  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.
