Numerical methods for matching for teams and Wasserstein barycenters

• Published in: ESAIM Mathematical Modelling and Numerical Analysis Vol. 49; no. 6; pp. 1621 - 1642
• Main Authors: Carlier, Guillaume, Oberman, Adam, Oudet, Edouard
• Format: Journal Article
• Language: English
• Published: Les Ulis EDP Sciences 01.11.2015; Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP
• Series: Special Issue - Optimal Transport
• Subjects: 49M29; 90C05; Algorithms; Center of gravity; duality; Image processing; Linear programming; Matching; Matching for teams; Mathematics; Numerical methods; numerical methods for nonsmooth convex minimization; Optimization; Optimization and Control; Wasserstein barycenters; matching for teams; linear programming; Wasserstein barycenters; duality; numerical methods for nonsmooth convex minimization
• ISSN: 0764-583X; 2822-7840; 1290-3841; 1290-3841; 2804-7214
• DOI: 10.1051/m2an/2015033

Equilibrium multi-population matching (matching for teams) is a problem from mathematical economics which is related to multi-marginal optimal transport. A special but important case is the Wasserstein barycenter problem, which has applications in image processing and statistics. Two algorithms are presented: a linear programming algorithm and an efficient nonsmooth optimization algorithm, which applies in the case of the Wasserstein barycenters. The measures are approximated by discrete measures: convergence of the approximation is proved. Numerical results are presented which illustrate the efficiency of the algorithms.