  This paper is concerned with minimax conditional independence testing. In
contrast to some previous works on the topic, which use the total variation
distance to separate the null from the alternative, here we use the Wasserstein
distance. In addition, we impose Wasserstein smoothness conditions which on
bounded domains are weaker than the corresponding total variation smoothness
imposed, for instance, by Neykov et al. [2021]. This added flexibility expands
the distributions which are allowed under the null and the alternative to
include distributions which may contain point masses for instance. We
characterize the optimal rate of the critical radius of testing up to
logarithmic factors. Our test statistic which nearly achieves the optimal
critical radius is novel, and can be thought of as a weighted multi-resolution
version of the U-statistic studied by Neykov et al. [2021].
