Smoothed Distance Kernels for MMDs and Applications in Wasserstein Gradient Flows

Negative distance kernels $K(x,y) := - \|x-y\|$ were used in the definition of maximum mean discrepancies (MMDs) in statistics and lead to favorable numerical results in various applications. In particular, so-called slicing techniques for handling high-dimensional kernel summations profit from the simple parameter-free structure of the distance kernel. However, due to its non-smoothness in $x=y$, most of the classical theoretical results, e.g. on Wasserstein gradient flows of the corresponding MMD functional do not longer hold true. In this paper, we propose a new kernel which keeps the favorable properties of the negative distance kernel as being conditionally positive definite of order one with a nearly linear increase towards infinity and a simple slicing structure, but is Lipschitz differentiable now. Our construction is based on a simple 1D smoothing procedure of the absolute value function followed by a Riemann-Liouville fractional integral transform. Numerical results demonstrate that the new kernel performs similarly well as the negative distance kernel in gradient descent methods, but now with theoretical guarantees.