  Computationally solving multi-marginal optimal transport (MOT) with squared
Euclidean costs for $N$ discrete probability measures has recently attracted
considerable attention, in part because of the correspondence of its solutions
with Wasserstein-$2$ barycenters, which have many applications in data science.
In general, this problem is NP-hard, calling for practical approximative
algorithms. While entropic regularization has been successfully applied to
approximate Wasserstein barycenters, this loses the sparsity of the optimal
solution, making it difficult to solve the MOT problem directly in practice
because of the curse of dimensionality. Thus, for obtaining barycenters, one
usually resorts to fixed-support restrictions to a grid, which is, however,
prohibitive in higher ambient dimensions $d$. In this paper, after analyzing
the relationship between MOT and barycenters, we present two algorithms to
approximate the solution of MOT directly, requiring mainly just $N-1$ standard
two-marginal OT computations. Thus, they are fast, memory-efficient and easy to
implement and can be used with any sparse OT solver as a black box. Moreover,
they produce sparse solutions and show promising numerical results. We analyze
these algorithms theoretically, proving upper and lower bounds for the relative
approximation error.
