Gaussian Herding across Pens: An Optimal Transport Perspective on Global Gaussian Reduction for 3DGS

Thank you for your insightful and positive feedback. We are encouraged that you found the paper well-written, the ideas and explanations clear, and the results impressive. We also appreciate your recognition of the visualizations, algorithms, and formulas in supporting understanding. Below, we address your comments point by point and will incorporate all suggestions into the final version.

• Run time comparison. We have included a theoretical analysis in Remark 1 of the manuscript. The total time cost is $O(n\log n + \rho s^2\log n)$. In response to your suggestion, we have also added empirical comparisons of runtime and memory usage against other post-processing compaction baselines (see table below). Our method achieves the fastest runtime with comparable memory consumption, resulting in negligible overhead when integrated into the rendering pipeline—while also delivering the highest PSNR.

• Comparison to Mini-splatting. In line with the your suggestion, we include Mini-Splatting for a more comprehensive comparison. The only difference between MiniSplatting and MiniSplatting-D+GHAP (our GMR pruning integrated into MiniSplatting) lies in the pruning step—our method replaces the default pruning with our proposed GMR-based approach. As shown in the results, this substitution significantly improves performance, thereby highlighting the effectiveness of our proposed pruning strategy.

• Design of the loss/cost function and refinement settings. We would like to clarify that we do not claim our loss function is optimal. The goal of our paper is to show that 3DGS compaction can be framed as a GMR problem. Our proposed loss serves as an initial exploration in this direction and has shown promising results in practice. We conducted additional ablation studies.

  • Ablation study on loss function to improve the LPIPS quality. We evaluated the impact of adding LPIPS to the loss during appearance optimization. As shown below, increasing the LPIPS weight improves LPIPS scores, but slightly reduces PSNR and SSIM:

  Metric/Loss	SSIM↑	PSNR↑	LPIPS↓
  original (0.8 * L1 + 0.2 * ssim)	0.818	23.312	0.242
  0.8 * L1 + 0.1 * ssim + 0.1 * lpips	0.793	23.0572	0.239
  0.6 * L1 + 0.2 * ssim + 0.2 * lpips	0.805	22.993	0.207

  • Ablation study on other cost functions. Since a Gaussian primitive includes both mean and covariance, considering only position or covariance during compaction is insufficient. Per your suggestion, we tested adding opacity to the cost function. However, this yielded no notable improvement:

  Metric/Cost	SSIM↑	PSNR↑	LPIPS↓
  original cost	0.818	23.312	0.242
  + opacity	0.814	23.266	0.250

• Ablation study on joint geometry and appearance fine tuning. We also tested multi-stage compaction during appearance optimization. As shown below, jointly refining geometry and appearance over multiple stages does not yield better performance:

• Clarification on the interpretation in Figure 4. We agree that the grass in Figure 4 (Bicycle) looks better before compaction, and we will revise our wording for clarity. The intended point is that our GHAP method is a post‑processing method: when the starting representation (here, Atom‑GS) lacks a strong densification strategy, compaction can further attenuate fine textures (e.g., grass), leading to a noticeable quality drop. In contrast, when GHAP is applied to a backbone with a more effective densification strategy (e.g., Mini‑Splatting‑D), it better preserves the visual quality on the Bicycle scene, as shown in Figure 8 of the supplementary material.
• Clarification on the comparison with random subsampling. The random subsampling setting is not intended as a competitive baseline, but rather as an ablation to highlight the advantage of our structured compaction strategy. Its basic procedure is similar to GHAP: after training, it randomly removes a fixed percentage of Gaussians from the trained 3DGS model and then performs appearance optimization. We will clarify the purpose of this experiment and describe the procedure more explicitly in the revised manuscript to avoid confusion. We have included comparison with other baselines such as Mini‑Splatting during rebuttal and the results show that our method can achieve SOTA performance.
• Discussion on large Gaussian primitives that overlap with neighboring blocks. In our current formulation, we assign each Gaussian to a single block based on its mean. This is mainly due to the observation that the eigenvalues of the covariances are very small, hence have limited impact.
• Code release. Due to rebuttal format constraints, we are unable to share a GitHub link at this stage. In the final version, we will release the source code along with example .ply files (before and after GHAP) and provide instructions for visualization and reproduction.
• Incorporating suggestions for paper writing. We have added the number of Gaussian primitives in Table 2. We will reformat Algorithm 2 in the revised version to improve clarity and readability in the following ways: (i) restructuring the input/output section with a cleaner layout; (ii) refining indentation to better reflect the logical hierarchy of the steps; (iii) clarifying sub-step details and notations for improved understanding. We will also adjust the overall width and formatting to ensure the pseudocode fits well and is easy to follow.

