Constrained Mass Optimal Transport

TL;DR

This paper introduces constrained optimal transport, extending classical methods by incorporating constraints on density and momentum, and proposes algorithms with convergence proofs and numerical validation.

Contribution

It defines a new constrained optimal transport problem using a fluid dynamics approach and develops algorithms to solve the resulting saddle point problems.

Findings

Algorithms successfully solve constrained optimal transport problems.

Convergence of the proposed algorithms is theoretically proven.

Numerical experiments demonstrate effectiveness and applicability.

Abstract

Optimal mass transport, also known as the earth mover's problem, is an optimization problem with important applications in various disciplines, including economics, probability theory, fluid dynamics, cosmology and geophysics to cite a few. Optimal transport has also found successful applications in image registration, content-based image retrieval, and more generally in pattern recognition and machine learning as a way to measure dissimilarity among data. This paper introduces the problem of constrained optimal transport. The time-dependent formulation, more precisely, the fluid dynamics approach is used as a starting point from which the constrained problem is defined by imposing a soft constraint on the density and momentum fields or restricting them to a subset of curves that satisfy some prescribed conditions. A family of algorithms is introduced to solve a class of constrained saddle point problems, which has convexly constrained optimal transport on closed convex subsets of the Euclidean space as a special case. Convergence proofs and numerical results are presented.