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On the Uniqueness of Fréchet Means for Polytope Norms

Roan Talbut, Andrew McCormack, Anthea Monod

Abstract

Fréchet means are a popular type of average for non-Euclidean datasets, defined as those points which minimise the average squared distance to a set of data points. We consider the behaviour of sample Fréchet means on normed spaces whose unit ball is a polytope; this setting is rarely covered by existing literature on Fréchet means, which focuses on smooth spaces or spaces with bounded curvature. We study the geometry of the set of Fréchet means over polytope normed spaces, with a focus on dimension and probabilistic conditions for uniqueness. In particular, we provide a geometric characterisation of the threshold sample size at which Fréchet means have a positive probability of being unique, and we prove that this threshold is at most one more than the dimension of our space. We are able to use this geometric characterisation to compute the unique Fréchet mean sample threshold in the case of the $\ell_\infty$ and $\ell_1$ norms.

On the Uniqueness of Fréchet Means for Polytope Norms

Abstract

Fréchet means are a popular type of average for non-Euclidean datasets, defined as those points which minimise the average squared distance to a set of data points. We consider the behaviour of sample Fréchet means on normed spaces whose unit ball is a polytope; this setting is rarely covered by existing literature on Fréchet means, which focuses on smooth spaces or spaces with bounded curvature. We study the geometry of the set of Fréchet means over polytope normed spaces, with a focus on dimension and probabilistic conditions for uniqueness. In particular, we provide a geometric characterisation of the threshold sample size at which Fréchet means have a positive probability of being unique, and we prove that this threshold is at most one more than the dimension of our space. We are able to use this geometric characterisation to compute the unique Fréchet mean sample threshold in the case of the and norms.
Paper Structure (19 sections, 28 theorems, 102 equations, 8 figures, 1 algorithm)

This paper contains 19 sections, 28 theorems, 102 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Keywords: polytope geometry, Fréchet means, polytope norms, uniqueness, small samples.
  2. Introduction
  3. Notation.
  4. Polytope Geometry
  5. Polytopes
  6. Polytope Norms
  7. Fréchet Means on Normed Spaces
  8. Large Sample Uniqueness
  9. Sample Threshold for Unique Fréchet Means
  10. Face Types
  11. Type Events
  12. Polytope Conditions for Unique Fréchet Means
  13. Defining Unique Fréchet Mean Sample Thresholds
  14. Calculating Uniqueness Sample Thresholds
  15. Uniqueness sample thresholds for the $\ell_{\infty}$ and $\ell_1$ norms
  16. ...and 4 more sections

Key Result

Lemma 3.1

Fix a normed space $(\mathbb{R}^k, \Vert \cdot \Vert_{})$ with induced metric $d$. For all $\mathbf{x}$, $d(\cdot, \mathbf{x})^2$ is convex, and in fact if, for some $t \in (0,1)$ we have then $d(\mathbf{\theta}_1, \mathbf{x}) = d(\mathbf{\theta}_2, \mathbf{x})$. The Fréchet function is also convex, and so Fréchet mean sets are convex.

Figures (8)

  • Figure 1: The hypercubes and cross-polytopes in dimension 2 and 3.
  • Figure 2: Fréchet mean sets (red) are contained in faces of balls that are centred about each data point (\ref{['cor:FMisPolytope']}).
  • Figure 3: A visualisation of the cones $S_i$ (defined in the proof of \ref{['thm:uniqueness_prob_convergence']}) in the case of the $\ell_1$ norm, and a sample such that every cone contains some data point. As each cone contains a data point, whenever there is a Fréchet mean within the unit ball of $\theta_*$, it must be a unique Fréchet mean.
  • Figure 4: A data configuration which occurs with positive probability and has non-unique $\ell_\infty$ Fréchet means.
  • Figure 5: A sample with face type $(F_1,\dots,F_n) = (\mathbf{e}_1+\mathbf{e}_2, \mathop{\mathrm{conv}}\nolimits \{ - \mathbf{e}_2 \pm \mathbf{e}_1 \}, \mathop{\mathrm{conv}}\nolimits \{ - \mathbf{e}_1 \pm \mathbf{e}_2 \})$, and its unique Fréchet mean highlighted in red.
  • ...and 3 more figures

Theorems & Definitions (73)

  • Definition 1.1
  • Definition 2.1: Polytope
  • Example 2.2: Hypercubes
  • Example 2.3: Cross-polytopes
  • Definition 2.4: Face, face lattice
  • Definition 2.5: Polar Polytope
  • Example 2.6: Hypercubes and Cross-polytopes
  • Definition 2.7: Polytope Normed Space
  • Definition 2.8
  • Definition 2.9: Subgradient, Directional Derivative
  • ...and 63 more