Subspace Detours Meet Gromov–Wasserstein
Round 1
Reviewer 1 Report
The article engages with the problem of optimal cost planning, and the subspace detour approach and the purpose of the paper is to extend this category of methods to the Gromov-Wasserstein problem, which is a particular type of optimal transport distance problem that involves the specific geometry of each distribution. After deriving the associated formalism and properties the authors provide an experimental illustration on a shape matching problem. The authors also discuss a specific cost for which they show connections with the Knothe-Rosenblatt rearrangement.
In particular, the authors depict how to adapt the definition of subspace optimal plans for different subspaces and based on this extension they are able to extend subspace detours from Muzellec and Cuturi with Gromov-Wasserstein costs.
The article deals with an interesting problem, with various applications and the techniques proposed seem to be suitable to be applied to other problems too. Moreover, the constructions proposed are non trivial and correct as far as I was able to check. Therefore I consider the article a worth contribution to the Algorithms journal with a potential number of applications to be further explored.
Author Response
We thank the reviewer for carefully reading the paper and for considering the article a worth contribution.
Reviewer 2 Report
My comments are in the attached file.
Comments for author File: Comments.pdf
Author Response
Please see the attachment.
Author Response File: Author Response.pdf