Gaussian Herding across Pens: An Optimal Transport Perspective on Global Gaussian Reduction for 3DGS
Authors: Tao Wang
Mengyu Li
Geduo Zeng
Cheng Meng
Qiong Zhang
Venue: NeurIPS 2025 spotlight ()
Keywords: 3D Gaussian splatting
model compaction
Gaussian mixture reduction
composite transport divergence
optimal transport
Primary area: applications
TL;DR: We introduce GHAP-3DGS, an agnostic optimal transport-based global Gaussian reduction method for 3DGS that achieves high rendering fidelity with only 10% Gaussians and outperforms state-of-the-art compaction techniques.

Abstract: 3D Gaussian Splatting (3DGS) has emerged as a powerful technique for radiance field rendering, but it typically requires millions of redundant Gaussian primitives, overwhelming memory and rendering budgets. Existing compaction approaches address this by pruning Gaussians based on heuristic importance scores, without global fidelity guarantee. To bridge this gap, we propose a novel optimal transport perspective that casts 3DGS compaction as global Gaussian mixture reduction. Specifically, we first minimize the composite transport divergence over a KD-tree partition to produce a compact geometric representation, and then decouple appearance from geometry by fine-tuning color and opacity attributes with far fewer Gaussian primitives. Experiments on benchmark datasets show that our method (i) yields negligible loss in rendering quality (PSNR, SSIM, LPIPS) compared to vanilla 3DGS with only 10\% Gaussians; and (ii) consistently outperforms state-of-the-art 3DGS compaction techniques. Notably, our method is applicable to any stage of vanilla or accelerated 3DGS pipelines, providing an efficient and agnostic pathway to lightweight neural rendering.

Gaussian Herding across Pens: An Optimal Transport Perspective on Global Gaussian Reduction for 3DGS
Tao Wang
Mengyu Li Geduo Zeng' Cheng Meng 131
Qiong Zhang 131
Center for Applied Statistics, Institute of Statistics and Big Data, Renmin University of China
Department of Statistics and Data Science, Tsinghua University
3 Big Data and Responsible Artificial Intelligence for National Governance, Renmin University of China wang_ tao@ruc. edu. cn sengyuli@tsinghua edu.. en geduozengêruc edu. cn chengmengêruc edu. qiong. wwwnngruc..n..... edu..
Abstract
3D Gaussian Splatting (3DGS) has emerged as a powerful technique for radiance field rendering, but it typically requires millions of redundant Gaussian primitives, overwhelming memory and rendering budgets. Existing compaction approaches address this by pruning Gaussians based on heuristic importance scores, without global fidelity guarantee.
bridge this gap, we propose novel optimal transport perspective that casts 3DGS compaction global Gaussian mixture reduction. Specifically, we first minimize the composite transport divergence over a KD- tree partition to produce compact geometric representation, and then decouple appearance from geometry by fine-tuning color and opacity attributes with far fewer Gaussian primitives. Experiments benchmark datasets show that method (i) yields negligible loss in rendering quality (PSNR, SSIM, LPIPS) compared to vanilla 3DGS with only 10% Gaussians; and (ii) consistently outperforms state- of-the-art 3DGS compaction techniques. Notably, our method applicable to any stage of vanilla or accelerated 3DGS pipelines, providing efficient and agnostic pathway to lightweight neural rendering. The code is publicly available at petps://githhe //tghttt..../...////////g wwwm/rrnkennoet.....
https: /
PUP-3DGS
3DGS
GHAP- 3DGS (ours)
PSNR: 25,374, SSIM: 0.886, LPIPS: 0.188
PSNR: 27.614, S5IM: 0.520, LPIPS: 0.155
PSNR: 27,217, SSIM M.8.86 LPIPS 0.211
Figure Visual comparison. When reducing the number of Gaussians by 90%, our method outperforms other compaction techniques, such as PUP-3DGS, and remains competitive with the original 3DGS.
