While I think the paper makes a good contribution, there are some limitation at the present stage: - [Remark 3.1] While it has been done in previous works, I think that a deeper understanding of those cases where modelling the pushforward P in (8) as a composition of perturbation in an RKHS does not introduce an error, would increase the quality of the work. Alternatively, trying to undestand the kind of error that this parametrization introduces would be valuable too. - The analysis does not cover explicitly what happens when the input measures \beta_i are absolutely continuous and one has to rely on samples. How does the sampling part impact the bound? - The experiments are limited to toy data. There is a range of problems with real data where barycenters can be used and it would be interesting to show performance of the method in those settings too.
- The experiments are limited to toy data. There is a range of problems with real data where barycenters can be used and it would be interesting to show performance of the method in those settings too.