Thank you for your thoughtful feedback. We're encouraged that you found our use of GMR interesting and the KD-tree partitioning effective in reducing computation. We also appreciate your recognition of the clarity and coherence of our presentation. Below, we provide point-by-point responses to your comments and will incorporate all of your suggestions into the final version.

• Additional results with suggested baselines. We have included five additional baselines: Trimming-the-Fat, Mini-Splatting, MesonGS, LocoGS, and 3DGS-MCMC. We excluded Fast Compression of 3D Gaussian Splatting from our comparisons because it is specifically tailored for post-processing compression based on properties of 3DGS, without aiming to compact the number of Gaussians. We also did not include ContextGS, as it does not provide the modified version of the diff-gaussian-rasterization module necessary for reproduction. We would like to emphasize that, for a fixed number of Gaussians, MiniSplatting-D+GHAP (our GMR pruning integrated into MiniSplatting) achieves the best performance among all methods. Our method combined with 3DGS also demonstrates comparable results. Importantly, the only difference between MiniSplatting and MiniSplatting-D+GHAP lies in the pruning step—our method replaces the default pruning with our proposed GMR-based approach. As shown in the results, this substitution significantly improves performance, thereby highlighting the effectiveness of our proposed pruning strategy.

• RD curve comparison. Due to space constraints, we only include RD curves of PSNR for comparisons against LightGaussian, PUP‑3DGS, Trimming‑the‑Fat, and MesonGS on the Tanks&Temples datasets. These baselines were selected to ensure a fair comparison, as they apply post-compaction alone without incorporating other technical innovations. The results show that our method consistently outperforms these approaches across all cases.

• Discussion on potential limitation of the method. Thank you for raising this point. Our method's performance depends on the quality of the initial Gaussian representation. We plan to study different backbones and scene types (e.g., texture-heavy or far-field content), and explore perceptual losses to improve LPIPS fidelity under extreme retention.

Thanks for giving us excellent scores on originality and significance. We respond to your comments below and will incorporate all of your suggestions into the final version.

• Highlight of our technical contribution. We respectfully disagree with the assessment that the technical contribution of our method is limited. We clarify and highlight the key innovations of our work:
  • Novel perspective. We are the first to reinterpret Gaussian primitives in 3DGS as components of a Gaussian mixture, using their density functions to model scene geometry. This contrasts with prior compaction methods that ignore geometric structure and often produce distortions (see Fig. 2), whereas our approach preserves geometric fidelity, offering a new and impactful direction for 3DGS.
  • Adaptation of GMR to 3DGS. We are the first to adapt GMR to 3DGS. We introduce a novel cost function that yields closed-form, low-cost updates. We also develop a block-wise GMR algorithm guided by a KD-tree, enabling efficient large‑scale scene compaction. These strategies are non-trivial and bridges theory with practical scalability.
  • Plug & play design. Our method is post-hoc and compatible with any existing 3DGS pipeline, making it highly practical and broadly applicable. With minimal overhead, our approach achieves SOTA compaction performance, both in quality and efficiency.
• Clarification on Table 2. During the rebuttal process, we found a configuration error in the LightGaussian baseline: it was mistakenly retrained for an additional 30k steps without pruning, exceeding the intended 5k-step budget. This led to significantly more Gaussians (e.g., 1.6M vs. 157k on average), and an unfair comparison. We have corrected the setup and present the updated results below. Our method achieves superior overall performance. We thank the reviewer and apologize for the oversight.