1 Introduction
Real-time 3D scene reconstruction and rendering dynamically generates photorealistic 3D represen- tations from sensor data (e.g., multi-view images, LiDAR) with minimal latency, enabling critical applications in augmented/virtual reality (AR/VR), autonomous navigation, and immersive me- dia [1, 2]. The current state-of-the-art, 3D Gaussian Splatting (3DGS) [3], iteratively learns 3D)
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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anisotropic Gaussian primitives with color and opacity attributes to model these scenes. During rendering. these 3D Gaussians are projected 2D screens and a-blended to achieve real-time photorealistic synthesis.
However, 3DGS faces significant efficiency challenges: its iterative densification process often produces millions of redundant Gaussians for complex scenes [4, 5, 6]. This inefficiency leads high memory/storage costs and increased per-frame rendering time, limiting deployment on resource-constrained platforms like mobile and AR/VR devices [7].
A common strategy improve ef- ficiency is compaction [8, 9, 10, 7]reducing the number of Gaussians while preserving rendering fidelity. Fewer primitives result in reduced storage needs and faster rendering, im- proving per-frame performance. This approach is viable because 3DGS den- sification inherently generates redun- dant primitives [4, 5, 6]. Existing methods achieve compaction via prun- ing random subset selection [11, 12,
Original Gaussien Mixture
Pruning
Global Reduction (ours)
Denaity
Figure 2: Comparison of heuristic pruning and our method. The original mixture (left) with 10 components is reduced to 5% using either pruning (middle) our method (right), Our method better preserves the overall structure.
5, 13, 14. 15. 6, 4, 16]. These strategies naïvely discard Gaussian primitives; while simple to im- plement, they are often ineffective. In particular, they tend to lose critical structural details (as shown in Fig. 1) or distort the underlying geometry (as shown in Fig. 2). Such losses can degrade rendering quality, particularly in regions with fine-scale features or complex material properties. These limitations motivate our key question:
How to design an efficient compaction method that preserves 3D spatial and structural geometry?
To address this, we frame compaction as an optimization problem, where the goal is to approximate the original 3DGS representation with fewer Gaussians. Our solution leverages a statistical perspective, treating the scene's geometric structure probabilistic model and employing principled reduction techniques to preserve fidelity.
(i) Geometric compaction via GMR. We first observe that the geometry 3DGS representa- tiondefined by the positions, covariances, and opacities of its Gaussian primitivescan be interpreted as a Gaussian mixture model (GMM). Here, the mixture density is a convex combination of individual Gaussian densities, weighted by their opacities. This formulation naturally connects 3DGS compaction to Gaussian Mixture Reduction (GMR), a well-studied problem in statistics where high-order GMM is approximated by with fewer components while minimizing divergence measure.
(ii) Appearance optimization. While geometry is compacted via GMR, the appearance of the scenegoverned by the color attributes of the Gaussiansmust also be preserved. To achieve this, we decouple the optimization of geometry (position, covariance) and appearance (color, opacity). After EMsed compaction, we fine-tune the color and opacity of the reduced set of Gaussians using the standard 3DGS training pipeline. This two-stage strategy ensures that the compacted model maintains both geometric accuracy and photorealistic rendering quality.
For GMR, we minimize the composite transportation divergence [17], which is rooted in optimal transport theory [18, 19] and allows an effective algorithm. We tailor this GMR algorithm (detailed in Section 3. 3.2.1) so that it scales well in scenes like 3DGS with extensive amount of Gaussians. Crucially, our GMR optimizer does not merely select subset of existing Gaussians; instead, it creates new primitives that dynamically adjust their positions and covariances better approximate the underlying geometry (as shown in Figure 2). Our method is fundamentally functions a plug-and-play module that can enhance both the standard 3DGS pipeline and any of its variants (Section 4), applicable at any stage of training to boost computational efficiency.
sesorithm-agnostie
To summarize, contributions

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We open a new pathway to view 3DGS representations as Gaussian mixture and perform compaction from the perspective of Gaussian mixture reduction via optimal transport. This contrasts with prior compaction methods that ignore geometric structure and often produce distortions, whereas our approach preserves geometric fidelity, offering a new and impactful direction for 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 block-wise GMR algorithm guided by KD-tree. enabling efficient largescale scene compaction. These strategies are non-trivial and bridges theory with practical scalability.
