Relative Translation Invariant Wasserstein Distance

Binshuai Wang, Qiwei Di, Ming Yin, Mengdi Wang, Quanquan Gu, Peng Wei

Abstract:

Motivated by the Bures distance, we introduce a new family of distances, \emph{relative translation invariant Wasserstein distances}, denoted by $RW_p$, as an extension of the classical Wasserstein distances $W_p$ for $p \in [1, +\infty)$.
We establish that $RW_p$ defines a valid metric and demonstrate that this type of metric is more intrinsic than the classical Wasserstein distance.
A bi-level algorithm is designed to compute the general $RW_p$ distance between arbitrary discrete distributions.
Moreover, when $p = 2$, we show that the optimal coupling matrix is invariant under distributional translation in the discrete setting, and we further propose two algorithms, the $\mathrm{RW}_2$-LP algorithm and the $\mathrm{RW}_2$-Sinkhorn algorithm, to improve the numerical stability of computing $W_2$ distance and the optimal coupling matrix solutions.
Finally, we conduct three experiments to validate our theoretical results and algorithms.
The first two experiments report that the $\mathrm{RW}_2$-LP algorithm and the $\mathrm{RW}_2$-Sinkhorn algorithm, both with and without normalization, can significantly reduce the numerical errors compared to standard algorithms.
The third experiment shows that $RW_p$ algorithms are computationally scalable and applicable to the retrieval of similar thunderstorm patterns in practical applications.

Submission Type: Long submission (more than 12 pages of main content)

Changes Since Last Submission:

The revised manuscript includes the following major updates:

1. **Introduction.**
We slightly revised the key question to emphasize the notion of intrinsic differences between distributions. We also added a footnote to clarify the distinction between the proposed $RW_p$ formulation and the notion of translation invariance.

2. **Expanded related work.**
We enriched the related work section by incorporating additional comparison methods, particularly translation-invariant distances and related optimal transport formulations. These include the Gromov–Wasserstein (GW) and entropic Gromov–Wasserstein (EGW) distances, robust optimal transport, the Bures metric, the Procrustes–Wasserstein distance, and sliced Gromov–Wasserstein methods.

3. **New Section 2.2.**
We added a dedicated subsection introducing the Bures metric and its relevance to our formulation. Changed Bures–Wasserstein distance to Bures distance. Added a footnote to clarify the distinction between the Bures distance, the Wasserstein distance, and the Bures–Wasserstein distance.

4. **Revisions to Section 3.2.**
We expanded the discussion of the one-dimensional case ($d=1$), with additional detailed results provided in Appendix A.4. We also clarified the treatment of the higher-dimensional ($d \ge 2$) and Gaussian settings.

5. **Revisions to Section 3.3.**
We strengthened the discussion of normalization and clarified its connection to the Bures metric.

6. **Updates to Section 5.1 and Appendix D.1**
We included normalization-based variants as additional baselines in the experimental evaluation.

7. **Updates to Appendix C.**
We mentioned the Procrustes–Wasserstein distance as well as sliced optimal transport methods.

8. **Revisions to Section 5.2. and Appendix D.2**
We included $FSS$, $W_{1c}$, $W_{4c}$, the Gromov–Wasserstein (GW) and entropic Gromov–Wasserstein (EGW) distances as additional baselines. We also included the experimental settings used for computing the $GW$ and $EGW$ distances.
We added one subsection for the quantitative Evaluation.
We have updated the $W_2$ distances results using normalized coordinates.

9. **Revisions for the camera-ready paper.**
Fixed typos, grammar issues, duplicated or redundant sentences. De-anonymized the author information, added the acknowledgments, and double-checked and reformatted the references. Added the github repo link. Removed the extra package and updated the author block.

Code: https://github.com/DRKWang/rw_metric

Assigned Action Editor: ~Rémi_Flamary1

Submission Number: 6929