recent years as alternatives, thanks to their efficiency and robustness. In these works, the researchers discovered the connections between different clustering problems and the OT problem through the Wasserstein barycenter (WB) formulation which computes a “mean” of one or multiple distributions. However, most of them deliver the results as *soft assignments* that need to be further discretized.

In this paper, we propose to compute the Wasserstein barycenter based on Monge OT and explore its natural connections to different clustering problems that prefer *hard assignments*. We base our OT solver on variational principles and coin our method as variational Wasserstein barycenters. We study the metric properties of WBs and use them to explain and solve different clustering-related problems such as regularized K-means clustering, co-clustering, and vector quantization and compression. We also show its immunity to unbalanced measures and its extension to measures on spherical domains. We discuss our method from different angles through comparison with other barycenter methods. We show the advantages of Monge OT-based barycenters in solving geometric clustering problems. We are among the first few that compute Monge barycenters and discover its connections to clustering problems.

Related Work and Our Contributions {#sec:related}
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Computational clustering algorithms date back to 