Our method is post-hoc and compatible with any existing 3DGS pipeline, making it highly practi- cal and broadly applicable. With minimal overhead, our approach achieves SOTA compaction performance, both in quality and efficiency.
Empirical results demonstrate that our method preserves rendering quality 10% retention ratio.
2
Related Works
Compaction Techniques in 3DGS. 3DGS compaction seeks to minimize the Gaussian count while preserving image quality. Existing work falls into two operations: densification (where to add) and pruning (what to drop). Most current approaches rely on per-Gaussian heuristic scores.
For densification, Taming3DGS [10] ranks candidate Gaussians by combining gradient, pixel cover- age, per-view saliency, and core attributes including opacity, depth, and scale; Color-cued GS [20] considers the jiew-independent spherical harmonics coefficient gradient to better capture color cues; and GaussianPro [9] guides growth using depth and normal maps. the pruning phase, LightGaus- sian [11], Mini-Spa Siini-Splatiing [12]. RadSplat [5], and AtomGS [13] compute importance score from each Gaussian's s accumulated ray contribution, typically a mix of volume, opacity, transmittance, and hit count, and discard the lowest-ranked Gaussians. Gradient-aware variants prune by per-Gaussian gradients (Trimming- -eeiiing-te---aee the-Fat) [14] or the second-order sensitivity score derived from the Hessian matrix (PUP-3DGS) [15]. The score also be trained via learnable mask, in LP-3DGS [6], Compact3DGS [4], and HAC [16]. Moreover, multi-view consistency criteria discard Gaussians unseen by keyframes [21] or visible only in real but virtual views [22]. For comprehensive survey of 3DGS compression and compaction, we refer readers to [8, 23).
Despite their success, most existing strategies evaluate each Gaussian independently, leaving open the question of whether the retained set is truly the best global surrogate. Our work addresses this gap from probabilistic perspective via Gaussian mixture reduction.
[24] formulate 3DGS as Markov chain Monte Carlo process and use stochastic gradient Langevin dynamics to migrate dropped Gaussians onto retained ones, partially recycling lost information. However, their update remains pairwise and lacks convergence guarantees under principled divergence. Moreover, this scheme is coupled with the original 3DGS pipeline, limiting its generalizability to other variants.
3DGS from Probabilistic Distribution Point of View. Kheradmand
3 Method: Gaussian Herding across Pens
3.1 Probabilistic Scene Representation
Let (b;; μu, Σ) = || 1 exp(-(x - μd) E n)) the PDF Gaussian distribution with mean y and covariance Σ. A Gaussian mixture with n components is a distribution with density:
(x = Σ aqol1; E,),
where are the mixture weights satisfying a; > 0'. In the context of 3DGS, let E R be spatial coordinates and fzx represent the implicit surface function (i.e., geometric shape and opacity that
In statistics, Gaussian mixtures are defined with the constraint > a; = to ensure the area under the density function is 1. However, we relax this constraint and consider unnormalized Gaussian mixtures, as in 3DGS, where the integral under the geometric surface need not equal
3

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KD-tree pariiion
Blockotse GHil Dooraiie Σ
Appearance opt. Geomanie Bratur R
SDGS training modula
B
Osemetiee
Apurentro
GHAP
I
Featurn
feature
Latopa
ps
Q C
Gcussions
(6g. Ps.
Figure 3: An illustration of the proposed GHAP approach. The process begins with full-resolution 3DGS training to obtain initial geometric and appearance features. These Gaussians are then spatially partitioned using KD-tree and grouped into blocks blocks-analogon analogous to sheep pens. We then perform blockwise Gaussian Mixture Reduction (GMR) to approximate the geometric shape within each block using much smaller number of Gaussians. This step is analogous to the popular kernel herding method [25]. Finally, a lightweight appearance refinement step further optimizes the appearance feature of the reduced set. This multi-stage pipeline progressively guides the Gaussians in each block-analogous to herding across pens-toward compact and high-fidelity representation.
excludes color). The training process in 3DGS learns opacity parameter d;, location parameter 14: and shape parameter Σ, such that
On Rz f(x), Vo E X,
where X' the 3D scene volume. Therefore, the geometry of the 3D scene can be effectively represented by Gaussian mixture. Then, each of these Gaussian primitive is associated with its own color o; Both the geometry and appearance attributes are important for high quality rendering.
