Improved Linear Programs for Discrete Barycenters

Abstract: Discrete barycenters are the optimal solutions to mass transport problems for a set of discrete measures. Such transport problems arise in many applications of operations research and statistics. The best known algorithms for exact barycenters 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 reduced linear program that contains dramatically fewer variables compared to previous formulations. Second, we recall a proof from the literature, which lends itself to a linear program that has not been considered for computations. We show that this second formulation is the best model for data in general position. Third, we combine the two programs into a single hybrid model that retains the best properties of both formulations for partially structured data. We study these models through an analysis of their scaling in size, the hardness of the required preprocessing, and computational experiments. In doing so, we show that each of the improved linear programs becomes the best model for different types of data.