It's worth noting in the literature review the similar works formulating...

Answer: The authors thank the reviewer for their point. These two references will be added in the Introduction.

The proof of the metric properties is not well displayed in the main text, as this is the main highlight of this work.

Answer: The proof of the metric property is proposed in Appendix D. We apologize that we cannot present it in the main text due to the page limit. The formal statement is based on the definition of equivalence classes among mm-spaces (Remark D.1), and the main technique of the proof relies on the relation between PGW and GW (see Appendix D.3 where we prove the triangle inequality). Due to the page limit, we cannot present these concepts in the main text, but we have added these comments in the main text (right before the statement of Proposition 3.4) to give the reader some insights about the main ideas behind the proof.

Further comparison with KL version was lacking...

Answer:

• Comparison with the KL version: In Experiment 5.1, we explicitly compare PGW and UGW (which uses the KL divergence) in outlier detection. Both UGW and PGW can detect outlier points when parameters are suitably selected. However, since UGW utilizes the Sinkhorn solver, it tends to favor a mass-splitting transportation plan. As a result, we can still observe some matching between clean data and outlier data. This phenomenon is also observed implicitly in other experiments.

• Scalability: In Appendix K, we will include the time complexity of our algorithms, which is derived from our convergence analysis in the same appendix. In particular, we refer to the "computational complexity" section in the author rebuttal for details.

  This can be further improved if the gradient computation step (optimal partial transport solving step) is replaced by the Sinkhorn method:

  $$\mathcal{O}\left(\frac{\max^2(2L_1, \max(|\mu|, |\nu|) \max(2|C^X|^2+ 2|C^Y|,2\lambda))n^2}{\epsilon^2} \frac{1}{\epsilon} \ln(n) n^2\right).$$

In the experiment section, the selection of hyper-parameters is not clear.

Answer: We refer to the "parameter selection" section in the author rebuttal for a discussion of the parameter settings of the PGW method.

In the three experiments presented in the main text, we require full matching for the measure that has less mass. Thus, we set $\lambda$ to be sufficiently large (see Lemma E.2). Similarly, $\rho$ is set to be $\min(|\mu|,|\nu|)$ in MPGW. Additionally, we set $\rho_1=\rho_2$ to be suitably large for UGW. In PU learning experiment, presented in appendix H, $\lambda$ in PGW and $\rho$ in MPGW are required to be set to be sufficiently large. In UGW, we test different $\rho_1,\rho_2$ such that the transported mass equals the prior $\pi=0.2$.

In particular:

• In the point cloud matching experiment (Section 5.1 and Appendix M.2), we explain how $\lambda$ (for PGW), $\rho_1, \rho_2$ (in UGW), and $\rho$ (for MPGW) are selected in lines 1077-1081.
• In the point cloud interpolation experiment (Section 5.3 and Appendix M.1), we explain how $\lambda$ is selected in line 1043.
• In the shape retrieval experiment (Section 5.2 and Appendix N), we explain the setting of $\lambda$ (for PGW), $\rho_1, \rho_2$ (for UGW), and $\rho$ (for MPGW) in lines 1093-1097.
• In the PU learning experiment (Section P), we explain the settings of $\lambda$ (for PGW) and $\rho$ (for MPGW) in lines 1198-1202. The setting of $\rho_1, \rho_2$ for UGW is explained in lines 1189-1190.

Could you add the parameter sensitivity analysis in the experiment section? The table under line 271 doesn't have a title, and in (a) the performance of MPGW is on Dataset II which looks abnormal to me. Is this the actual experiment result or just a typo?

Answer:

• We've add one section to explain the parameter sensitivity in the experiments into the paper. In particular, $\lambda$ in PGW, $\rho$ in MPGW, $\rho_1,\rho_2$ in UGW has a (sufficient) threashold, whenever the parameter is greather than the threashold, the performance demonstrated in this paper will be achieved.

• We apology for the title missing of the table below 271. The title should be "accuracy and wall-clock comparision in shape retrival problem." and it has been added.

One weakness with the methods in the paper...

Answer: The authors apologize for the misunderstanding regarding the English term "Partial Gromov-Wasserstein solver" in the main text. As discussed in the paper, GW and its variants UGW/MPGW/PGW are non-convex problems. To the best of our knowledge, both classical algorithms (Frank-Wolfe algorithm) and the Sinkhorn algorithm converge to local minima rather than global minima. Thus, in the optimal transport community, it is generally accepted to say these methods are "solvers" or that these methods "solve the GW/UGW/MPGW problem", rather than stating "these methods are computational algorithms that achieve a local minimum" (see, e.g., [45] abstract and introduction section; [44] section 3.1). We have added a footnote in the introduction section to explain this, maintaining both readability and rigor, and ensuring consistency with previous works.

