Robust clustering tools based on optimal transportation
A robust clustering method for probabilities in Wasserstein space is introduced. This new "trimmed $k$-barycenters" approach relies on recent results on barycenters in Wasserstein space that allow intensive computation, as required by clustering algorithms. The possibility of trimming the most discrepant distributions results in a gain in stability and robustness, highly convenient in this setting. As a remarkable application we consider a parallelized estimation setup in which each of $m$ units processes a portion of the data, producing an estimate of $k$-features, encoded as $k$ probabilities. We prove that the trimmed $k$-barycenter of the $m\times k$ estimates produces a consistent aggregation. We illustrate the methodology with simulated and real data examples. These include clustering populations by age distributions and analysis of cytometric data.