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Closing the Gap: Efficient Algorithms for Discrete Wasserstein Barycenters

Jiaqi Wang, Weijun Xie

TL;DR

This work addresses computing discrete Wasserstein barycenters for finite-support measures, a problem that is NP-hard even for sparse instances. It develops a polynomial-time approximation scheme (PTAS) that delivers a $(1+\alpha)$-approximate barycenter in time polynomial in $(nk)^{1/\alpha}$ and the ambient dimension $d$, improving the prior 2-approximation. The authors introduce a candidate-support reduction framework via a restricted MOT formulation, providing randomized and deterministic variants, with a tighter guarantee in the equal-weight setting. They validate the approach with extensive experiments on synthetic data and real-world datasets (e.g., MNIST and sign-language images), showing near-optimal barycenters and practical scalability. Overall, the paper advances efficient, provably-accurate algorithms for discrete Wasserstein barycenters and outlines promising avenues for extending the framework to broader OT-related problems.

Abstract

The Wasserstein barycenter problem seeks a probability measure that minimizes the weighted average of the Wasserstein distances to a given collection of probability measures. We study the discrete setting, where each measure has finite support-- a regime that frequently arises in machine learning and operations research. The discrete Wasserstein barycenter problem is known to be NP-hard, which motivates us to study approximation algorithms with provable guarantees. The best-known algorithm to date achieves an approximation ratio of two. We close this gap by developing a polynomial-time approximation scheme (PTAS) for the discrete Wasserstein barycenter problem that generalizes and improves upon the 2-approximation method. In addition, for the special case of equally weighted measures, we obtain a strictly tighter approximation guarantee. Numerical experiments show that the proposed algorithms are computationally efficient and produce near-optimal barycenter solutions.

Closing the Gap: Efficient Algorithms for Discrete Wasserstein Barycenters

TL;DR

This work addresses computing discrete Wasserstein barycenters for finite-support measures, a problem that is NP-hard even for sparse instances. It develops a polynomial-time approximation scheme (PTAS) that delivers a -approximate barycenter in time polynomial in and the ambient dimension , improving the prior 2-approximation. The authors introduce a candidate-support reduction framework via a restricted MOT formulation, providing randomized and deterministic variants, with a tighter guarantee in the equal-weight setting. They validate the approach with extensive experiments on synthetic data and real-world datasets (e.g., MNIST and sign-language images), showing near-optimal barycenters and practical scalability. Overall, the paper advances efficient, provably-accurate algorithms for discrete Wasserstein barycenters and outlines promising avenues for extending the framework to broader OT-related problems.

Abstract

The Wasserstein barycenter problem seeks a probability measure that minimizes the weighted average of the Wasserstein distances to a given collection of probability measures. We study the discrete setting, where each measure has finite support-- a regime that frequently arises in machine learning and operations research. The discrete Wasserstein barycenter problem is known to be NP-hard, which motivates us to study approximation algorithms with provable guarantees. The best-known algorithm to date achieves an approximation ratio of two. We close this gap by developing a polynomial-time approximation scheme (PTAS) for the discrete Wasserstein barycenter problem that generalizes and improves upon the 2-approximation method. In addition, for the special case of equally weighted measures, we obtain a strictly tighter approximation guarantee. Numerical experiments show that the proposed algorithms are computationally efficient and produce near-optimal barycenter solutions.

Paper Structure

This paper contains 17 sections, 8 theorems, 48 equations, 6 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Literature review
  3. Summary of contributions
  4. Preliminary results and approximation scheme
  5. Optimal Support
  6. The dual representation of MOT formulation and candidate support reduction
  7. Approximation algorithms for the Wasserstein barycenter problem with general weights
  8. Sampling algorithm with repetition
  9. Deterministic counterpart based on multi-subset enumeration
  10. Approximation algorithms for the Wasserstein barycenter problem with equal weights
  11. Subset sampling algorithm without repetition
  12. Deterministic counterpart based on subset enumeration
  13. Numerical experiments
  14. Synthetic nested ellipse dataset
  15. MNIST Dataset
  16. ...and 2 more sections

Key Result

Lemma 1

The support $S^*$ of the optimal barycenter of problem eq:def_barycenter is given by

Figures (6)

  • Figure 1: Ten images from the nested ellipses dataset.
  • Figure 2: Barycenters produced by different algorithms for nested ellipse.
  • Figure 3: Barycenters produced by different algorithms for MNIST.
  • Figure 4: Barycenters produced by different algorithms for gesture 1.
  • Figure 5: Barycenters produced by different algorithms for gesture 2.
  • ...and 1 more figures

Theorems & Definitions (19)

  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • Theorem 1
  • proof
  • Remark
  • Theorem 2
  • proof
  • Remark
  • ...and 9 more