3.2 Compaction via Optimal Transport
Motivated by the observation that many 3DGS algorithms [4, 5, 6] produce a significant number of redundant Gaussians during training, we improve rendering efficiency through compactionreducing the number of Gaussian primitives to achieve lower memory usage and faster rendering while preserving visual fidelity. The process consists of two key phases:
1. Geometric Compaction via GMR: Leveraging our probabilistic interpretation, we formulate compaction as Gaussian Mixture Reduction (GMR) [26], approximating the original Gaussian mixture with redundant components by one with fewer components. This yields compacted geometric representation: (x) Σタ= (zz((;; Jú, Eÿ), where m << n. This step modifies only the Gaussian positions (fzz) and covariances (Ez). leaving appearance attributes unchanged.
2. Appearance Optimization:
The reduced Gaussians are initialized with appearance attributes (col- ors, opacities) and fine-tuned for optimal rendering performance. This step optimizes appearance only. maintaining geometric consistency.
Our approach decouples geometry and appearance optimization while using standard 3DGS training to preserve quality. Our training pipeline is visualized in Figure and describe the details below.
3.2.1 Geometric Compaction via GMR
Following Zhang et al. [17], formulate compaction minimizing the composite transportation divergence (CTD) between two Gaussian mixtures:
Definition (Composite transportation divergence). Let cl...) be a divergence between two Gaussians. The composite transportation divergence (CTD) between Gaussian mixtures on (x) = E1=1 a; pú; Σx) and (x. = Σn a, d(;(;; uly Ex) with cost function c(.,.) is
Te(dn, )'m) = inf
[,,0(((;1,,,)) E(), E3))
Σ
Fij
ai
Tij = d
1=1

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The CTD generalizes optimal transport [18] to mixtures, treating each component as discrete distribution the space of Gaussian distributions. The cost function measures the cost of moving one unit of Gaussian from one location to another, and Tij measures the corresponding amount of mass that is being moved. The total cost is proportional the cost and the mass, and the divergence is the smallest transportation cost to move the original mixture to the target mixture. The reduced mixture becomes
{āz Bg. E,}
arg min Te(ons P'm)
With this formulation, the Gaussian mixture after compaction has optimal guarantee. The solution also guides the choice of m to balance compactness and fidelity.
As shown in Zhang et al. [17]. (1) can be solved using the effective iterative algorithm in Algorithm 1. k-means vari-
The algorithm reduces ant in Gaussian space: 1) The assignment step follows the same principle as tradi- tional k-means, but replaces the L dis- tance between vectors with a cost function
Algorithm 1 GMR via k-means Clustering
[0))m 2: for t=l.... do
1: Initialize {at "j 13=1
3: Assignment Step:
. between Gaussian distributions. 2) The update step generalizes the cluster cen- ter computation: In traditional k- k-means, centers are updated as arithmetic averages (barycenters w.r.t. L distance) of vectors in each cluster. this algorithm, cen- ters become barycenters of Gaussians in each cluster, minimized w.r.t. cost function -...)). Thus, standard k-means emerges a special case when using I.2 distance on vectorized Gaussian parameters.
