Multi-Marginal Optimal Transport Defines a Generalized Metric

Optimal transport (OT) is rapidly finding its way into machine learning. Favoring its use are its metric properties. Indeed, many problems admit solution guarantees only for objects embedded in a metric space, and the use of non-metrics can make their solving more difficult. 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, the uniqueness and structure of transports, applications, practical algorithms, and the choice of cost functions. There is a lack of discussion on the metric properties of MMOT, which critically limits its theoretical and practical use. Here, we prove that 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 prove MMOT's metric properties. Finally, we show that the generalized triangle inequality that MMOT satisfies cannot be improved.