We study the existence and uniqueness of the barycenter of a signed distribution of probability measures on a Hilbert space. The barycenter is found, as usual, as a minimum of a functional. In the case where the positive part of the signed measure is a singleton, we can show also uniqueness. In the one-dimensional case, we characterize the quantile function of the unique minimum as the orthogonal projection of the L^2-barycenter of the quantiles on the cone of nonincreasing functions in L^2(0,1). Further, we provide a stability estimate in dimension one and a counterexample to uniqueness in R^2. Finally, we address the consistency of the barycenters and we prove that barycenters of a sequence of approximating measures converge (up to subsequences) to a barycenter of the limit measure.
Generalized Wasserstein Barycenters
Tornabene, Francesco;Veneroni, Marco
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2025-01-01
Abstract
We study the existence and uniqueness of the barycenter of a signed distribution of probability measures on a Hilbert space. The barycenter is found, as usual, as a minimum of a functional. In the case where the positive part of the signed measure is a singleton, we can show also uniqueness. In the one-dimensional case, we characterize the quantile function of the unique minimum as the orthogonal projection of the L^2-barycenter of the quantiles on the cone of nonincreasing functions in L^2(0,1). Further, we provide a stability estimate in dimension one and a counterexample to uniqueness in R^2. Finally, we address the consistency of the barycenters and we prove that barycenters of a sequence of approximating measures converge (up to subsequences) to a barycenter of the limit measure.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
