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Barycentric Weak Inner-Product Gromov-Wasserstein

arXiv · AI, language, vision and robotics · article · Aug 25, 2026 · UTC

Gromov-Wasserstein (GW) compares distributions through relations within each space. This pointwise comparison can be too sensitive in one-to-many settings, where several target outcomes refine one source state and their mean carries the geometry of interest. We introduce a weak GW framework that compares source relations with relations between the target conditional laws induced by a coupling. For inner-product relations, we retain the conditional means $m_π(x)=\mathbb{E}_π[Y\mid X=x]$. The resulting barycentric weak inner-product GW (wIGW) satisfies $\mathrm{wIGW}_{\mathrm{bar}}^2(μ,ν)=\inf_{

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