Relative-Translation Invariant Wasserstein Distance

Thank you for your valuable feedback and constructive suggestions. Below, we address the weaknesses and questions you raised:

1. We acknowledge your concern regarding the time complexity claim in Remark 1. While it is true that solving $RW_p$ may require multiple Sinkhorn subproblems, we argue that ${RW}_p$ offers greater robustness compared to $W_p$, especially in datasets with noise or variability. While the computational complexity of ${RW}_p$ may be higher in certain cases, the additional cost is justified by the robustness and adaptability it provides. This makes the method particularly valuable in scenarios where $W_p$ might fail to capture meaningful relationships due to instability or bias. We will clarify these points in the revised manuscript.

2. In Section 3.2, we discuss the relationship between the ${RW}_p$ metric and the $L_p$ norm. Specifically: ${RW}_1$ is robust to noisy datasets, making it well-suited for scenarios with high variability. ${RW}_2$ is better suited for cleaner datasets, offering a balance between computational feasibility and precision. Higher-order ${RW}_p$ values apply stronger penalties to larger distances, making them useful for applications requiring stricter enforcement of outlier effects.

3. We acknowledge the simplicity of the example in Section 5.1 and agree that for Gaussian distributions, closed-form solutions for Wasserstein distances are available. To better showcase the utility of our method, we have included additional experiments involving non-Gaussian distributions in D.1. These examples demonstrate the numerical benefits of debiasing the centers and the robustness of ${RW}_p$ in more complex settings.

Thank you for pointing out the use of $\eta$ and the notation of $\bar{\mu}$ and $\bar{\nu}$. We will remove $\eta_2$ from Algorithm 1 for consistency. Additionally, the definitions of $\bar{\mu}$ and $\bar{\nu}$ are already provided in the notations paragraph in Section 1. We will cross-reference this section in the relevant parts of the manuscript to ensure clarity for readers.

Thank you again for your constructive feedback and suggestions. These insights will help improve the clarity, rigor, and practical relevance of our manuscript. Please let us know if further clarification or additional experiments would be helpful.

Thanks for your efforts for reviewing the paper.

The straightforward option may not be applicable in all cases. We provide a counterexample in Section C.3 to illustrate that this does not hold for the $RW_1$ metric. Please check the supplementary materials for more information.

Thank you for your valuable feedback and constructive suggestions. Below, we address the weaknesses and questions you raised:

1. The GW problem is a highly nonconvex optimization problem and is particularly sensitive to the choice of the initial point. Due to this sensitivity and the fundamental differences in their problem formulations, we believe that a direct comparison with our proposed method may not yield meaningful insights. However, we acknowledge the relevance of GW as a benchmark and will consider further clarifying these distinctions in the manuscript.

2. The convergence rate of the proposed method matches that of the original Sinkhorn algorithm, as established by Theorem 4.1 on page 70 of reference [1]. Specifically, the convergence depends on the contraction factor $\lambda(K)$, which is determined by $\eta(K)$. For a translation vector $s$ and the translated cost matrix $K'$, we can show that $\eta(K) = \eta(K')$. This follows from the fact that: $|| x_i - y_j + s ||_2^2 + || x_k - y_l + s||_2^2 - || x_i - y_l + s ||_2^2 - || x_k - y_j + s||_2^2 = || x_i - y_j ||_2^2 + || x_k - y_l ||_2^2 - || x_i - y_l ||_2^2 - || x_k - y_j ||_2^2.$ Thus, $\eta(K) = \eta(K')$, and the convergence rate remains unaffected by the translation $s$. Therefore, the proposed method's convergence rate is equivalent to that of the original Sinkhorn method, regardless of the translation.

3. As mentioned in the paper (line 252), the $RW_p$ metric is not designed to handle rotations, where it is possible to find a counterexample again rotation-invariant.

4. The thunderstorm snapshot data used in the pattern detection experiment is not labeled. Consequently, we cannot provide accuracy metrics for the retrieved patterns. However, we agree that labeled datasets could provide more quantitative evaluations in future experiments.

5. The code for the experiment section is publicly available. We will ensure this information is explicitly stated in the manuscript for ease of access.

Thank you again for your constructive feedback and suggestions. These insights will help improve the clarity, rigor, and practical relevance of our manuscript. Please let us know if further clarification or additional experiments would be helpful.

[1] Peyré and Cuturi, computational optimal transport, 2019

Thanks for your efforts for reviewing the paper.

For the questions:

1. The reviewer’s straightforward approach may not apply universally, particularly in cases such as $p = 1$. A counterexample is provided in Section C.3, where two distributions are used to demonstrate that the optimal solution is not invariant to translation when the metric is $RW_1$. Please refer to the supplementary materials for further details.

2. This is an excellent question. The primary reason for the dropout near zero is that as $s$ increases, it leads to division-by-zero issues, causing significant numerical errors, however, which also make the program faster to compute. This results in faster convergence and reduced runtime. You may check more details by testing the uploaded files.

3. This is the same as point 2.

4. The computational error is calculated as the difference between the Wasserstein-2 distances obtained using the Sinkhorn/$RW_2$-Sinkhorn method and the true Wasserstein-2 distances computed via linear programming. As shown in Figure 3, the errors are nearly identical, with slight variations around $1 \times 10^{-5}$.

5. The unstable images are also a result of division-by-zero issues, as previously mentioned, demonstrating that the $RW_2$-Sinkhorn method is more stable than the regular Sinkhorn approach.

Thank you for your valuable feedback. Below, we address the points raised in your review:

For the weaknesses: Thank you for highlighting this question. Could you please clarify what specific properties you are referring to outside the quotient set? While the main results focus on the behavior within the quotient set, further exploration of properties beyond this set may require additional assumptions or analysis. We would appreciate a more detailed explanation to better address your concerns.

We acknowledge your observation regarding the connection to Remark 2.19 in Peyré and Cuturi (2019). While the decomposition of Quadratic ROT for shifting distributions aligns with this observation, our contribution focuses on leveraging this decomposition to extend the analysis specifically for the case of the relative transplantation invariant Wasserstein distance. The cited reference is mentioned in the subsequent paragraph of Corollary 2. We can consider removing this part or reframing it to emphasize how our approach offers a complementary perspective, thereby enriching the understanding of the decomposition.

Regarding your comment on the order of $\sqrt{n}$, it is not entirely clear what aspect of the computational complexity you are referencing. If your comment pertains to the dependency on the $\ell_2$-norm or the shifting cost matrix, could you provide more details? Our experimental results demonstrate computational advantages of the proposed Sinkhorn variant, particularly in scenarios where the shifting parameter is bounded. A clearer explanation will help us refine and strengthen our arguments in this context.

Thank you for pointing out the typographical errors. We will make the corrections.

We appreciate your feedback and suggestions, as they will help improve the clarity and rigor of our work. Please let us know if further clarification is needed on any of the points raised.