I am reading the survey of the relationships between metrics of distributions (see https://arxiv.org/pdf/math/0209021.pdf for the paper).
The general results show that for general distributions, we cannot upper bound the total variation by Wasserstein distance. Many answers in MO give the same intuitive counterexample: consider a discrete distribution and a continuous distribution.
However, assuming the two underlying distributions are both continuous (for example consider the simpliest when they are both Gaussian mixtures with zeros mean or just Gaussians with zero mean), can we find such an upper bound?
Any related suggestions or comments are welcomed!
Best Answer
No. One should realize that the transportation and the total variation distances metrize two quite different topologies. Even if the measures are equivalent (i.e., absolutely continuous with respect to each other), one can still easily have examples when the transportation distance is arbitrarily close to 0, whereas the total variation distance is arbitrarily close to 2. Bounding the Radon-Nikodym derivatives is not of much help either as, once again, one can easily have examples of measures arbitrarily close in the transportation metric, whereas their total variation distance will be the one dictated by the density bounds. However, if you talk about measures from a certain specific class, then the situation may be different.