NeurIPS 2020
### Partial Optimal Transport with applications on Positive-Unlabeled Learning

### Meta Review

Overall, the reviewers were quite satisfied with the paper, which provides a conditional gradient algorithm to solve a Gromov-partial Wasserstein problem and theory and experiments to support the algorithms.
However I, the area chair, and the senior area chair have discussed this paper in detail and we have a more mitigated opinion. I particular we found several claims to be misleading and they should be changed for the final version to be accepted :
- "but when it comes with exact solutions, almost no partial formulation of neither Wasserstein nor Gromov-Wasserstein are available yet." This is not true (partial OT is a standard linear program) so this claim must be removed;
- The authors should not claim (such as in the current conclusion) to have introduced the "dummy" point technique for partial optimal transport, which is classical (R#1 gave an example of reference but this is common practice, see e.g. this other reference Pele, Werman, ICCV 2009 "Fast and Robust Earth Moverâ€™s Distances"). Overall, the contributions from the algorithmic optimal transport point of view are rather minor, and we advise the authors to emphasize more on their application to PU learning instead.
Other remarks:
- please check the grammar (in particular the abstract has several mistakes)
- the paper by Solomon et al 2016 "Entropic Metric Alignment for Correspondence Problems" is very relevant in paragraph 2 of the introduction;
- what do you mean by "xi" is bounded? (a fixed scalar is always bounded);
- the broader impact section is not meant to discuss technical facts, but rather should be about the potential societal impact of the work, if any (see author's guidelines).