A hierarchical approach to optimal transport

Abstract

A significant class of variational models in connection with matching general data structures and comparison of metric measure spaces, lead to computationally intensive dense linear assignment and mass transportation problems. To accelerate the computation we present an extension of the auction algorithm that exploits the regularity of the otherwise arbitrary cost function. The algorithm only takes into account a sparse subset of possible assignment pairs while still guaranteeing global optimality of the solution. These subsets are determined by a multiscale approach together with a hierarchical consistency check in order to solve problems at successively finer scales. While the theoretical worst-case complexity is limited, the average-case complexity observed for a variety of realistic experimental scenarios yields a significant gain in computation time that increases with the problem size. © 2013 Springer-Verlag.