# Interpretation of upper bound on the Wasserstein Distance

I am trying to interpret the 2-Wasserstein distance and the upper bound on it. Let's say I have 2-Wasserstein distance between two distributions to be $$x$$, and I have an upper bound on it which gives me $$100x$$. How do I interpret this upper bound i.e. is this too loose or how good is it?

For example, if the value of $$x$$ is $$0.01$$ then $$100x$$ would be $$1$$ but is $$1$$ a significant value? Is $$1$$ as bad as $$100$$ if we look on a relative scale?

I am not sure what is the best way to evaluate the upper bound and interpret it.

• Can you cite the "2-Wasserstein" distance". How are you getting an upper bound? May 5 '21 at 0:48
• I am looking for an interpretation of the upper bound in general, the derivation of the upper bound depends on the application, shouldn't be relevant here. May 5 '21 at 14:31
• How can this be answered in general? How good or bad an upper bound is depends on what you want to do with it. So to get answers you need to include the relevant context. E.g. what are you trying to accomplish and how does the Wasserstein distance relate to it.
– g g
May 7 '21 at 12:58
• If we take accuracy loss as a metric for example, then we can easily interpret that. An upper bound of 100x in the case when actually loss x=0.01% is pretty ok. We wouldn't need a context here because we know what good or bad is in absolute terms. Is it possible to have the same interpretation for the 2-Wasserstein distance? May 8 '21 at 20:11
• @Dushyant Sahoo In general this would depend on the variation of the random variables. For an example see my answer here: stats.stackexchange.com/a/295729/150025 and consider the case where the two variance-covariance matrices are not the same. In that hypothetical example, how significant the 100x upper bound would depend on the variance-covariance matrices of the two Gaussians. May 9 '21 at 17:23