Supervised Optimal Transport

TL;DR

Supervised Optimal Transport (sOT) introduces elementwise constraints into optimal transport problems, enabling applications requiring prohibited couplings, with efficient algorithms and demonstrated effectiveness in color transfer and barycenter problems.

Contribution

This paper formulates a constrained optimal transport framework called sOT, proves its equivalence to an $l^1$ penalized problem, and develops efficient algorithms for its entropy regularized version.

Findings

sOT effectively enforces elementwise constraints in transport plans

Demonstrated superior performance in color transfer tasks

Revealed a unique reverse and portion selection mechanism in barycenter problems

Abstract

Optimal Transport, a theory for optimal allocation of resources, is widely used in various fields such as astrophysics, machine learning, and imaging science. However, many applications impose elementwise constraints on the transport plan which traditional optimal transport cannot enforce. Here we introduce Supervised Optimal Transport (sOT) that formulates a constrained optimal transport problem where couplings between certain elements are prohibited according to specific applications. sOT is proved to be equivalent to an $l^1$ penalized optimization problem, from which efficient algorithms are designed to solve its entropy regularized formulation. We demonstrate the capability of sOT by comparing it to other variants and extensions of traditional OT in color transfer problem. We also study the barycenter problem in sOT formulation, where we discover and prove a unique reverse and portion selection (control) mechanism. Supervised optimal transport is broadly applicable to applications in which constrained transport plan is involved and the original unit should be preserved by avoiding normalization.