PAPER: IntroductionThe comparison of heterogeneous data distributions is a fundamental task in computer vision, computational biology and machine learning. Most existing approaches rely on using a suitable ground cost function such as an available metric. A classic example is the Wasserstein distance which seeks an optimal transport (OT) between two given distributions. However, often the given distributions are in heterogeneous spaces, where a readily available ground cost function between these spaces does not generally exist. Additional effort may be required to learn appropriate cost functions (Cuturi & Avis, 2014;Heitz et al., 2021). However, even if there exists a natural embedding into a Proceedings of the 42 nd International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s). . Joint (aligning) transfer of two metric spaces (gray surfaces with surface distance) to a fixed reference space, namely to the sphere and the torus by our method, where the color "yellow" corresponds to higher values, see Subsection 6.1.canonical joint metric space, this metric may not be suited to accurately gauge differences in their samples. Examples of such heterogeneous settings are, e.g., the comparison of graph-or mesh-valued data such as 3d shapes or manifolds. This paper introduces a novel framework based on OT which enables the joint comparison and visualization of heterogeneous datasets by optimally transferring them into an a-priori fixed metric space, see Figure 1 for an illustrative example.
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REVIEW
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**Summary Of The Paper**  
The paper proposes a novel framework termed “Relaxed Embedded Wasserstein Distance” (REWD) for the joint comparison and visualization of heterogeneous datasets. The central idea revolves around using an unbalanced Optimal Transport (OT) formulation with Gromov-Wasserstein (GW) penalization to align and embed data from different metric spaces into a fixed reference space. This approach aims to enable accurate comparisons of heterogeneous data by enforcing near-isometric mappings. The authors formulate a multi-marginal OT problem, derive theoretical guarantees (existence of minimizers, convergence properties), and introduce a bi-convex relaxation amenable to numerical computation via a modified Sinkhorn algorithm. They evaluate the method on diverse tasks, including 3D shape alignment, Gaussian Mixture Model (GMM) alignment, and feature space alignment, reporting competitive results against baselines like JMDS and SCOT.

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**Strengths**  
1. **Mathematical Rigor**: The paper presents a solid theoretical foundation, including existence of minimizers, convergence behavior as the regularization parameter $\lambda \rightarrow \infty$, and formal relationships with established concepts like embedded Wasserstein distances and GW barycenters. These derivations are backed by lemmas and propositions with proper proofs in the appendix.  

2. **Conceptual Contribution**: The framework extends the scope of joint embedding from Euclidean to arbitrary metric spaces, addressing a gap in existing methods like JMDS, which are confined to Euclidean geometries. This is a notable conceptual advance.  

3. **Algorithm Design**: The proposed bi-convex relaxation and associated Sinkhorn-like algorithm offer a computationally feasible path toward solving the complex multi-marginal OT problem. The algorithm’s design is grounded in well-established OT solvers, making it plausible and scalable for moderate-sized instances.  

4. **Empirical Evaluation**: The paper conducts extensive empirical evaluations across multiple domains — 3D shapes, GMMs, and feature spaces — demonstrating the versatility of the method. Visual and quantitative comparisons with state-of-the-art methods like JMDS and SCOT suggest that the proposed approach performs competitively.  

5. **Clarity and Structure**: The paper is well-organized, logically structured, and provides sufficient background on OT and related methodologies. The experimental setup is described in enough detail to facilitate replication, although reproducibility details remain sparse.

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**Weaknesses**  
1. **Insufficient Discussion of Scalability and Practical Limitations** (Severe): While the algorithm is designed for a fixed reference space $Z$, the paper does not discuss the scalability of the method with respect to the size of $Z$ or the cardinality of the input measures. For instance, the complexity of the 4-plan optimization grows rapidly with the size of $Z$, yet the paper offers no guidance on managing this trade-off. Additionally, the reliance on a fixed-grid discretization limits applicability to high-dimensional or non-grid-based reference spaces.

2. **Limited Quantitative Analysis of Improvements Over Baselines** (Major): Although the paper compares the proposed method with JMDS and SCOT qualitatively, the improvements are mostly illustrated through visual inspection. The reported metrics (e.g., GW and Wasserstein distances) are summarized in tables, but the significance of these improvements is not assessed statistically (e.g., p-values, confidence intervals, or hypothesis testing). This undermines the strength of the empirical claims.

