Relative Optimal Transport
Peter Bubenik, Alex Elchesen
TL;DR
This work develops a comprehensive theory of relative optimal transport on metric pairs $(X,d,A)$, where $A$ serves as a reservoir for mass and total masses may differ or be infinite. It builds a robust foundation using Riesz cones, relative measure spaces, and Lip(X,A) duality, culminating in relative Monge–Kantorovich and Kantorovich–Rubinstein dualities and a relative $1$-Wasserstein norm $ rm{\mu-\nu}_{KR}=W_1(\mu,\nu)$. The authors introduce 1-finite and locally 1-finite Radon measures on metric pairs, establish representation theorems linking these measures to Lipschitz functionals, and define relative $W_p$ distances through cost functions $ar{d}$ and $d_p$. Existence of optimal couplings is guaranteed under boundedly compact hypotheses, and a full duality framework ties primal transport to operator norms on Lipschitz spaces. The resulting framework unifies unbalanced transport with Lipschitz duality, providing tools potentially impactful for applications such as topological data analysis and beyond, where mass may be created or annihilated relative to a reservoir.
Abstract
We develop a theory of optimal transport relative to a distinguished subset, which acts as a reservoir of mass, allowing us to compare measures of different total variation. This relative transportation problem has an optimal solution and we obtain relative versions of the Kantorovich-Rubinstein norm, Wasserstein distance, Kantorovich-Rubinstein duality and Monge-Kantorovich duality. We also prove relative versions of the Riesz-Markov-Kakutani theorem, which connect the spaces of measures arising from the relative optimal transport problem to spaces of Lipschitz functions. For a boundedly compact Polish space, we show that our relative 1-finite real-valued Radon measures with relative Kantorovich-Rubinstein norm coincide with the sequentially order continuous dual of relative Lipschitz functions with the operator norm. As part of our work we develop a theory of Riesz cones that may be of independent interest.