Wasserstein distances define a metric between probability measures on arbitrary metric spaces, including meta-measures (measures over measures). The resulting Wasserstein over Wasserstein (WoW) distance is a powerful, but computationally costly tool for comparing datasets or distributions over images and shapes. To lower the computational burden, we propose to leverage the isometry between the 1d Wasserstein space and the quantile functions. For this purpose, we introduce a general sliced Wasserstein framework for arbitrary Banach spaces. Due to the 1d Wasserstein isometry, this framework defines a sliced distance between 1d meta-measures via infinite-dimensional projections, parametrized by Gaussian processes. Combining this 1d construction with a second slicing of the underlying domain yields the double-sliced Wasserstein (DSW) metric for general meta-measures. Numerical experiments on datasets, shapes, and images validate DSW as a scalable substitute for the WoW distance. Moreover, our functional framework can be used to introduce a novel slicing of the Gromov-Wasserstein (GW) distance, which allows to compare shapes and heterogeneous data. Former GW slicings are restricted to the Euclidean geometry and lose the desired invariance to isometries, strongly limiting their application in practice. Our novel sliced GW lower bound significantly reduces the numerical effort while remaining invariant to isometric transformations and allowing the comparison of arbitrary geometries.

Unlike classical optimal transport, weak transport costs depend nonlinearly on the

conditionallawofcouplings. Thisfeatureisessentialinproblemsinvolvingbarycenter,

conditional moments, and martingale-type constraints. Meanwhile, such conditional

dependence makes ordinary Wasserstein geometry insufficient and calls instead for an

adapted Wasserstein viewpoint. In this paper, we investigate the entropy-regularized

weak optimal transport via gradient flows in adapted Wasserstein space.

We derive, from the formal tangent structure of adapted Wasserstein space and the

projection onto the set of couplings with prescribed marginals, a coupled McKean–

Vlasov SDE. A novel and subtle term is a projection that, at each Y-location, averages

a weak-transport force that already depends on the conditional law of Y given X,

thereby preserving marginals while retaining the nonlinear weak-transport structure.

Under mild integrability and regularity assumptions, we prove weak existence and

uniqueness in law for this projected McKean–Vlasov equation. We then prove that the

flow converges, in the adapted Wasserstein topology, to the unique minimizer of the

entropic weak optimal transport problem. We also describe a particle approximation

and illustrate the dynamics on optimal transport and martingale optimal transport

examples.

Arxiv : https://arxiv.org/pdf/2605.30560

The Rochet-Choné problem is a free boundary problem connected to optimal transport, under the constraint that the solution must be convex. A principal challenge for numerical methods is the phenomenon of "bunching," where the optimal gradient map collapses regions of positive Lebesgue measure onto lower-dimensional manifolds. We present a highly scalable numerical approach that achieves high-resolution simulations in 3D. Extensions to general interaction terms $b(x,y)$ are also discussed.