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∈(0,2) have exceptional properties which
allow for their efficient computation. First, the MMD of Riesz kernels
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+N2) to O((M+N)log(M+N)) for two empirical
measures with M and N support points. 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(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 large scale
applications. We demonstrate the efficiency of our model by image generation on
MNIST, FashionMNIST and CIFAR10