for to n do
P (nm)
5:
Assign component to cluster Cj that mini-
mizes c(0(:), ((3 (-)), .. 5(((-1))
6:
end for
7:
Update Step:
8:
for j = to m do
b (mm)
9:
Compute new cluster center: p
(t) 5(0
arg min Eiec, (u, E,), ol M4, E))
10:
end for
11:
if no change in assignments then
While the standard GMR algorithm pro- vides optimal theoretical guarantees, its
12:
for jorj= to m do Compute weight: ā₃- = Eiec, as
D (n)
13:
direct application to 3DGS compaction 14:
end for
proves computationally prohibitive. Al-
15:
break
though the algorithm must converge in fi- nite steps [17] the assignment step in- volves nm evaluations of the cost function 17:
16:
end if
end for
R2(10)3, and even after 95% reduction, each iteration would still require at least 10 operations and memory storage. For the update step. the computational cost for the cluster center depends on the pre-specified cost function c(.,)). The KL. divergence considered in Zhang et al. [17] suffers from (a) significant overhead of computing Ops" log n)) covariance matrices inversions, and (b) numerical instability due to small eigenvalues of covariance matrices. To overcome this challenge, we introduce two key optimizations designed for 3DGS:
per iteration. In typical 3D scenes, the number of Gaussians scales
Blockwise GMR via KD-Tree: To improve computational and memory efficiency during training, we partition the scene into spatially blocks and perform GMR within each block. As demonstrated in Remark 1, this blockwise approach yields significant computational savings. While both KD- trees [27] and Octrees [28] are effective for spatial partitioning 3D space, we employ KD-tree for two key advantages. First, it produces more balanced partitions across regions. Second, it avoids unnecessary subdivisions in sparse regions that would waste computational resources.
Our KD-tree is constructed solely from Gaussian centers { (justified by the observed small eigenvalues of covariance matrices). Each split uses the median coordinate value, creating 2d blocks at depth d. We set d = (ogg(n/ss) to ensure blocks contain at most Gaussians with < n, then reduce each block to m = ps components (p retention ratio).
Remark (Computational Cost Comparison). Our blockwise approach reduces the per-iteration computational cost from (pn²²) to Ops²)) per block. 4 With 2depth = O n) blocks in total, the
PI particular, we find that in experiment, the algorithm converges in only around iterations. 'The a, = ((b,,) ) (O.h.) )) there exist C 2 such that an 2 Cb2 (a,, S Cb..).
'aach block reduces from Gaussian components ps.
5

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overall complexity becomes (ps log For rypical values of n. = 10", 10", s- 102 and 0.05, this reduces the cost from 10 approximately 10 operations-a a substantial improvement. The savings become even more pronounced for larger n. Furthermore, the reduction steps can be executed in parallel across blocks, offering additional computational speedup.
Efficient Cost Function: We introduce novel cost function that overcomes the limitations of the KL divergence used [17] by being computationally efficient without sacrificing approximation quality. Our proposed divergence is:
c (1, 2), oi: 22)) ||u pov-pia 2' -
which offers three significant advantages: First, it preserves distributional similarity, Gaussian distributions are uniquely determined by their mean and covariance. Second, the assignment step requires only efficient vector and matrix norm computations. Third, the update step simplifies to calculating weighted averages, thereby avoiding the computationally expensive covariance matrix inversions required by the KL divergence:
1 = Exec, aitl
5(1) = Liec, azEi
Eiec, ai
Eiec,
3.2.2 Appearance Optimization
Following geometric compaction, we initialize the appearance attributes (opacity and color) of the compacted Gaussian primitives. For each primitive in the reduced mixture, we assign the appearance parameters from its closest counter- part in the original Gaussian mixture. Using these initial values, we then optimize the appear- ance attributes through backpropagation within the standard 3DGS training pipeline used in the first stage. We optimize the opacity instead of di- rectly using the values from the GMR algorithm
Algorithm 2 GHAP: 3DGS Compaction via Block- wise GMR
1: Input: Trained 3DGS model for T steps to obtain {(as, Mág E,, ) fi= retention ratio p
2: Output: Compacted {(ā), P. Ej, K
3: Stage 1: Geometric Compaction
4: 1. Build KD-tree from Gaussians {us ) with depth d = [logg(n/s)
Ond log n)
5: 2. For each leaf block By: D O(nmT )
6: Run Algorithm 1 to reduce to ps Gaussians
7: Stage 2: Appearance Optimization
because its output weights do not necessarily sat- isfy the constraint that opacity must be between O and 1. Fine-tuning the opacity leads to better visualization performance.
