{"paper":{"title":"A fixed-point approach to barycenters in Wasserstein space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"stat.CO","authors_text":"C. Matr\\'an, E. del Barrio, J.A. Cuesta-Albertos, Pedro C. \\'Alvarez-Esteban","submitted_at":"2015-11-17T11:41:02Z","abstract_excerpt":"Let $\\mathcal{P}_{2,ac}$ be the set of Borel probabilities on $\\mathbb{R}^d$ with finite second moment and absolutely continuous with respect to Lebesgue measure. We consider the problem of finding the barycenter (or Fr\\'echet mean) of a finite set of probabilities $\\nu_1,\\ldots,\\nu_k \\in \\mathcal{P}_{2,ac}$ with respect to the $L_2-$Wasserstein metric. For this task we introduce an operator on $\\mathcal{P}_{2,ac}$ related to the optimal transport maps pushing forward any $\\mu \\in \\mathcal{P}_{2,ac}$ to $\\nu_1,\\ldots,\\nu_k$. Under very general conditions we prove that the barycenter must be a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.05355","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}