This is a very specific question about this optimal transport paper, which addresses estimating the $2-$Wasserstein distance. The authors write the transport cost explicitly for a specific type of coupling, and then proceed to omit terms in that expression when actually giving an estimate of the Wasserstein distance and I have no (good) idea why. They do this as follows (you don't need to fully understand what the terms mean, just why they exclude the 2nd sum in the estimate):
Then they proceed to estimate the argmin of the above transport cost using:
where $P$ = $H$ = the barycenter, which is defined as
since the objective function of the barycenter problem is similar to the expression in prop. 4.2 (this is shown in proposition 4.3). This is (somewhat) reasonable. However, then they go on to say the following:
where the cost is just
Obviously this is just the first of the two sum terms and I'm confused as to why this is a valid idea. I know they've made the statement that the second term is an "intra-cluster variance term" and the first is a "transport term", but don't they sum up to be the transport cost? So we shouldn't be ignoring one of them in our estimate? Thanks if you're able to understand what's going on here, or have any of the insight I'm (clearly) missing.