8: 1. Initialize appearance for each oy:
9: Cj + C:- and aj < a;- where i"
arg millg((]] ||ı Mjil2
10: 2. Fine-tune {āz, J using standard 3DGS
rendering pipeline for T steps
3.3 Training Details with GHAP Algorithm
Integrating these components, present our
complete training pipeline in Algorithm 2, called Gaussian Herding Across Pens (GHAP). The process begins with standard 3DGS optimization for T steps, followed by blockwise GMR. We then freeze the geometric parameters (μ and Σ) while fine-tuning the appearance attributes (opacity a and color e) through additional T-step 3DGS optimization. This procedure be applied iteratively throughout training as needed.
4 Experiments
4.1 Experimental Setup
Datasets. For comprehensive evaluation of GHAP algorithm, we three real-world datasets: Tanks
& Temples [29], Mip-NeRF 360 [30], and Deep Blending [31], which cover varying levels of detail, lighting conditions, and scene complexities. For each dataset, we adopt the same scenes in [8].
Tanks & Temples:
We evaluate two unbounded outdoor scenes, "Truck" and "Train", both featuring centered viewpoints.
Mip-NeRF 360: We test mix of indoor and outdoor scenes, including "Bicycle", "Bonsai", "Counter", "Flowers", "Garden", "Kitchen", "Room", "Stump", and "Treehill", all with centered viewpoints.

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We include two indoor scenes, "Dr. Johnson" and "Playroom", where the viewpoint is directed outward.
Deep Blending:
Baselines. To evaluate the effectiveness of our proposed method, compared it against four strong compaction techniques: LightGaussian [11], PUP-3DGS [15], Trimming the Fat [14), and MesonGS [32], as well as four end-to-end 3DGS variants: Mini-Splatting(-D) [12], AtomGS [13], 3DGS-MCMC [24], and LocoGS [33]. Notably, Mini-SplattingDD and AtomGS were employed backbone models in our approach, while the other variants were used for direct comparative evaluation against the compaction methods. A consistent evaluation protocol was established to ensure fair and reliable conclusions. For post-training compaction methods applicable pre-trained models-including GHAP, LightGaussian, PUP-3DGS, Trimming the Fat, and MesonGS-w initialized all from the same backbone model (trained using vanilla 3DGS, Mini- Mini-Splatting-D. or AtomGS for 15k iterations) and applied their respective compaction procedures directly, excluding any compression- specific modules. All models subsequently underwent identical fine-tuning for 15k iterations to achieve the target retention ratio. For the other end-to-end variants (MiniSplatting, 3DGS-MCMI and LocoGS), we executed training for 30k iterations under their default configurations. Detailed experimental steps for each method can be found in the appendix.
Evaluation Metrics. We assess 3DGS compaction using standard metrics for rendering quality. We report: (1) PSNR, measuring pixel-level accuracy; (2) SSIM, evaluating perceptual similarity based on luminance, contrast, and structure; and (3) LPIPS, capturing perceptual distance via a learned model. Higher PSNR/SSIM and lower LPIPS indicate better quality. For each method, we also report the corresponding number of Gaussian primitives.
4.2 Quantitative Results
We first show that our approach outperforms other SOTA compaction techniques. Second, we demonstrate the effectiveness of our GHAP compaction method as plug-in module within 3DGS and its variants. We present the experimental results followed by our key findings.
Comparison with SOTA. We compare our method against a comprehensive set of baselines, which can be categorized into two groups:
End-to-End Compact 3DGS Variants: Mini- Sini-tplatting [12], DDGS-MCMC [24], and LocoGS [33]. Note that for LocoGS, the final number of Gaussians not user-conteollable parameter.
Post- Dest-Training Compaction Baselines: LightGaussian [11], PUP-3DGS [15]. Trimming the Fat [14], and MesonGS [32]. These methods all use a standard 3DGS backbone and apply pruning- techniques for compaction.