In the abstract it is stated the methods solve the PGW problem...

Answer: The claim that our method solves the PGW problem follows the convention of English explanations in previous works [44], [45]. We've added a footnote in the introduction section to maintain consistency with related works and provide a rigorous explanation. See the answer for weaknesses for details.

I think that some early reference...

Answer: These two references will be added to the paper. Equation (3), Optimal partial transport, is a classical unbalanced OT, and many works discuss its theory and applications. We apologize for only citing the classical reference from the authors who introduced this concept (Mccann and Figalli) initially.

Some minor typos...

Answer: The authors thank the reviewer, and we have fixed the typos.

While the paper provides a comparison with existing methods...

Answer: The baseline methods we selected follow from the paper [45] [44] and Beier et al., 2023. As the main topic of this paper is to introduce a partial GW formulation that defines a metric for two positive measures in different spaces, we only implemented our method in the classical experiments from related works and compared our method with existing baselines in these papers.

The paper does not extensively discuss ...

Answer: The authors agree that the choice of $\lambda$ is important, whether in the classical partial OT setting or the partial GW setting. We refer to the parameter selection section in AU for details. This section will be added to the paper.

There is limited discussion on the scalability...

Answer: In Section O, we provide the wall-clock time comparison for the PGW solvers, with the size of tested data varying from 100 to 10,000. In Section K, we've discussed the convergence rate of the proposed algorithms. We refer to the "computational complexity" section and Equation (TC) in the Author Rebuttal for the theoretical time complexity. We will add this conclusion as a direct result in Appendix K and the main text.

The regularization coefficient lambda is referenced in Equations (9), (10), and (14)...

Answer: In Algorithm 1, $\lambda$ is incorporated in the gradient computation. See Equations (59) and (60) in Step 1. Additionally, $\lambda$ is incorporated in the line search step; see Equations (65) and (66).

Similarly, in Algorithm 2, $\lambda$ is applied in the gradient computation step (see Equation (61)) and the line search step (see lines 886-887).

If lambda is indeed utilized in these algorithms...

Answer: Selection of $\lambda$ highly depends on the task where PGW is applied. When full matching for one measure is desired, by Lemma E.2, we should set $\lambda$ to be sufficiently large (i.e., $2\lambda=\max( (C^X)^2+(C^Y)^2)$). We refer to the "parameter selection" section in the author rebuttal for details.

Intuitively, $2\lambda$ plays the role of an "upper bound" on transportation cost. Suppose $(|x_1 - x_2|^2 - |y_1 - y_2|^2)^2 > 2\lambda$; in this case, we will not apply the pairing $x_1 \to y_1, x_2 \to y_2$ or $x_2 \to y_1, x_1 \to y_2$.

If we aim to transport less mass and destroy/create more mass, a smaller $\lambda$ is required. If we aim to transport more mass, we need to set a larger $\lambda$.

In practical applications, especially in high-dimensional spaces...

Answer: To the best of our knowledge, the computation of GW/MPGW/UGW/PGW only depends on the size of the dataset and is independent of the dimension of the space. The dimension of the space may affect the computation of the cost matrices $C^X$ and $C^Y$; however, theoretically, the computation cost in this step is always $O(n^2 + m^2)$.

Are there potential extensions of the PGW metric that could handle non-metric spaces...

Answer: In general, GW/UGW/MPGW/PGW can be defined in a gauged measure space, where the gauge function (symmetric, L2 function) is a generalized version of the metric function. The main idea of GW/MPGW/UGW/PGW is to adapt the similarity of each space to build a measure that describes the similarity between data in two different spaces. We require a structure to describe the similarity for each pair of points in each space. Such a structure can be a metric or a gauge mapping. If such a structure is missing, then it is beyond the scope of the GW method.

It is unclear what the paper implies by "mathematically this equivalence relation is not verified." in line 955?

Answer: The authors thank the reviewer's point. The sentence "mathematically this equivalence relation is not verified" will be removed. The section "relation between PGW and MPGW" in author rebuttal will be added into the paper to clarify the relation between PGW and MPGW.

For a given \lambda=\lambda_0 in PGW ....

