[@lloyd1982least; @forgy1965cluster] for solving K-means problems. From then, researchers have proposed different formulations and algorithms such as spectral clustering

and density-based clustering

. Mixture modeling, especially Gaussian mixture modeling, is also considered to be a robust solution to clustering problems

. Hierarchical clustering

and co-clustering

also attracted much attention in the machine learning community. [@xu2005survey] surveys some classic clustering algorithms.

The term “*geometric clustering*” appeared in the early literature, such as [@murtagh1983survey; @quigley2000fade], referring to clustering samples into subspaces according to their location in the metric space, usually the Euclidean space. In [@applegate2011unsupervised], the authors discuss the connection between K-means and another famous problem – the OT distance, or the Wasserstein distance.

The transportation problem has attracted many mathematicians since its very birth. Monge first raised the problem [@monge1781memoire] as finding a measure-preserving map between probability measures; Kantorovich extended the problem to finding a joint probability measure [@kantorovich1942translocation]; Brenier further connected the OT problem to fluid dynamics and convex geometry [@brenier1991polar]. It’s early applications include comparing 1D histograms for image retrieval [@rubner2000earth]. Thanks to efficient OT solvers, e.g., [@cuturi2013sinkhorn],

OT has become a popular tool in machine learning with which we compare distributional data.

Meanwhile, by regarding the OT distance 