Unpaired Point Cloud Completion via Unbalanced Optimal Transport

We sincerely thank the reviewer for carefully reading our manuscript and providing valuable feedback. Moreover, we appreciate the reviewer for considering our work addresses "limitations in existing heuristic-driven methods" by "bridging OT theory and unpaired point cloud completion". We hope our responses to be helpful in addressing the reviewer's concerns.

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Q.[Questions For Authors] Why is InfoCD the most suitable for completion tasks? Please provide a theoretical explanation. MSNet and SpareNet previously used EMD to calculate the distance between point sets. Would EMD based on optimal transport be better than InfoCD?

A. Thank you for providing the insightful comments. Following the reviewer's suggestion, we conducted an additional ablation study using the EMD as the cost function for our UOT-UPC model. The results are as follows (Table 7):

• Comparison of EMD cost function with other cost functions on the USSPA benchmark, assessed by L1 Chamfer Distance $cd^{l 1} \times 10^2$ ($\downarrow$).

Our results show that InfoCD consistently outperforms EMD across all evaluated categories. Additionally, training with EMD incurs significantly higher computational costs due to its iterative computation procedure, as shown below:

• Train time comparison on 'lamp' category from the USSPA [1].

InfoCD is designed to prevent multiple points from being matched to a single point, encouraging a more evenly distributed alignment between point sets, compared to EMD (Fig 4 in [1]). This advantage arises from the contrastive nature of the InfoCD cost function [1], which introduces an additional repulsion effect between negative pairs. This property aligns well with the goals of point cloud completion, where generating globally coherent and evenly spaced completions is critical. Empirically, this is supported by the higher-fidelity completions of the UOT-UPC model using InfoCD in Fig 2 and 10, where the completions exhibit more globally evenly separated completions.

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Reference

• [1] Lin, Fangzhou, et al. "InfoCD: a contrastive chamfer distance loss for point cloud completion." NeurIPS 2023.

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Q. [Theoretical Claims] ... The paper acknowledges that the UOT-UPC training process may exhibit instability, resembling the mode collapse phenomenon observed in GAN training. However, no theoretical analysis is provided to explain why UOT-UPC training might be unstable.

A. We appreciate the reviewer for providing constructive comments. As the reviewer said, we believe the observed training instability is due to the inherent difficulty of finding a Nash equilibrium in the min-max optimization, similar to challenges encountered in GAN training. In this regard, several works investigated GAN instability both theoretically [1] and empirically [2]. We agree that extending this analysis to the UOT-UPC setting would be a valuable future research direction.

Reference

• [1] Mescheder, Lars, Andreas Geiger, and Sebastian Nowozin. "Which training methods for GANs do actually converge?." ICML 2018
• [2] Salimans, Tim, et al. "Improved techniques for training gans." NeurIPS 2016.

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Q.[Other Comments Or Suggestions] The results of the proposed approach should be added to Figure 11.

A. We agree with the reviewer that including UOT-UPC results in Fig 11, i.e., providing a completion result on the exact same incomplete point cloud, would allow a clearer qualitative comparison. However, we were unable to find the random seed necessary for reproducibility in the ACL-SPC model. As an alternative, we chose to present multiple completion results on the KITTI dataset in Fig 10. Across multiple incomplete point clouds, our model consistently produces higher-fidelity completions, exhibiting better global structure and more evenly distributed points.

We sincerely thank the reviewer for carefully reading our manuscript and providing valuable feedback. We are especially grateful for the reviewer’s recognition of our contribution, noting that “this paper pushes forward the state of the art of point cloud completion by considering the unpaired problem and designing a strong method based on optimal transport, which is also robust to class imbalances.” Moreover, we appreciate the reviewer's positive assessment of our cost function analysis and empirical evaluations under class imbalance.

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Q. [Other Comments Or Suggestions] You define the "unbalanced optimal transport map" on line 129, but does this necessarily exist?

A. Thank you for the great question. In this work, we assume that the incomplete and complete point cloud distributions are absolutely continuous with respect to the Lebesgue measure (Lines 76-77). This is a mild assumption, as it is satisfied whenever the distributions admit probability density functions (pdfs). Under this assumption, the existence of the unbalanced optimal transport map is guaranteed, as stated in Thm 3.3 in [1].

Reference

• [1] Liero, Matthias, Alexander Mielke, and Giuseppe Savaré. "Optimal entropy-transport problems and a new Hellinger–Kantorovich distance between positive measures." Inventiones mathematicae 211.3 (2018): 969-1117.

We sincerely thank the reviewer for carefully reading our manuscript and providing valuable feedback. Moreover, we appreciate the reviewer for considering "the most important contribution is that this paper is the first to bring UOT to unsupervised point cloud completion". We hope our responses to be helpful in addressing the reviewer's concerns.

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Q. [Other Strengths And Weaknesses] The paper's clarity could be improved by explicitly defining important benchmarks before using them in the text. For example, Section 3.1 first compares UOT-UPC with USSPA without first introducing USSPA.'

A. We appreciate the reviewer for providing valuable suggestions. Following the reviewer's advice, we moved the Related Works section (Sec 4) before the Method (Sec 3). This reordering allows us to introduce USSPA before presenting experimental results in Section 3.1.

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Q. [Other Comments Or Suggestions] The name of the approach is misspelled in the bolded section of the third paragraph in the introduction. Also, the beginning of Section 3.1 defined the set $X$ as: $X = \{x_i \mid x_i \in X, i = 1, \dots, N\}$ for incomplete point clouds and the set $Y$ as: $Y = \{y_j \mid y_j \in Y, j = 1, \dots, M\}$ for complete point clouds. However, the cost function comparison uses $n$ for complete and $m$ for incomplete point clouds. I believe it would be a bit clearer to change the variables used for the Task formulation section.

A. We appreciate the reviewer for the careful comment. (1) We corrected the typo in the bolded section of the third paragraph in the introduction. (2) Following the reviewer's advice, we revised the variable notation in the cost function definitions to ensure consistency. Specifically, we now consistently use the $n$ for the incomplete point clouds $x$ and $m$ for the complete point clouds $y$ throughout the paper.