Paper ID: | 4437 |
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Title: | Large-scale optimal transport map estimation using projection pursuit |

Reviewers have read the rebuttal with interest and have slightly raised their scores. The paper provides a nice addition to the current literature dealing with the estimation of W using projections onto subspaces, such as https://arxiv.org/abs/1902.00434 https://arxiv.org/abs/1901.08949 https://arxiv.org/abs/1903.03784 which the paper needs to refer and discuss if it were published. The paper is on the fence. minor comments: - in the intro you describe accurately that GANs involve computing a transport map (l.21), and provide several examples. Then, in l.27, this becomes computing an optimal transport map. These two things are however different. - l.33 "These methods, however, are not able to provide the explicit 34 form of the OTM". I am not sure the method proposed here by the authors does so as well, so there is a logical problem here. In any case, any solver outputting a coupling can generally output a suboptimal map through the so-called "barycentric projection", this has been used several times. - "The composition of all the one-dimensional maps serves as the final estimate of the target OTM" I am not sure I understand what is meant by composition. Isn't it rather a sum? - "In addition, the existing projection-based approaches usually suffer from slow convergence or even not convergent". Not sure what is meant here by convergence. - I am not sure I understand the point of section 4.1, since Sliced W was never presented as something defined to approximate OT in dimensions higher than 2 (and neither is the approach presented here in fact). Wouldn't it be better to present it as a form of implicit regularizer? Numbers in table 1 do not strike me as meaningful.