**Summary:**

The paper introduces a novel framework for the joint comparison and visualization of heterogeneous datasets by optimally transferring them into a fixed metric space. The proposed method, based on optimal transport (OT) with GW marginal penalization, enables the comparison of measures on distinct metric spaces. The approach is demonstrated through several numerical examples, including the joint embedding of 3D shapes, feature spaces, and Gaussian mixture models.

**Strengths:**

1. The proposed method extends the applicability of OT to heterogeneous spaces by introducing GW penalization in the unbalanced OT formulation, allowing for the comparison of measures on distinct metric spaces.
2. The method provides a general solution for the joint embedding of heterogeneous datasets into a fixed metric space, which is demonstrated through various examples.
3. The paper includes theoretical results, such as the proof of the functional's minimizer and its relation to the embedded Wasserstein distance, which adds to the method's robustness and theoretical foundation.

**Weaknesses:**

1. The method relies on the appropriate definition of the target metric space, which can be a limitation in practical applications.
2. The method's effectiveness in high-dimensional spaces (as indicated by computational restrictions on working with a grid in R^d) might limit its applicability to real-world datasets, especially those with many dimensions.
3. The method's reliance on the computation of transport plans, which can be computationally intensive, might impact scalability and practicality for large datasets.

**Questions:**

1. How does the method handle the computational challenges associated with high-dimensional spaces, especially in terms of the grid size and the computational cost of the multi-marginal OT problem?
2. What strategies can be employed to optimize the method's performance for large datasets, considering the computational limitations mentioned?
3. Can the method be extended to handle multiple input spaces simultaneously, or is the current formulation limited to two input spaces?

**Soundness:**

3 (good)

The paper presents a sound theoretical framework and demonstrates its practical applicability through various examples. The method's soundness is supported by both theoretical analysis and numerical experiments, although the potential limitations in computational efficiency and scalability need further investigation.

**Presentation:**

3 (good)

The paper is well-structured and presents the proposed method clearly. The inclusion of relevant previous work and detailed explanation of the method's components enhance the paper's clarity. However, the paper could benefit from more detailed discussion on the method's practical implications and limitations.

**Contribution:**

4 (excellent)

The paper contributes significantly to the field by extending the applicability of OT to heterogeneous spaces and providing a general solution for the joint embedding of heterogeneous datasets. The theoretical underpinnings and practical demonstrations enhance the method's novelty and impact.

**Rating:**

6 (marginally above the acceptance threshold)

The paper demonstrates a solid contribution to the field with a well-presented method and sound theoretical and practical aspects. However, the method's limitations, especially in computational efficiency and scalability, need to be addressed for broader applicability.

**Paper Decision:**

- Decision: Accept
- Reasons: The paper presents a valuable contribution to the field of optimal transport with a method that extends its applicability to heterogeneous spaces. While the method's limitations are acknowledged, the paper's soundness, contribution, and presentation warrant acceptance. The reviewers recommend further investigation into computational efficiency and scalability for real-world applications.