Lifting functionals defined on maps to measure-valued maps via optimal transport

TL;DR

This paper explores how to extend functionals from maps to measure-valued maps using optimal transport, enabling convexification and handling uncertainty in mappings.

Contribution

It introduces a method to lift functionals to measure-valued maps via optimal transport, connecting convexification with multi-marginal problems and addressing Jacobian-based functionals.

Findings

Largest convex lifting relates to a multi-marginal optimal transport problem.

Adding additivity restriction recovers classical Jacobian-dependent functionals.

The approach provides a new perspective on measure-valued map functionals.

Abstract

How can one lift a functional defined on maps from a space X to a space Y into a functional defined on maps from X into P(Y) the space of probability distributions over Y? Looking at measure-valued maps can be interpreted as knowing a classical map with uncertainty, and from an optimization point of view the main gain is the convexification of Y into P(Y). We will explain why trying to single out the largest convex lifting amounts to solve an optimal transport problem with an infinity of marginals which can be interesting by itself. Moreover we will show that, to recover previously proposed liftings for functionals depending on the Jacobian of the map, one needs to add a restriction of additivity to the lifted functional.