Our approach is evaluated in two configurations: 3DGS+GHAP and Minisplattnn=GHAAP The former uses the same vanilla 3DGS backbone and 10% retention rate as the pruning baselines for a direct comparison of compaction strategies. The latter replaces the built-in pruning step in Mini-Splatting with our GHAP algorithm to demonstrate its effectiveness on different backbone.
Quantitative results are summarized in Table 1, with methods grouped by their final primitive count. As expected, LocoGS achieves strong performance due its larger primitive count. Among methods with comparable primitive counts (Group 2). our 3DGS+GHAP achieves superior performance in SSIM and PSNR, with a marginally lower LPIPS score. The advantage of our compaction strategy is further evident when using the Mini-Splatting backbone. Our MiniSplattingGGH... other compaction- based approaches while often using fewer primitives. As shown in Fig. (left), this performance lead is consistent across a wide range of retention ratios, not just at 0.1.
outperforms
Runtime & Memory Usage Comparison. Crucially, the improved performance of method does not come at a computational cost. As depicted in Fig. 4 (middle), our method's 's runtime is faster than all baselines except the exceptionally swift Trimming the Fat. While our method exhibits slightly higher memory footprint (in Fig. 4 right) during compaction due to pairwise distance computation in each KD-tree block, the difference is not substantial (less than an order of magnitude).
As Plug-In In Compaction Method. Our method can be used as a plug-in compaction method in various 3DGS training algorithms. This demonstrates the broad applicability of our proposed method. To verify this, apply our compaction method within various 3DGS pipelines. Specifically,
7

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Table 1: Quantitative comparison of rendering quality against SOTA methods. Our method (GHAP), applied to two different backbones (3DGS and Mini-Splatting) consistently matches or surpasses pruning-based baselines, even at a low retention rate (10%). while maintaining competitive runtime and memory usage. The best results for given number of Gaussians are shown in bold; second best are underlined.
Thnks&Temple
MipNcRF-360
Deep Blending
Method
SSIM
LPIPS]
SSIM
PSNR1
LPIPS)
Gaussians
SSIM1
PSNK
LPIPS]
Vanilla 3DGS
0.853
23.785
0.169
1577
0.813
27.554
0,221
2627
0.907
29,816
0.238
2475
LocoGS
0.843
23.655
0.191
571
0.798
27.049
0.257
674
0.903
29.972
0.261
529
3DGS (ours)
0.818
157
29.647
0.264
LightCaassinn
0.756
22.113
0.306
158
0.735
25.674
0.331
263
0.869
28.010
0.327
248
PUP-3DGE
0,767
21.519
0.280
158
0.753
25.333
0.309
262
0.895
29.153
0.274
248
Trimming the
0.776
21.535
156
263
28.00 08.056
0,302
Mesoo
08111
20.714
0.208
157
0.773
24.924
0.264
263
0.896
28. 693
0.264
248
MCMC
0.779
22. 141
0.282
157
0.763
25.952
0.309
263
0.885
28.976
0.298
248
MiniSplating
0.799
0.265
78
26/022
0.289
MiniSplatiig GHAP (ouns)
0,835
23.232
0.198
79
0.802
27.090
0.250
112
0.909
30.042
0.254
127
25
PSNR1 (dB)
Timel (s)
20
Memory, (GB)
4.5
15-
22
10
3.5
"LINDCS
Neaon22
10
20
40 10
20
30
40
10
20
30
50
Retention Ratio (%)
Retention Ratio (%)
Retention Ratio (%)
Figure 4: Comparison of compaction methods on the Tanks & Temples dataset. Left: Rate-
Distortion (RD) curves; middle: computational time, and right: memory consumption.
consider three representative variants as the backbone: 3DGS [3], AtomGS [13], and Mini- Mini-Splatting D [12]. Each of them employs a distinct densification strategy, and our method can be directly embedded into the pipeline without extensive engineering effort. For each backbone, we evaluate performance under two retention ratios (10% and 20%). Tab. 2 summarizes experimental results.
Table 2: Quantitative results with different backbones. The compaction performance of our GHAP method when used with different 3DGS variants as backbones under varying retention ratios.
