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On Heterogeneity in Wasserstein Space

Kisung You

Abstract

Data represented by probability measures arise as empirical distributions, posterior distributions, and feature-based representations of complex objects. We study heterogeneity in a population of probability measures through the expected value of a chosen transform of the pairwise Wasserstein distance. The resulting estimator is unbiased and, under simple moment conditions on the population law, is strongly consistent, asymptotically normal, and equipped with a consistent standard error. This also yields a simple comparison of two populations and remains stable under plug-in approximation when the measures are estimated. The associated empirical eccentricities identify the observations that contribute most strongly to heterogeneity within a sample.

On Heterogeneity in Wasserstein Space

Abstract

Data represented by probability measures arise as empirical distributions, posterior distributions, and feature-based representations of complex objects. We study heterogeneity in a population of probability measures through the expected value of a chosen transform of the pairwise Wasserstein distance. The resulting estimator is unbiased and, under simple moment conditions on the population law, is strongly consistent, asymptotically normal, and equipped with a consistent standard error. This also yields a simple comparison of two populations and remains stable under plug-in approximation when the measures are estimated. The associated empirical eccentricities identify the observations that contribute most strongly to heterogeneity within a sample.
Paper Structure (7 sections, 5 theorems, 105 equations, 3 figures, 1 table)

This paper contains 7 sections, 5 theorems, 105 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Theory and inference
  3. Interpretation in simple models
  4. Numerical illustrations
  5. Synthetic Gaussian measures
  6. Handwritten digits
  7. Discussion

Key Result

Lemma 2.1

Suppose that, for some $p\geq 1$ and constants $a,b\geq 0$, and fix $\mu_0\in\mathcal{P}_2(\mathbb{R}^d)$. Then there exists $C\geq 0$ such that, for all $\mu,\nu\in\mathcal{P}_2(\mathbb{R}^d)$, Consequently, if $E\{W_2(\mu,\mu_0)^p\}<\infty$, then $E|h(\mu,\nu)|<\infty$; if $E\{W_2(\mu,\mu_0)^{2p}\}<\infty$, then $E\{h(\mu,\nu)^2\}<\infty$. These moment conditions do not depend on the particula

Figures (3)

  • Figure 1: Synthetic Gaussian experiment. Panel (a) shows the locations of the generated measures; colours indicate the three groups A, B, and C. Panel (b) shows selected Gaussian measures from group C; grey ellipses denote typical observations and red ellipses denote observations with large empirical eccentricity. Panel (c) shows within-group heterogeneity estimates (points) with 95% Wald intervals (vertical bars); crosses indicate analytic reference values. Panel (d) shows empirical eccentricities $\widehat{g}_i$ for observations in group C; red points correspond to the atypical component and dark grey points to the main component.
  • Figure 2: Transform choice and plug-in stability in the synthetic experiment.
  • Figure 3: Most eccentric images for digits 5 (top row), 2 (middle row), and 7 (bottom row).

Theorems & Definitions (5)

  • Lemma 2.1: Moment domination
  • Theorem 2.2
  • Proposition 2.3: Variance estimation
  • Proposition 2.4: Two-sample comparison
  • Proposition 2.5: Plug-in stability