| Abstract: |
| The talk is about the barycenters of $N$ probability measures with respect to the $p$-Wasserstein metric ($p > 1$),
which generalizes the notion of Wasserstein barycenters for $p = 2$, introduced by Agueh and Carlier. Providing a
natural way to interpolate probability measures and computing a representative summary of input datasets, they are useful tools in data science, statistics, and image processing. This is a highly nonlinear problem but it can be rewritten as an equivalent multi-marginal optimal transport problem, paying with a (a priori) high dimension. Here we show that thanks to a new technique based on the geometric properties of the support of the optimal plan, the $p$-Wasserstein barycenters of absolutely continuous marginals are unique and absolutely continuous. This implies that the optimal MMOT plan is unique and can be parametrized as a graph over any marginal space (with a consequent dimension reduction). Some finer integrability properties of the density of the barycenter are also discussed. We present also some examples in one dimension, with emphasis on the statistical meaning of the $p$-Wasserstein barycenters and on the two natural limits $p\to1$ and $p\to\infty$. This is based on joint works
with G. Friesecke and T. Ried and L. Portinale. |
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