• Comparison with baselines. We have included 5 additional baselines: Trimming-the-Fat, Mini-Splatting, MesonGS, LocoGS, and 3DGS-MCMC. RadSplat is excluded due to unavailability of public code. For a fixed number of Gaussians, MiniSplatting-D+GHAP (MiniSplatting with our pruning) achieves the best performance. Notably, the only difference from MiniSplatting is the pruning step—replacing it with our method yields significant gains, underscoring the effectiveness of our GMR strategy.

• Effectiveness of KD-tree partition. On the NeRF-synthetic-mic scene, our ablation shows that increasing KD-tree depth reduces memory and runtime, while PSNR first improves then drops. This suggests finer partitions help GMR compact regions more effectively, but overly fine splits may fragment primitives and hurt quality.

• RD curve. We compare PSNR RD curves on Tanks&Temples against LightGaussian, PUP‑3DGS, Trimming‑the‑Fat, and MesonGS-baselines using post-compaction only for fair comparison. Our method consistently outperforms them across all cases.

Thank you for your constructive feedback. We’re encouraged that you view 3DGS compaction as a relevant and timely research problem and found our method both novel and effective. Below, we address your comments and will incorporate all of your suggestions into the final version.

• Computational cost of the method.
  • Time complexity. As described in L184, the overall time complexity of our method is $O(n\log n + \rho s^2\log n)$, where $\rho$ is the retention ratio, $n$ is the initial number of Gaussians, $s (\ll n)$ is the maximum number of Gaussians in each block. We use median splits to depth $d = \lfloor \log_2(n/s)\rfloor$, resulting in $2^{\text{depth}} = O(\log n)$ leaf blocks, each with at most $s$ Gaussians. Within each block, we reduce $s$ to $\rho s$ Gaussians. This gives $O(\rho s^2\log n)$ overall cost. Including KD‑tree construction $O(n\log n)$, the total time cost becomes $O(n\log n + \rho s^2\log n)$.
  • Memory complexity. We will add the following analysis of memory consumption in the revision. The peak memory cost is $O(\rho s^2)$, dominated by the distance matrix in GMR. Global GMR requires $O(n \times \rho n)$ memory to store a full distance matrix; with our KD‑tree partitioning, the peak memory reduces to $O(s \times \rho s)$. In practice, batched distance evaluation or parallel processing further reduces workspace.
  • Empirical results. Following your suggestion, we also provide empirical comparisons of runtime and memory usage with other post-processing compaction baselines in the table below. Our method achieves the fastest runtime with comparable memory consumption, resulting in negligible overhead when integrated into the rendering pipeline. Moreover, our method has the best PSNR.

• Comparison with 3DGS-MCMC[24]. In line with your suggestion, we include 3DGS-MCMC for a more comprehensive comparison. MCMC is an end-to-end 3DGS variant rather than a compaction method. Since it allows controlling the number of Gaussians, we also compare it with our method under the same Gaussian budget. As shown in the table, even compared to end-to-end approaches, our post-processing compaction method still achieves superior performance.

• Potential Limitations and possible future directions. Thanks for your suggestion. We will integrate the following potential limitations and future directions in the "Conclusion" section: First, the performance of our method depends on the quality of the initial Gaussian representation. We will further study backbones and scene types (e.g., texture‑heavy or far‑field content) and integrate perceptual losses to strengthen LPIPS‑oriented fidelity under extreme retention. Second, our block‑wise KD‑tree strategy brings strong scalability; going forward we plan to add overlap‑aware or multi‑scale partitioning to better handle large Gaussians spanning neighboring blocks and further improve efficiency. Third, we will move from fixed hyperparameters to auto‑tuned schedules guided by validation RD metrics and visibility statistics. Finally, future work also includes extending our method to dynamic 3DGS for real-time temporal scenes and developing adaptive reduction strategies for task-specific needs.
• Incorporating writing suggestions. Thank you for your suggestion and careful observations. We will incorporate the writing suggestions and improve the readability of Figure 3 by enlarging the font sizes and adjusting the layout of the flowchart. We will also correct the typo in Figure 4. Regarding the PSNR values in Figure 4, we have confirmed that they are not switched. Unlike Table 2, which reports the average PSNR across views, Figure 4 presents the PSNR for a single representative view of each image. We will clarify this distinction in the final version of the paper. Additionally, we will include difference maps to better highlight subtle changes.