent if their associated MMDs are equal. We characterize the equivalence of kernels in terms of their variogram and deduce that MMDs are in one to one correspondance with negative semi-definite functions. As a consequence, we provide a full characterization of translation invariant MMDs on R d that are parametrized by a spectral measure and a semi-definite symmetric matrix. Furthermore, we investigate the connections between translation invariant MMDs and Wasserstein distances on R d. We show in particular that convergence with respect to the MMD associated with the Energy Kernel of order α ∈ (0, 1) implies convergence with respect to the Wasserstein distance of order β < α. We also provide examples of kernels metrizing the Wasserstein space of order α ≥ 1.
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