Answer: Answer is No. See example 1 and "relation between PGW and MPGW" section in AR.

the steps of the FW algorithm...

Answer: The formulation PGW and MPGW are related and thus the FW iteration steps of the two problem are similar. They have difference in Gradient computation steps and line search step. The algorithms for PGW requires $\lambda$ in these two steps. The solver for $MPGW$ requires $\rho$ in the gradient computation step.

The paper is poorly written due to the following reasons: ...

Answer: The authors thank the reviewer for their points

• We have not include the two variants of the FW solver in the main text due to the page limit. However, for clarity, in the abstract and in the introduction section, we have modified the text:
  • Abtract: "We then propose variants of the Frank-Wolfe algorithm for solving the PGW problem."
  • Introduction: line 74 "Based on this relation, we propose two (mathematical equivalent) variants of Frank-Wolfe algorithms for the discrete PGW problem. One is presented in the main text, and the other is discussed in Appendix G." We refer section "relation between the two FW algorithms" and "motivation of algorithm 2" in Author Rebuttal.
• We refer "relation between the two FW algorithms" section in Author Rebuttal for details. This section will be added into the paper.

The paper provides very less discussion...

Answer: The modified statement of Proposition L.1, which demonstrates the relation between PGW and MPGW, will be added into the main text. Especially,
$PGW_\lambda(\mu,\nu)=MPGW_\rho(\mu,\nu)+\lambda(|\mu|^2+|\nu|^2-2|\rho|^2)$ where $\rho$ is determined by $\lambda$.

There are multiple grammatical and spelling errors...

Answer:The authors thank the reviewer. We have fixed the typos and grammar mistakes.

The main paper contained experiments...

Answer:

• Datasets: both synthetic and real datasets are applied in this paper:
  • In the shape retrieval experiment (Section 5.2), the dataset is given in [51].
  • In the point cloud interpolation experiment (Section 5.3), the dataset is from Github.
  • In the positive unsupervised learning experiment (Section P), the datasets include MNIST, EMNIST, Amazon, Webcam, and DSLR (See [60]).
• We refer "Positive label unsupervised learning" section in Author Rebuttal to see the explanation about why PGW, MPGW admit same performance in this particular task. The discussion will be added in the main text as a numerical evidence of Proposition L.1.

In (14), let \beta = minimum entry in cost tensor M.

Answer: Note, $M_{i,j,i',j'}=|C^X_{i,i'}-C^Y_{j,j'}|^2,\forall i,i'\in[1:n], j,j'\in[1:m]$. Thus $\min M=|C^X_{1,1}-C^Y_{1,1}|^2=|0-0|^2=0$. In this case, $\lambda=0$. zero measure will be one optimal solution. Our algorithm will return $\gamma=0_{n\times m}$. Run the code link for a numerical example.

Why does MPGW obtains 0.0813 and 0 mean accuracy in Table (a)...

Answer:

• We refer "limitations of MPGW" section in Author Rebuttal to see the why MPGW does not has good performance in shape retrieval experiment. In one sentence, when one shape is similar to part of another shape (e.g. "square" is similar to part of "house"), MPGW value will be closed to 0.
• For the PU learning experiment, (table (2)(4), we refer section "PU learning" section in Author Rebuttal for details. In this experiment, we only need transportation plans from the GW-based methods. In the experimental setting, full matching is required for PGW/MPGW, and based on Proposition L.1 and Lemma E.2, MPGW and PGW yield the same solution sets in this scenario.

I thank the authors for their rebuttal. Please find my (further) questions below.

1. Regarding global response Statement 1 example 1: Given $lambda=0$, solve PGW and obtain the solution $\gamma_{PGW}$. Then set $\rho=|\gamma_{PGW}|$. Now, would the first order critical points of MPGW contain $\gamma_{PGW}$?

2. Regarding the authors' response "both synthetic and real datasets are applied in this paper:"

• By the statement "In the shape retrieval experiment (Section 5.2), the dataset is given in [51].", are the authors claiming that this experiment use real-world dataset?
• By the statement "In the point cloud interpolation experiment (Section 5.3), the dataset is from Github (link)", are the authors claiming that this experiment use real-world dataset?
• Regarding experiments in Section P, I have already acknowledged that they are the only real-world experiments in the whole manuscript.

3. Regarding, "In (14), let \beta = minimum entry in cost tensor M", I agree with the authors that the optimal solution will be a zero measure if lambda is sufficiently large. However, does the author response contract their Lemma E.2.

