Title: Partial Gromov Wasserstein Metric
Venue: NeurIPS 2024
Accepted: no
Keywords: Optimal transport

machine learning
Abstract: The Gromov-Wasserstein (GW) distance has gained increasing interest in the machine learning community in recent years, as it allows for the comparison of measures in different metric spaces. To overcome the limitations imposed by the equal mass requirements of the classical GW problem, researchers have begun exploring its application in unbalanced settings. However, Unbalanced GW (UGW) can only be regarded as a discrepancy rather than a rigorous metric/distance between two metric measure spaces (mm-spaces). In this paper, we propose a particular case of the UGW problem, termed Partial Gromov-Wasserstein (PGW). We establish that PGW is a well-defined metric between mm-spaces and discuss its theoretical properties, including the existence of a minimizer for the PGW problem and the relationship between PGW and GW, among others. We then propose two variants of the Frank-Wolfe algorithm for solving the PGW problem and show that they are mathematically and computationally equivalent. Moreover, based on our PGW metric, we introduce the analogous concept of barycenters for mm-spaces. Finally, we validate the effectiveness of our PGW metric and related solvers in applications such as shape matching, shape retrieval, and shape interpolation, comparing them against existing baselines.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces the an unbalance Gromov-Wasserstein distance, which adopted the formulation of unbalance optimal transport into the Gromov-Wasserstein with total variance (TV) penalty instead of KL. This new distance allows comparison of probability measures from different space with partial amount of mass, and so they name it Partial Gromov-Wasserstein (PGW). They further prove the metric properties of this distance and proposed two algorithms to solve PGW: Frank-Wolfe algorithms and Line search method. In experiment section, they test PGW with different tasks including: shape-matching, shape retrieval and barycenter problem.\n\nSTRENGTHS:\nThis paper fills a gap in the topic of unbalance (GW) problems by introducing a TV-relaxed unbalanced GW, they call it Partial Gromov-Wasserstein (PGW). This work shows the theoretic metric properties of this distance with a solid proof, making a contribution within this topic. They proposed two solvers to compute this distance and show diverse experimental applications. The experiment results show that this new variance of unbalance GW performs effectively with data containing outliers, aligning with the anticipated performance of unbalanced Wasserstein or unbalanced Gromov-Wasserstein methods on noisy data.\n\nWEAKNESSES:\n- It's worth to note in literature review the similar works formulating partial Waserstein as a metric with TV constraints [1] [2].\n- The proof of the metric properties is not well displayed in the main text, as this is the main highlight of this work.\n- Further comparison with KL version was lack as regards to robustness against outliers. And also, further discussion on the scalability (very large dataset) will be appreciated.\n- In experiment section, the selection of hyperparameter is not clear. The justification of choosing $\\lambda$ value for both PGW and MPGW method was not presented. \n\n[1] Raghvendra, Sharath, Pouyan Shirzadian, and Kaiyi Zhang. \"A New Robust Partial $ p $-Wasserstein-Based Metric for Comparing Distributions.\" arXiv preprint arXiv:2405.03664 (2024).\n\n[2] Nietert, Sloan, Rachel Cummings, and Ziv Goldfeld. \"Robust estimation under the Wasserstein distance.\" arXiv preprint arXiv:2302.01237 (2023).\n\nQUESTIONS:\n- Could you add the parameter sensitivity analysis in experiment section? \n- The table under line 271 doesn't have a title, and in (a) the performance of MPGW is $0$ on Dataset II which looks abnormal to me. Is this the actual experiment result or just a typo?\n\nLIMITATIONS:\nNone", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2024-07-12"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper considers an unbalanced version of the Gromov-Wasserstein distance, where the discrepancy terms correspond to the total variation between certain product measures for the given marginal and the marginal of the solution, respectively. Different (re)formulations of this distance is considered in both the discrete case and for general measures, existence of optimal solution and metric properties are shown for certain cost functions, as well as numerical methods for computing (local) optimal solutions. Finally a number of numerical experiments are considered.\n\nThe paper is well written and extensive. In the main paper the results are stated and with proofs in the  appendix.\n\nSTRENGTHS:\nThe paper is strong in terms of both content and the presentation. In particular the results about the metric properties of the PGW problem. It is also a quite complete paper in terms that is contains substantial results on theory, computational methods, and numerical simulations.\n\nWEAKNESSES:\nOne weakness with the methods in the paper is that it only provides local optimal solutions. This is a problem with many non-convex problems, but in some cases global solutions can be guaranteed. In the abstract it is stated the methods solve the PGW problem. This is probably to strong statement since they are not guaranteed to converge to the global solution.\n\nQUESTIONS:\n1) In the abstract it is stated the methods solve the PGW problem. This is probably to strong statement since they are not guaranteed to converge to the global solution. \n\n2) I think that some early references on the partial/unbalanced OT problem are missing. Neither of the following two papers are mentioned in the introduction. Both papers contain the problem formulation (3).\n\nGeorgiou, Tryphon T., etal. \"Metrics for power spectra: an axiomatic approach.\" IEEE Transactions on Signal Processing 57.3 (2008): 859-867.\n\nPiccoli, Benedetto, and Francesco Rossi. \"Generalized Wasserstein distance and its application to transport equations with source.\" Archive for Rational Mechanics and Analysis 211 (2014): 335-358.