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abstract: 'We propose to compute Wasserstein barycenters (WBs) by solving for Monge maps with variational principle. We discuss the metric properties of WBs and explore their connections, especially the connections of Monge WBs, to K-means clustering and co-clustering. We also discuss the feasibility of Monge WBs on unbalanced measures and spherical domains. We propose two new problems – regularized K-means and Wasserstein barycenter compression. We demonstrate the use of VWBs in solving these clustering-related problems.'
bibliography:
- 'main.bib'
---

Introduction {#sec:intro}
============

Clustering distributional data according to their spatial similarities has been a core issue in machine learning.

Numerous theories and algorithms for clustering problems have been developed to help understand the structure of the data and to discover homogeneous groups

in their embedding spaces

. Clustering algorithms also apply to unsupervised learning problems that pass information from known centroids to unknown empirical samples

. Occasionally, researchers regard clustering as finding the optimal semi-discrete correspondence between distributional data or vice versa.

Optimal transportation (OT) techniques have gained increasing popularity in the past two decades for measuring the distance between distributional data as well as aligning them together. Rooted in the OT theories, several OT-based clustering algorithms have emerged in 