We deal with the task of sampling from an unnormalized Boltzmann density $\rho_D$ by learning a Boltzmann curve given by energies $f_t$ starting in a simple density $\rho_Z$. First, we examine conditions under which Fisher-Rao flows are absolutely continuous in the Wasserstein geometry. Second, we address specific interpolations $f_t$ and the learning of the related density/velocity pairs $(\rho_t,v_t)$. It was numerically observed that the linear interpolation, which requires only a parametrization of the velocity field $v_t$, suffers from a "teleportation-of-mass" issue. Using tools from the Wasserstein geometry, we give an analytical example, where we can precisely measure the explosion of the velocity field. Inspired by Máté and Fleuret, who parametrize both $f_t$ and $v_t$, we propose an interpolation which parametrizes only $f_t$ and fixes an appropriate $v_t$. This corresponds to the Wasserstein gradient flow of the Kullback-Leibler divergence related to Langevin dynamics. We demonstrate by numerical examples that our model provides a well-behaved flow field which successfully solves the above sampling task.

In inverse problems, many conditional generative models approximate the posterior measure by minimizing a distance between the joint measure and its learned approximation. While this approach also controls the distance between the posterior measures in the case of the Kullback–Leibler divergence, the same in general does not hold true for the Wasserstein distance. In this paper, we introduce a conditional Wasserstein distance via a set of restricted couplings that equals the expected Wasserstein distance of the posteriors. Interestingly, the dual formulation of the conditional Wasserstein-1 distance resembles losses in the conditional Wasserstein GAN literature in a quite natural way. We derive theoretical properties of the conditional Wasserstein distance, characterize the corresponding geodesics and velocity fields as well as the flow ODEs. Subsequently, we propose to approximate the velocity fields by relaxing the conditional Wasserstein distance. Based on this, we propose an extension of OT Flow Matching for solving Bayesian inverse problems and demonstrate its numerical advantages on an inverse problem and class-conditional image generation.

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 \smash{$O(\sqrt{d/P})$}, 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.

In this work we reduce undersampling artefacts in two-dimensional (2D) golden-angle radial cine cardiac MRI by applying a modified version of the U-net. The network is trained on 2D spatio-temporal slices which are previously extracted from the image sequences. We compare our approach to two 2D and a 3D deep learning-based post processing methods, three iterative reconstruction methods and two recently proposed methods for dynamic cardiac MRI based on 2D and 3D cascaded networks. Our method outperforms the 2D spatially trained U-net and the 2D spatio-temporal U-net. Compared to the 3D spatio-temporal U-net, our method delivers comparable results, but requiring shorter training times and less training data. Compared to the compressed sensing-based methods kt-FOCUSS and a total variation regularized reconstruction approach, our method improves image quality with respect to all reported metrics. Further, it achieves competitive results when compared to the iterative reconstruction method based on adaptive regularization with dictionary learning and total variation and when compared to the methods based on cascaded networks, while only requiring a small fraction of the computational and training time. A persistent homology analysis demonstrates that the data manifold of the spatio-temporal domain has a lower complexity than the one of the spatial domain and therefore, the learning of a projection-like mapping is facilitated. Even when trained on only one single subject without data-augmentation, our approach yields results which are similar to the ones obtained on a large training dataset. This makes the method particularly suitable for training a network on limited training data. Finally, in contrast to the spatial 2D U-net, our proposed method is shown to be naturally robust with respect to image rotation in image space and almost achieves rotation-equivariance where neither data-augmentation nor a particular network design are required.