4. In the appendix L.1 example, does $MPGW_\rho(X_i,X_j)$ = 0 for all values of rho? What about $\rho=|\gamma_{PGW}|$?

Overall, I have found the author response underwhelming. Hence, for now, I am reducing my score by one.

4. In the appendix L.1 example, does $MPGW_\rho(X_i,X_j)=0$ for all values of $\rho$? What about $\rho = |\gamma_{PGW}|?$

Answer: Yes, $MPGW_\rho(X_i,X_j)=0$ for all values of $\rho$, including $\rho = |\gamma_{PGW}|$.

Below, we provide the reasoning to clarify this point. In this example:

$\mathbb{X}^1=(\mathbb{R}^3,\|\cdot\|,\sum_{i=1}^{1000} \alpha \delta_{x_i}),$ $\mathbb{X}^2=(\mathbb{R}^3,\|\cdot\|,\sum_{i=1}^{800} \alpha \delta_{x_i}),$ $\mathbb{X}^3=(\mathbb{R}^3,\|\cdot\|,\sum_{i=1}^{400} \alpha \delta_{x_i}),$

where $\alpha=1e-3$, $\lambda=1e+1$, and $x_i \in [0,1]^2$ for all $i$.

Let $\gamma_{1,2}$ be the optimal solution for $PGW_\lambda(X_1,X_2)$, then we have $|\gamma_{1,2}|=0.8$. Similarly, we define $\gamma_{1,3}$ and $\gamma_{2,3}$, where $|\gamma_{1,3}|=0.4$ and $|\gamma_{2,3}|=0.4$. The numerical results are consistent with our Lemma E.2. We can verify the following:

$MPGW_{0.8}(\mathbb{X}^1,\mathbb{X}^2)=0, \text{ where } |\gamma_{1,2}|=0.8,$ $MPGW_{0.4}(\mathbb{X}^1,\mathbb{X}^3)=0, \text{ where } |\gamma_{1,3}|=0.4,$ $MPGW_{0.4}(\mathbb{X}^2,\mathbb{X}^3)=0, \text{ where } |\gamma_{2,3}|=0.4.$

For the case of arbitrary $\rho > 0$, we may first consider the problem $MPGW_{\rho}(\mathbb{X}^1,\mathbb{X}^2)$. Let $\gamma^*=\sum_{i=1}^{800}\alpha\delta_{(x_i,x_i)}$ denote the transportation plan induced by the Monge mapping $x_i \mapsto x_i$ for all $i \in [1:800]$. Then, $\frac{\rho}{|\gamma^*|}\gamma$ will be an optimal solution for $MPGW_{\rho}(\mathbb{X}^1,\mathbb{X}^2)$. Consequently, $MPGW_{\rho}(\mathbb{X}^1,\mathbb{X}^2) = 0$. We may apply similar reasoning to see that $MPGW_{\rho}(\mathbb{X}^1,\mathbb{X}^3),MPGW_{\rho}(\mathbb{X}^2,\mathbb{X}^3)=0$ for any $\rho\in[0,0.4]$ as well, where 0.4 is the largest value we can choose for $\rho$ in these two MPGW problems.

Run this link (cell 1 and cell 3) to see a numerical example. Note that, as we set the tolerance of PGW/MPGW algorithms to 1e-5, the values of $MPGW_{0.8}(\mathbb{X}^1,\mathbb{X}^2)$, $MPGW_{0.4}(\mathbb{X}^1,\mathbb{X}^3)$, and $MPGW_{0.4}(\mathbb{X}^2,\mathbb{X}^3)$ are of the order $\mathcal{O}(1e-5)$, rather than exactly 0.

We appreciate the reviewer’s engagement with our work and the response to our rebuttal. While we respect the reviewer's perspective, we have concerns that the current evaluation may not fully reflect the scientific and technical merits of our paper. The original review did not identify any theoretical flaws, instead raising minor clarification questions that we have thoroughly addressed. We also recognize the reviewer's comments regarding the paper's organization and the request for additional experiments on real-world data. However, these concerns do not appear to undermine the fundamental theoretical contributions of our work, which we feel may have been overlooked. Furthermore, we would like to highlight that the scope of our numerical experiments is consistent with prior work on Gromov-Wasserstein (GW) and unbalanced GW. We would welcome further discussion rooted in specific scientific reasoning and detailed feedback, which would enhance the constructive nature of this review process.

