  We propose conditional flows of the maximum mean discrepancy (MMD) with the
negative distance kernel for posterior sampling and conditional generative
modeling. This MMD, which is also known as energy distance, has several
advantageous properties like efficient computation via slicing and sorting. We
approximate the joint distribution of the ground truth and the observations
using discrete Wasserstein gradient flows and establish an error bound for the
posterior distributions. Further, we prove that our particle flow is indeed a
Wasserstein gradient flow of an appropriate functional. The power of our method
is demonstrated by numerical examples including conditional image generation
and inverse problems like superresolution, inpainting and computed tomography
in low-dose and limited-angle settings.
