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Sample complexity of optimal transport barycenters with discrete support

Léo Portales, Edouard Pauwels, Elsa Cazelles

TL;DR

The paper addresses statistical guarantees for empirical sparse optimal transport (OT) barycenters, deriving uniform generalization bounds of order $O\big(\sqrt{N/n}\big)$ where $N$ is the barycenter's maximum support and $n$ is the per-target-sample size. The authors develop a framework that covers multiple OT divergences, including $W_p^p$, $W_{\epsilon,p}^p$, $SW_p^p$, and max-$SW_p^p$, by leveraging semi-dual representations and empirical process theory to control the dual variables. They show that the $O(\sqrt{N/n})$ rate is uniform across the number of measures $L$ and the regularization parameter $\epsilon$, with dimension-free constants, and discuss tightness via lower bounds and the behavior in $N$, including implications for K-means and constrained K-means. The results provide practical guidance for the sample complexity of sparse OT barycenters, informing algorithm design and theoretical understanding in high-dimensional settings and enabling reliable performance when working with discrete or discretized target measures.

Abstract

Computational implementation of optimal transport barycenters for a set of target probability measures requires a form of approximation, a widespread solution being empirical approximation of measures. We provide an $O(\sqrt{N/n})$ statistical generalization bounds for the empirical sparse optimal transport barycenters problem, where $N$ is the maximum cardinality of the barycenter (sparse support) and $n$ is the sample size of the target measures empirical approximation. Our analysis includes various optimal transport divergences including Wasserstein, Sinkhorn and Sliced-Wasserstein. We discuss the application of our result to specific settings including K-means, constrained K-means, free and fixed support Wasserstein barycenters.

Sample complexity of optimal transport barycenters with discrete support

TL;DR

The paper addresses statistical guarantees for empirical sparse optimal transport (OT) barycenters, deriving uniform generalization bounds of order where is the barycenter's maximum support and is the per-target-sample size. The authors develop a framework that covers multiple OT divergences, including , , , and max-, by leveraging semi-dual representations and empirical process theory to control the dual variables. They show that the rate is uniform across the number of measures and the regularization parameter , with dimension-free constants, and discuss tightness via lower bounds and the behavior in , including implications for K-means and constrained K-means. The results provide practical guidance for the sample complexity of sparse OT barycenters, informing algorithm design and theoretical understanding in high-dimensional settings and enabling reliable performance when working with discrete or discretized target measures.

Abstract

Computational implementation of optimal transport barycenters for a set of target probability measures requires a form of approximation, a widespread solution being empirical approximation of measures. We provide an statistical generalization bounds for the empirical sparse optimal transport barycenters problem, where is the maximum cardinality of the barycenter (sparse support) and is the sample size of the target measures empirical approximation. Our analysis includes various optimal transport divergences including Wasserstein, Sinkhorn and Sliced-Wasserstein. We discuss the application of our result to specific settings including K-means, constrained K-means, free and fixed support Wasserstein barycenters.

Paper Structure

This paper contains 36 sections, 13 theorems, 74 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Sample averaged approximation and sparse barycenters
  3. Statistical guaranties for optimal transport barycenters
  4. Contribution
  5. Notations
  6. Organization of the paper
  7. Introduction to optimal transport and barycenters
  8. Optimal transport divergences
  9. Optimal transport barycenters with sparse support
  10. Discretization and computational complexity
  11. Sample complexity for sparse optimal transport barycenters
  12. Discussion of the results
  13. Sample complexity of optimal transport divergences
  14. Barycenters with sparse support
  15. Tightness of the result
  16. ...and 21 more sections

Key Result

Theorem 3.1

Let $\mu^1,\ldots,\mu^L\in \mathcal{M}_1(\mathbb{B}_{R})$, for $R>0$. For some integer $n$, let $\mu_{n}^1,\ldots,\mu_n^L$ be empirical measures supported over $n$ i.i.d random variables of respective law $\mu^\ell$, for all $\ell\in [\![1,L]\!]$. Let $\mathbb{D}=W_p^p,\:W_{\epsilon,p}^p$ (for some where the finite constant values $C_{p,R}$ and $C_{d,N}$ depend on the divergence $\mathbb{D}$ and

Figures (2)

  • Figure 1: The graphs show the evolution of the estimation error (in log-scale) as the sample size $n$ increases. Each experiments are performed $40$ times. The mean value of the estimation errors is plotted along with the $95\%$ confidence interval. The theoretical upper bound $n\mapsto1/\sqrt{n}$ is plotted in red.
  • Figure 2: The graphs show the estimation error (in log-scale) as the support cardinality $N$ increases, for $n=500$. Each experiments are performed $10$ times. The mean value of the estimation errors is plotted along with the $95\%$ confidence interval. The theoretical bounds (Corollary \ref{['cor:sparse_ot_bar']} and Lemma \ref{['Lem:lowerbound']}) are plotted in red.

Theorems & Definitions (31)

  • Theorem 3.1: Generalization error bound
  • Remark 3.2: On the hypothesis of Theorem \ref{['th:main']}
  • proof : Sketch of proof
  • Remark 3.3
  • Corollary 3.4: Sparse optimal transport barycenters
  • Remark 3.5
  • proof
  • Corollary 3.6: Debiased Sinkhorn barycenters
  • Lemma 4.1
  • Proposition A.1
  • ...and 21 more