1. "Regarding global response Statement 1 example 1:..."

Answer: Yes.

In this case, if we select the zero measure, which is an optimal solution for PGW, and set $\rho=|\gamma|=0$, the global solution coincides with the first-order critical point of MPGW. In fact, the space $\Gamma_\leq^\rho(\mu,\nu)$ contains only a single element for MPGW.

Otherwise, as explained in the author rebuttal, if we choose $\gamma=\delta_{(x_i,y_j)}$, where $i \in [1:n]$ and $j \in [1:m]$, which is also an optimal solution for $PGW_0(\mathbb{X},\mathbb{Y})$, and we set $\rho=|\gamma|=1$, then $\gamma=\delta_{(x_i,y_j)}$ would be one of the first-order critical points of $MPGW_1(\mathbb{X},\mathbb{Y})$ and a solution to the MPGW problem.

2. "Regarding the authors' response "both synthetic and real datasets are applied in this paper..."

Answer: Our statement regarding our use of both synthetic and real-world data remains to be correct. To clarify:

• In the shape retrieval experiment (Section 5.2), “Dataset I” is derived from [51] and “Dataset II” is entirely synthetic. Dataset I is also synthetic; we do not mean to claim otherwise, only to outline the sources of data in our experiments.
• The data for the point cloud interpolation experiment comes from [41]. This dataset is also synthetic, but nevertheless used in prior GW papers for method evaluation.
• MNIST and EMNIST are real-world yet small-scale problems commonly used for demonstration purposes, and the Amazon, Webcam, and DSLR are clearly real-world datasets.

3. "Regarding, "In (14), let \beta = minimum entry in cost tensor M", I agree with the authors that the optimal solution will be a zero measure if lambda is sufficiently large. However, does the author response contract their Lemma E.2.

Answer: No, it does not contradict with Lemma E.2.

Here, we point out that the reviewer might have misinterpreted our response or there might have been a confusion. The optimal solution will be a zero measure if $\lambda$ is sufficiently small (e.g., $\lambda \leq\min(M)$). When $\lambda$ is sufficiently large, based on our Lemma E.2., the $|\gamma|$ will achieve its maximum, that is $|\gamma|=\min(|\mu|,|\nu|)$.

Then, in the setting of Example 1 of the global response, it does seem that the solution set of PGW and MPGW can be identical - with appropriate value of $\rho$.

Answer: There may have been some confusion here. In Example 1, the solution sets of PGW and MPGW are NOT identical for any $\rho\in [0, \min{(|\mu|,|\nu|)}]$.

• The solution set of $PGW_0(\mu, \nu)$ contains $\{\delta_{(x_i, y_j)} : i\in[1:n], j\in[1:m]\} \cup \{0\}$.
• When $\rho>0$, the zero measure is a solution for $PGW_0(\mu,\nu)$, but it is not a solution for $MPGW_\rho(\mu,\nu)$.
• When $\rho=0$, $\delta_{(x_i,y_j)}$ is a solution of $PGW_\lambda(\mu,\nu)$, but it is not a solution for $MPGW_0(\mu,\nu)$.

From the author response, I think we are on the same page. In the original review, I had written - "The main paper contained experiments only on synthetic dataset. Tables 2 and 4 in appendix discuss experiments on real-world datasets. MPGW obtains same generalization performance as PGW in both the tables."

Answer: Thank you for the clarification. We would once again like to note that the scope of our numerical experiments is consistent with existing literature on GW and unbalanced GW. If the reviewer is instead concerned with the similar performance of PGW and MPGW in Tables 2 and 4 in the appendix, this is discussed in the global response under the heading "Positive Label Unsupervised Learning", as stated earlier.

5. The response of one of my questions in my original review was a bit unclear. The question was "In (14), let \beta = minimum entry in cost tensor M. Then, what is the solution of PGW (16) when \beta > 2\lambda ?"

The solution of (16) seems to be zero when \beta > 2\lambda. If this is indeed the case, then the PGW distance between two distributions with marginals p and q, respectively, and with \lambda < \beta/2 will be \lambda(|q|^2 + |p|^2) (from 15) - which is a constant dependent on hyper-parameter and marginals. This restricts PGW's ability to distinguish the source distribution with target distributions of same marginal norm (|q|=|q_1|=|q_2|). No geometric information is being taken into account.

