  Maximum mean discrepancy (MMD) flows suffer from high computational costs in
large scale computations. In this paper, we show that MMD flows with Riesz
kernels $K(x,y) = - \|x-y\|^r$, $r \in (0,2)$ have exceptional properties which
allow their efficient computation. We prove that the MMD of Riesz kernels,
which is also known as energy distance, coincides with the MMD of their sliced
version. As a consequence, the computation of gradients of MMDs can be
performed in the one-dimensional setting. Here, for $r=1$, a simple sorting
algorithm can be applied to reduce the complexity from $O(MN+N^2)$ to
$O((M+N)\log(M+N))$ for two measures with $M$ and $N$ support points. As
another interesting follow-up result, the MMD of compactly supported measures
can be estimated from above and below by the Wasserstein-1 distance. For the
implementations we approximate the gradient of the sliced MMD by using only a
finite number $P$ of slices. We show that the resulting error has complexity
$O(\sqrt{d/P})$, where $d$ is the data dimension. These results enable us to
train generative models by approximating MMD gradient flows by neural networks
even for image applications. We demonstrate the efficiency of our model by
image generation on MNIST, FashionMNIST and CIFAR10.
