Improved Linear Programs for Discrete Barycenters

Abstract

Discrete barycenters are the optimal solutions to mass transport problems for a set of discrete measures. They arise in applications of operations research and statistics. The best known algorithms are based on linear programming, but these programs scale exponentially in the number of measures, making them prohibitive for practical purposes. In this paper, we improve on these algorithms. First, by using the optimality conditions to restrict the search space, we provide a better linear program that reduces the number of variables dramatically. Second, we recall a proof method from the literature, which lends itself to a linear program that has not been considered for computations. We exhibit that this second formulation is a viable, and arguably the go-to approach, for data in general position. Third, we then combine the two programs into a single hybrid model that retains the best properties of both formulations for partially structured data. We then study the models through both a theoretical analysis and computational experiments. We consider both the hardness of constructing the models and their actual solution. In doing so, we exhibit that each of the improved linear programs becomes the best, go-to approach for data of different underlying structure.