Answer: There still seems to be some confusion here. When $\beta=\min M$ as specified in the reviewer's original comment, this results in $\beta = 0$. To be precise, since $\lambda$ cannot be negative number, we first correct $2\lambda<\beta=0$ to $2\lambda\leq \beta=0$, i.e., $\lambda = 0$. As a result, $PGW_0(\mu, \nu)$ admits the zero measure as one solution, and the PGW distance will be $\lambda(|q|^2+|p|^2)=0$ as $\lambda=0$.

This restricts PGW's ability to distinguish the source distribution with target distributions of same marginal norm (|q|=|q_1|=|q_2|).

As stated in Proposition 3.4, PGW is a metric which can distinguish two measures if $\lambda>0$. In this case, $\lambda=0$ and thus PGW cannot distinguish the difference between the source and target distributions. The resulting analysis given by the reviewer simply reflects the fact that $\lambda=0$ is a bad choice of hyperparameter when we aim to apply PGW as a metric.

This statement contradicts the reviewer's original claim (from Response to authors #2):

Previously, I was wondering if the set of solution can be shown to be same for individual values of hyper-parameters. The authors have shown with example that this is not the case.

However, please note that my previous response was more in the form of a question to the authors (albeit without a question mark) rather than a definite claim about the paper's contribution.

As stated previously, the authors hoped to clarify exactly this point; namely, that the solution sets to the two problems are different, although particular solutions can coincide.

While this is true for specific values of hyper-parameter values, practically, one does not run the model on one hyper-parameter value. A common practice is to run the model on multiple hyper-parameter values (tuning/cross-validation) and choose one (based on some other criteria). Hence, can the solutions of PGW as one tunes PGW over $lambda$ be obtained by MPGW as one tunes over $\rho$?

I would also increase my score to the original score.

Let me clarify my question. In Example 1, will the solution of PGW be a first-order critical point of MPGW for an appropriate value of $\rho$?

Answer:

• In Example 1, we assert that the solution sets of $PGW_0(\mu,\nu)$ and $MPGW_\rho$ are different. We have already explained the distinction between these sets in our author rebuttal and in previous comment.

  In my earlier question, "Given $\lambda=0$, solve PGW and obtain the solution $\gamma_{PGW}$. Then set $\rho=|\gamma_{PGW}|$. Now, would the first-order critical points of MPGW contain $\gamma_{PGW}$?", the author responded "yes."

• The answer to this earlier question remains yes, as we previously explained. This particular $\gamma_{PGW}$ is also a solution for $MPGW_\rho(\mu,\nu)$ since $\rho=|\gamma_{PGW}|$. However, this does not imply that another solution of $PGW_0(\mu,\nu)$ will also be a solution for this specific $MPGW_\rho(\mu,\nu)$.

• Therefore, there is no contradiction to our claim "the solution sets of the two problems are different", although particular solutions might coincide.

In the global response, it was stated that PGW has the advantage of being a metric (whereas MPGW is not). Perhaps some experiments should be conducted to demonstrate the utility of PGW as a metric.

Answer: In the shape retrieval experiment, we obtained GW/UGW/MPGW/PGW costs to describe the similarity between two shapes. Since PGW and GW are metrics, they induce better performance than the other two methods. That is shown in table on page 8.

I am not sure why $\min M$ should be zero....

Answer: The definition of $M$ is provided in Eq (14), and the definitions of $C^X$ and $C^Y$ are given in Eq (11).

In particular $(C^X)_{i,i}=d_X(x_i,x_i')$ where $d_X$ is a metric.

Thus, when $i=i'$, $C^X_{i,i'}=0$. That is, diagonal elements of matrix $C^X$ are zero. Similarly, diagonal elements of matrix $C^Y$ are zero.

Based on the definition of $M$, we have $\min M=|C_{1,1}^X-C_{1,1}^Y|^2=|d_X(x_1,x_1)-d_Y(y_1,y_1)|^2=|0-0|^2=0.$

I am not sure what the phrase another solution of ...

Answer:

We have explained this in Author Rebuttal example 1 and in the comment “answers to new comments 2”.

When $\gamma_{PGW}$ is the zero measure, we set $\rho=|\gamma_{PGW}|=0$:

• The zero measure is a solution for $PGW_0(\mu,\nu)$ and also for $MPGW_0(\mu,\nu)$.
• Choose $i\in[1:n],j\in[1:m]$, $\delta_{(x_i,y_j)}$ is another solution for $PGW_0(\mu,\nu)$. But $\delta_{(x_i,y_j)}$ is not a solution for $MPGW_0(\mu,\nu)$.