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<p>Fréchet means, conceptually appealing, generalize the Euclidean expectation to general metric spaces. We explore how well Fréchet means can be estimated from independent and identically distributed samples and uncover a fundamental limitation: In the vicinity of a probability distribution <i>P</i> with nonunique means, independent of sample size, it is not possible to uniformly estimate Fréchet means below a precision determined by the diameter of the set of Fréchet means of <i>P</i>. Our findings thus confirm inevitable statistical challenges in the estimation of Fréchet means on metric spaces for which there exist distributions with nonunique means. Illustrating the relevance of our lower bound, examples of extrinsic, intrinsic, Procrustes, diffusion and Wasserstein means showcase either deteriorating constants or slow convergence rates of empirical Fréchet means for samples near the regime of nonunique means.</p>

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A lower bound for estimating Fréchet means

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年6月22日

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  • Shayan Hundrieser,
  • Benjamin Eltzner,
  • Stephan F. Huckemann

摘要

Fréchet means, conceptually appealing, generalize the Euclidean expectation to general metric spaces. We explore how well Fréchet means can be estimated from independent and identically distributed samples and uncover a fundamental limitation: In the vicinity of a probability distribution P with nonunique means, independent of sample size, it is not possible to uniformly estimate Fréchet means below a precision determined by the diameter of the set of Fréchet means of P. Our findings thus confirm inevitable statistical challenges in the estimation of Fréchet means on metric spaces for which there exist distributions with nonunique means. Illustrating the relevance of our lower bound, examples of extrinsic, intrinsic, Procrustes, diffusion and Wasserstein means showcase either deteriorating constants or slow convergence rates of empirical Fréchet means for samples near the regime of nonunique means.