I am very happy to share our new work with @Theo_Lacombe_ and François-Xavier Vialard on the existence of optimal maps for the Gromov-Wasserstein problem!.

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We also conduct experiments on the quadratic 1D case, that is very little understood so far. In particular, we show that there are cases where the optimal plan is (strictly) a bimap (gif at the top of the thread), which supports the tightness of our theorem!.

We then rely on a theorem that concerns costs that are "invariant in some directions". The proof is quite visual and consists in gluing together a bunch of optimal maps while guaranteeing the overall measurability.

The key point is to observe that these GW quadratic problems can be rephrased as OT problems whose cost are parametrized by an (unknown) matrix M. Funnily enough, for (ii), the more M is degenerate, the more structure we get on the solution!.

We study two costs: (i) inner product, and (ii) squared euclidean distance, for which we respectively prove that always exists (i) an optimal map, and (ii) an optimal map or bimap.

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RT @PicardoMichel: We are excited to share our work. Incredible people, Incredible work! Proud of all of you….@DardRobin @LeprinceErwan @ju….

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