Multi-Marginal Optimal Transport Defines a Generalized Metric

The Optimal transport (OT) problem is rapidly finding its way into machine learning. Favoring its use are its metric properties. Many problems admit solutions that guarantee only for objects embedded in metric spaces, and the use of non-metrics can complicate them. Multi-marginal OT (MMOT) generalizes OT to simultaneously transporting multiple distributions. It captures important relations that are missed if the transport is pairwise. Research on MMOT, however, has been focused on its existence, uniqueness, practical algorithms, and the choice of cost functions. There is a lack of discussion on the metric properties of MMOT, which limits its theoretical and practical use. Here, we prove that (pairwise) MMOT defines a generalized metric. We first explain the difficulty of proving this via two negative results. Afterwards, we prove key intermediate steps and then MMOT's metric properties. Finally, we show that the generalized triangle inequality of MMOT cannot be improved.