Fabian Altekrüger

Johannes Hertrich, Christian Wald, Fabian Altekrüger et al.

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.

Fabian Altekrüger, Johannes Hertrich

Exploiting image patches instead of whole images have proved to be a powerful approach to tackle various problems in image processing. Recently, Wasserstein patch priors (WPP), which are based on the comparison of the patch distributions of the unknown image and a reference image, were successfully used as data-driven regularizers in the variational formulation of superresolution. However, for each input image, this approach requires the solution of a non-convex minimization problem which is computationally costly. In this paper, we propose to learn two kind of neural networks in an unsupervised way based on WPP loss functions. First, we show how convolutional neural networks (CNNs) can be incorporated. Once the network, called WPPNet, is learned, it can be very efficiently applied to any input image. Second, we incorporate conditional normalizing flows to provide a tool for uncertainty quantification. Numerical examples demonstrate the very good performance of WPPNets for superresolution in various image classes even if the forward operator is known only approximately.