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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^2\)</EquationSource> </InlineEquation>-barycenter of the quantiles on the cone of nonincreasing functions in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^2(0,1)\)</EquationSource> </InlineEquation>. Further, we provide a stability estimate in dimension one and a counterexample to uniqueness in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbb {R}^2\)</EquationSource> </InlineEquation>. 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.</p>

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Generalized Wasserstein Barycenters

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年11月7日

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  • Francesco Tornabene,
  • Marco Veneroni,
  • Giuseppe Savaré

摘要

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 \(\mathbb {R}^2\) . 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.