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Robust Wasserstein barycenter

Zixiong Cheng, Hang Liu

TL;DR

This paper proposes the robust Wasserstein barycenter (RWB) based on a recent concept of the robust optimal transport and demonstrates that the RWB exhibits superior robustness compared to the classical Wasserstein barycenter.

Abstract

In this paper, we address a fundamental limitation of the classical Wasserstein barycenter -- its sensitivity to outliers and its reliance on finite first/second moment assumptions. To overcome these issues, we propose the robust Wasserstein barycenter (RWB) based on a recent concept of the robust optimal transport. Theoretical guarantees, including existence and consistency, are established for the proposed RWB. Through extensive numerical experiments on both simulated and real-world data -- including image processing and financial time series analysis -- we demonstrate that the RWB exhibits superior robustness compared to the classical Wasserstein barycenter.

Robust Wasserstein barycenter

TL;DR

This paper proposes the robust Wasserstein barycenter (RWB) based on a recent concept of the robust optimal transport and demonstrates that the RWB exhibits superior robustness compared to the classical Wasserstein barycenter.

Abstract

In this paper, we address a fundamental limitation of the classical Wasserstein barycenter -- its sensitivity to outliers and its reliance on finite first/second moment assumptions. To overcome these issues, we propose the robust Wasserstein barycenter (RWB) based on a recent concept of the robust optimal transport. Theoretical guarantees, including existence and consistency, are established for the proposed RWB. Through extensive numerical experiments on both simulated and real-world data -- including image processing and financial time series analysis -- we demonstrate that the RWB exhibits superior robustness compared to the classical Wasserstein barycenter.
Paper Structure (15 sections, 4 theorems, 24 equations, 13 figures, 2 tables, 1 algorithm)

This paper contains 15 sections, 4 theorems, 24 equations, 13 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fréchet mean
  4. Optimal transport
  5. Robust Wasserstein barycenter
  6. Robust Wasserstein distance
  7. Robust Wasserstein barycenter
  8. Computation algorithms
  9. Support of RWB
  10. Free-support methods
  11. Numerical results
  12. Comparison of image processing
  13. Comparison of numerical data
  14. Real data analysis
  15. Conclusions

Key Result

Proposition 1

For any $k\geq 1$, the functional $F(\nu) = \mathbb{E}\left[W_p^{(\lambda)}(\nu, \Lambda)\right]^k$ associated with any random probability measure $\Lambda$ on ${\cal W}_p^{(\lambda)}(\mathcal{X})$ admits a minimizer $\nu^*$.

Figures (13)

  • Figure 1: A random nested ellipse image with random outlier points in the upper-right corner.
  • Figure 3: Distribution curves of WB and RWB.
  • Figure 4: Wasserstein distance between WB (and RWB) and the true WB.
  • Figure 5: Wasserstein distance between WB (and RWB) and the true WB in heavy-tailed case.
  • Figure 6: Distribution curves of WB and RWB in heavy-tailed case.
  • ...and 8 more figures

Theorems & Definitions (10)

  • Definition 1
  • Proposition 1: Existence
  • proof
  • Proposition 2: Consistency
  • proof
  • Remark 1
  • Remark 2
  • Theorem 1
  • Theorem 2
  • proof