A data-driven linear-programming methodology for optimal transport
This work provides a practical, scalable algorithm for optimal transport and Wasserstein barycenters, which are important in machine learning and data science, though the novelty is incremental as it combines existing ideas (mixture models, adaptive meshes) in a new way.
The paper presents a data-driven linear-programming methodology for optimal transport that uses adaptively refined meshes to decompose the problem into a sequence of finite linear programs, enabling efficient computation of Wasserstein barycenters. The method achieves closed-form solutions for factorized components and scales via adaptive refinement.
A data-driven formulation of the optimal transport problem is presented and solved using adaptively refined meshes to decompose the problem into a sequence of finite linear programming problems. Both the marginal distributions and their unknown optimal coupling are approximated through mixtures, which decouples the problem into the the optimal transport between the individual components of the mixtures and a classical assignment problem linking them all. A factorization of the components into products of single-variable distributions makes the first sub-problem solvable in closed form. The size of the assignment problem is addressed through an adaptive procedure: a sequence of linear programming problems which utilize at each level the solution from the previous coarser mesh to restrict the size of the function space where solutions are sought. The linear programming approach for pairwise optimal transportation, combined with an iterative scheme, gives a data driven algorithm for the Wasserstein barycenter problem, which is well suited to parallel computing.