\n \n3) Some minor typos: \nline 174: he --> the\nline 175: con --> cost\nline 574: spit --> split\n\nLIMITATIONS:\nNA", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2024-07-14"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces the Partial Gromov-Wasserstein (PGW) metric as a means to handle unbalanced Gromov-Wasserstein problems between non-probability mm-spaces. The authors develop and demonstrate two computationally efficient variants of the Frank-Wolfe algorithm for solving the PGW problem. They establish that PGW is a well-defined metric, providing theoretical proofs and applications in shape matching, shape retrieval, and interpolation. The metric and algorithms are validated through numerical experiments against established baselines, showcasing improved results in handling outlier data with a robust performance.\n\nSTRENGTHS:\n1. The paper presents a novel approach to defining a metric in the context of unbalanced Gromov-Wasserstein, which has been a challenging issue in the field.\n\n2. The quality of the research is high, evident from rigorous theoretical developments, proofs, and comprehensive experiments that validate the effectiveness of the PGW metric in practical applications.\n\n3. The paper is well-organized, with clear explanations of complex concepts. The use of examples and experimental results helps in understanding the practical implications and advantages of the PGW metric.\n\nWEAKNESSES:\n1. While the paper provides a comparison with existing methods, it could benefit from a broader range of comparative baselines, especially newer techniques that might provide a stiffer benchmark.\n\n2. The paper does not extensively discuss the sensitivity of the PGW algorithm to the choice of its parameter (e.g., the regularization coefficient - lambda), which is crucial for understanding its robustness and adaptability in diverse real-world scenarios.\n\n3. There is limited discussion on the scalability (or the time complexity) of the proposed methods, particularly in high-dimensional or large-scale settings, which is vital for their applicability in big data applications.\n\nQUESTIONS:\n1. The regularization coefficient lambda is referenced in Equations (9), (10), and (14) of $\\tilde{M}$. However, it appears that two of the algorithms do not explicitly incorporate lambda. Could you clarify whether these algorithms require the use of lambda?\n\n2. If lambda is indeed utilized in these algorithms, how does the selection of lambda influence the performance of the PGW metric? Are there guidelines or methods for choosing this parameter optimally based on the dataset characteristics?\n\n3. In practical applications, especially in high-dimensional spaces, what are the computational limitations of the proposed algorithms, and how might these be mitigated?\n\n4. Are there potential extensions of the PGW metric that could handle non-metric spaces, given the increasing interest in such spaces in various applications?\n\nLIMITATIONS:\nIncluding limitations on the scalability and time complexity, such as the maximum solvable problem size within one hour, would be beneficial for its applications.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2024-07-16"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a partial Gromov -Wasserstein (PGW) formulation, which relaxes the original constraints present in Gromov-Wasserstein (GW) formulation. In PGW, the marginal equality constraints of GW are replaced by marginal inequality constraints. Following existing works in partial GW setting, the paper additional imposes TV-based marginal regularization in the objective. The paper showed that the proposed PGW approach defines a metric between metric measure spaces. The PGW problem is solved via Frank-Wolfe (FW) and empirical results are shown on shape interpolation  in the main paper.\n\nSTRENGTHS:\n- The paper explores partial mass transportation setting in the GW problem. As  discussed around lines 149, an existing work [45] has also explored similar concepts in GW setting. [45], in particular, involves an additional hyperparameter (\\rho) for overall mass of the learned transport plan. The proposed problem (16) as well as [45] employs FW algorithm. \n    - Proposition L.1 in the appendix shows that if \\gamma is an optimal solution of proposed PGW problem, then \\gamma is also an optimal solution of the mass constrained PGW problem (MPGW) of [45] with \\rho=|\\gamma|. \n    - It is unclear what the paper implies by \"mathematically this equivalence relation is not verified.\" in line 955? How is the paper defining the term \"equivalence\" which is used multiple times in Appendix L. \n    - For a given \\lambda=\\lambda_0 in PGW, does there exist a \\rho=\\rho_0 for MPGW such that the set of first order critical points for PGW and MPGW are same?\n    - The steps of the FW algorithm for proposed PGW and MPGW [45] seem quite similar.\n\nWEAKNESSES:\n- The paper is poorly written due to the following reasons:\n    - The abstract and introduction states that the paper propose two variants of FW algorithm for solving the proposed PGW. However, the main paper does not describe two variants of FW algorithm. Only one variant is discussed in Sections 4-5. The other variant is described only in Appendix G. If the algorithm Typically, the main paper should be self contained, having the necessary details of the contributions claimed in the abstract and introduction. It should be noted that going through the supplementary material is a reviewer's discretion. \n    - The paper provides very less discussion on how its differences with MPGW [45] in the main paper (lines 148-149). While Appendix L contains this discussion, important parts of it should be discussed in the main paper. As mentioned above, going through the supplementary material is a reviewer's discretion. \n    - There are multiple grammatical and spelling errors throughout the paper. Eg. he -> the, con -> cost, etc. \n\n- 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. This should be discussed in the main paper as well.\n\nQUESTIONS:\n- In (14), let \\beta =  minimum entry in cost tensor M. Then, what is the solution of PGW (16) when \\beta > 2\\lambda ?\n\n- Why does MPGW obtains 0.0813 and 0 mean accuracy in Table (a) while other methods get > 0.89 and > 0.78 scores, respectively? It should be noted that in Tables 2 and 4 in appendix, MPGW performs at par with PGW.\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-23"}