Variational Analysis in the Wasserstein Hierarchy
Christophe Vauthier
TL;DR
This work extends variational analysis in optimal transport to the Wasserstein hierarchy P2^(n)(M) by lifting the geometry of a Riemannian manifold M through the Wasserstein functor. It constructs a rigorous framework of velocity plans, couplings, and tangent-like structures across multiple levels, yielding a precise characterization of constant-speed geodesics via optimal velocity plans and a notion of gradient for functionals on the hierarchy. Fully deterministic velocity plans are developed to provide a Hilbert-space-like tangent structure, and a robust geodesic theory is established, including parallel transport and both forward and inverse geodesic characterizations. The paper also develops a calculus for functionals on the hierarchical spaces, including subdifferentials, gradient rules, and analyses of higher-order Wasserstein distances, enabling gradient flows and variational optimization on hierarchical distributions with potential data-structural applications in multi-layer representations.
Abstract
Let $M$ be a complete connected Riemannian manifold. For $n \geq 0$, we endow the Wasserstein space $P^{(n)}_2(M) = P_2(\ldots P_2(M)\ldots)$, equipped with the Wasserstein distance $W_2$, with a variational structure that generalizes the standard variational structure on $P_2(M)$ provided by optimal transport theory. Our approach makes use of tools from category theory to lift the geometric structure of the manifold $M$ to the spaces $P^{(n)}_2(M)$, in order to establish in a principled way a rigorous theoretical framework for variational analysis on the space $P^{(n)}_2(M)$. In particular, we obtain a precise characterization of the constant speed geodesics of the space $P^{(n)}_2(M)$ in terms of optimal velocity plans. Moreover, we introduce a notion of gradient for functionals defined on $P^{(n)}_2(M)$, which allows us to study the differentiability and the convexity of various types of such functionals.