Constrained Mass Optimal Transport
This work addresses a theoretical extension of optimal transport with potential applications in machine learning and other fields, but it appears incremental as it builds on existing fluid dynamics formulations.
The paper tackles the problem of constrained optimal transport by imposing soft constraints on density and momentum fields, introducing a family of algorithms to solve saddle point problems with convergence proofs and numerical results.
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.