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A Lower Bound for Estimating Fréchet Means

Shayan Hundrieser, Benjamin Eltzner, Stephan F. Huckemann

Abstract

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 $P$. Implications were previously identified for empirical plug-in estimators as part of the phenomenon \emph{finite sample smeariness}. 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.

A Lower Bound for Estimating Fréchet Means

Abstract

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 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 . Implications were previously identified for empirical plug-in estimators as part of the phenomenon \emph{finite sample smeariness}. 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.
Paper Structure (10 sections, 8 theorems, 63 equations, 1 figure)

This paper contains 10 sections, 8 theorems, 63 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Result
  3. Slow Rates and Deteriorating Constants: Examples
  4. Extrinsic Means on Manifolds
  5. Intrinsic Means on Manifolds
  6. Procrustean Means for Shape Spaces
  7. Diffusion Means on Manifolds
  8. Wasserstein Barycenters on Euclidean Spaces
  9. Discussion
  10. Deferred Proofs

Key Result

Lemma 2.4

Under ass:nonUnique, it follows for all $P\in\mathcal{P}(\mathcal{X})$, all $t\in (0,1]$ and all $x\in \mathbb{M}(P)$ that with the perturbed measure $P_{x,t} := (1-t)P + t \delta_x$.

Figures (1)

  • Figure 1: Depicting the points from \ref{['ex:non-uniqueMeans']}: $E$ (resp. $F$) represents the midpoint between $A$ and $-B$ (resp. $A$ and $B$). The set of Wasserstein barycenters for the probability measure $\frac{1}{2}(\delta_{\frac{1}{2}(\delta_{A} + \delta_{-A})} + \delta_{\frac{1}{2}(\delta_{B} + \delta_{-B})})$ is given by $\{\frac{1}{2}(\delta_{E} + \delta_{-E}), \frac{1}{2}(\delta_{F} + \delta_{-F})\}$.

Theorems & Definitions (26)

  • Definition 2.1: Fréchet $\rho$-means
  • Remark 2.2: Honesty
  • Remark 2.3: Relation to standard Fréchet mean
  • Lemma 2.4: Instability of non-uniqueness
  • proof
  • Theorem 2.5: Uniform lower bound
  • Corollary 2.6
  • Remark 2.7: Comparison with tran2021smeariness
  • Remark 2.8: Complementary contributions
  • proof : Proof of \ref{['thm:minimaxLower']}
  • ...and 16 more