3. **Ambiguity Around Discretization Choices** (Moderate): The paper uses a fixed grid for $Z$ in many experiments, but the rationale for choosing specific grid resolutions (e.g., 20×20 or 60×60 grids) is unclear. No sensitivity analysis is provided to justify these choices or explore alternative discretization strategies.

4. **Lack of Detailed Explanation of Fixed Support Constraint** (Moderate): The decision to fix the support of the measures is a crucial design choice that affects the flexibility of the method. However, the paper does not adequately explain the consequences of this constraint, nor does it compare the expressive power of fixed-support vs. free-support methods in depth.

5. **Missing Reproducibility Information** (Minor): The paper lacks a dedicated reproducibility section, despite being submitted to a top-tier ML conference. Details such as code availability, hyperparameter tuning procedures, and software dependencies are only briefly referenced, making independent reproduction difficult.

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**Questions For The Authors**  
1. In Section 3, Equation (3) defines the GW penalized unbalanced OT problem. What theoretical basis justifies the use of GW as a penalty term, and how does it enforce near-isometric embeddings rather than introducing distortions?

2. In Proposition 3.4, the paper asserts that as $\lambda \to \infty$, $EW_\lambda$ approaches the embedded Wasserstein distance. Under what assumptions does this convergence hold, and how sensitive is this result to the choice of the reference space $(Z, d_Z)$?

3. In Section 5, the discrete formulation of the problem assumes fixed supports for the measures. What are the trade-offs between fixed and variable support approaches in terms of modeling capacity and computational tractability?

4. In Equation (9), the objective function involves a high-dimensional product space $X_1 \times Z_1 \times Z_2 \times X_2$. What approximations or optimizations are implemented to make this feasible in practice?

5. In the FAUST dataset experiment (Section 6.1), Table 1 summarizes the GW and Wasserstein distances between input shapes and embeddings. Were statistical tests (e.g., t-tests, ANOVA) performed to confirm the significance of the differences between the proposed method and baselines?

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**Limitations Not Addressed By The Authors**  
1. **Scalability Concerns**: The paper does not address the computational complexity of the proposed method with respect to the size of the input data or the reference space $Z$. For large-scale applications, the method may suffer from prohibitive time and memory requirements.

2. **Generalizability Beyond Grid-Based Reference Spaces**: The reliance on fixed-grid discretizations severely limits the method’s ability to generalize to arbitrary or high-dimensional reference spaces. The paper does not discuss how to extend the approach to such cases.

3. **Handling High-Dimensional Input Spaces**: The experiments are largely focused on low-dimensional input spaces (e.g., 2D or 3D shapes). The paper does not explore how the method behaves when dealing with high-dimensional input data, which is a common scenario in modern ML.

4. **Interpretability of Embeddings**: While the paper demonstrates successful alignment and embedding, it does not delve into the interpretability of the learned embeddings or how users might leverage them for downstream tasks (e.g., clustering, classification).

5. **Ethical and Societal Implications**: The paper does not include an ethics statement, which is increasingly expected in submissions to top-tier conferences, especially when dealing with real-world data or potentially biased datasets.

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**Soundness**  
**Score: 3 (Good)**  
The paper presents a mathematically sound framework with rigorous theoretical foundations and empirical validation. However, some gaps in scalability and robustness analyses reduce the overall soundness rating.

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**Contribution**  
**Score: 3 (Good)**  
The paper contributes a novel OT-based framework for joint embedding of heterogeneous data, extending prior work to non-Euclidean spaces. However, the novelty is somewhat diluted by similarities with existing methods like JMDS and the lack of comprehensive comparative analysis with recent advances.

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**Confidence**  
**Score: 4 (High Confidence)**  
The paper is well-written, technically correct, and the experiments are reasonably designed. The clarity of exposition and the presence of theoretical guarantees instill strong confidence in the validity of the claims.

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**Rating**  
**Score: 8 (Accept)**  
Despite some shortcomings in scalability, reproducibility, and detailed quantitative analysis, the paper presents a valuable and well-founded contribution to the field of OT-based joint embedding. Its theoretical rigor and empirical evaluation justify acceptance, though the authors should address the identified limitations in future revisions.

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**Brief Justification For Rating**  
The paper introduces a promising and theoretically grounded method for joint embedding of heterogeneous data using OT with GW penalization. Despite some technical and empirical shortfalls, the approach is innovative, well-explained, and empirically validated across multiple domains. The contributions are substantial enough to warrant acceptance, pending improvements in scalability, reproducibility, and deeper comparative analysis.

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