Monge-Kantorovich Optimal Transport Through Constrictions and Flow-rate Constraints

TL;DR

This paper develops a generalized optimal transport framework incorporating flow-rate constraints at constrictions, modeled as toll stations, using a Kantorovich-type formalism with applications in machine learning.

Contribution

It introduces a more general multi-marginal transport formulation with flow-rate constraints based on crossing times, extending previous work on Monge maps and kinetic energy minimization.

Findings

Formulated a convex constraint involving crossing times for flow-rate limits.

Cast the problem as a multi-marginal optimal transport with practical computational methods.

Extended the theoretical framework to more general transportation scenarios.

Abstract

We consider the problem to transport resources/mass while abiding by constraints on the flow through constrictions along their path between specified terminal distributions. Constrictions, conceptualized as toll stations at specified points, limit the flow rate across. We quantify flow-rate constraints via a bound on a sought probability density of the times that mass-elements cross toll stations and cast the transportation scheduling in a Kantorovich-type of formalism. Recent work by our team focused on the existence of Monge maps for similarly constrained transport minimizing average kinetic energy. The present formulation in this paper, besides being substantially more general, is cast as a (generalized) multi-marginal transport problem - a problem of considerable interest in modern-day machine learning literature and motivated extensive computational analyses. An enabling feature of our formalism is the representation of an average quadratic cost on the speed of transport as a convex constraint that involves crossing times.