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<p>Geodesic metric spaces support a variety of averaging constructions for given finite sets. Computing such averages has generated extensive interest in diverse disciplines. Here we consider the inverse problem of recognizing computationally whether or not a given point is such an average, exactly or approximately. In nonpositively curved spaces, several averaging notions, including the usual weighted barycenter, produce the same “mean set”. In such spaces, at points where the tangent cone is a Euclidean space, the recognition problem reduces to Euclidean projection onto a polytope. Hadamard manifolds comprise one example. Another consists of CAT(0) cubical complexes, at interior points of maximal cells: the recognition problem is harder for general points, but we present an efficient semidefinite-programming-based algorithm.</p>

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Recognizing Weighted Means in Geodesic Spaces

  • 期刊论文

  • 发表日期：

    2025年9月26日

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  • Ariel Goodwin,
  • Adrian S. Lewis,
  • Genaro López-Acedo,
  • Adriana Nicolae

摘要

Geodesic metric spaces support a variety of averaging constructions for given finite sets. Computing such averages has generated extensive interest in diverse disciplines. Here we consider the inverse problem of recognizing computationally whether or not a given point is such an average, exactly or approximately. In nonpositively curved spaces, several averaging notions, including the usual weighted barycenter, produce the same “mean set”. In such spaces, at points where the tangent cone is a Euclidean space, the recognition problem reduces to Euclidean projection onto a polytope. Hadamard manifolds comprise one example. Another consists of CAT(0) cubical complexes, at interior points of maximal cells: the recognition problem is harder for general points, but we present an efficient semidefinite-programming-based algorithm.