Backhone
Tanks& Temples
MipNeRF-360
Deep Blending
SSIM
PSNR
LPIPS.
& Gomssians
SSIM
PSNR1
LPPPS)
à Gaussians
SSIM
PSNR*
LPIPS)
& Gaussians
3DGS-300
10%
0.818
23.312
0.242
157
0.764
26.4/4
0.314
263
039015
29.647
0.264
248
20%
23.615
0.788
0.907
29.864
100%
0.853
23.785
0.169
1577
0.813
27.551
0.221
2627
0.907
29.816
0.238
2475
Mini- Simii-Splatingg... D
10%
0.835
23.232
0.198
313
0.802
27.090
0,250
357
0.909
30.042
0.254
331
20%
0.855
23:103
0.171
626
0.821
27.310
0.214
614
0.912
30.170
0.238
662
AtomsS
10%
0,793
22.988
0,274
189
0.764
26.535
0.307
293
0.899
29.347
0,282
271
20%
0.812
23.282
0.240
378
0.788
27.0 27.025
0.269
586
0.902
29.347
0.268
542
100%
23.289
0.796
27.135
2928
0.902
29.314
The guantitative results in Tab. 2 demonstrate that method effectively preserves the backbone models' visual quality, even at an extreme retention rate of 10%. Notably, on some scenes (highlighted in bold), the compacted model's performance surpasses that of the uncompacted backbone.
Beyond quantitative metrics, we provide a qualitative analysis by visualizing multiple scenes before and after compaction in Fig. 5. As evidenced in the figure, our method successfully preserves rendering quality across most scenes while using only 10% of the Gaussian primitives. Interestingly, in certain cases, our compaction not only preserves but surpasses the original quality. A representative example is the "Kitchen" scene -Mini-SplattingDD backhone). We conjecture this improvement occurs because Mini-Splatting lacks pruning mechanism, often generating over saturated
8

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Before
After (o 0.1)
Before
After (p 0.1)
3DC3 300
truck SSIM-0.92, PSNR:27.6M.
truck SSIM: 0.898, PSNR: 27.217, UPIS:00211
playroom SSIM: 0,943, PSNR:
playroom SSIM: 0.939, PSNR:
LPIPS:0.155
34,042, LPIS: 0.195
542, LPIPS 0.5
Mini- Mini- Splatting D
LLETIT
drgohnson 9400
drjohnson S5IM 0,942,
kitchen SSIM: 0,941, 24.476, LPIPS.0.101
kitchen SSIM: 0.946, PSNR: 31.449, LPIPS: 0.106
PSNR-32.621, LPIPS.0.677
PSNR:32.77, LPIP5:0.194
Atom-GS
bicycle P5N9:19.904, LPIPS:0.317
PSNR:34.018, LPIP5:0.133
PSNR:32 449, LPIPS:0.175
P5NR:19.461, L11
Figure 5: Visual Quality Before and After Compaction. Visual results for various scenes under different 3DGS backbones, compacted to 10% of their primitives using our GHAP method. Our approach preserves rendering quality with negligible loss. some cases (e.g.. "Kitchen"), compaction even improves quality by regularizing over saturated Gaussian distribution.
set of Gaussians that introduce visual artifacts (e.g., the unnatural shadows in the lower-left region). Our method acts as a global regularizer, mitigating this issue by reducing unnecessary density while improving the overall expressiveness and preserving the underlying 3D structure. Naturally, our approach is inherently limited by the quality of its input. the original model suffers from significant artifacts due to lack of primitives in certain regions, compaction cannot resolve these fundamental issues. This limitation is demonstrated in the "Bicycle" scene, where artifacts present in the Atom-GS backbone persist after compaction.
4.3
Ablation Studies
Influence of KD Tree Depth. To validate the effectiveness of the KD-tree partitioning strategy, we provide ablation study on it. Due to the large scale of the three primary datasets, low KD-tree depths result in an excessive number points per block, making it infeasible to run the GMR algorithm. Therefore, conduct ablation experiments the smaller mic scene from NeRF-Synthetic. Results in Fig. show that increasing KD-tree depth reduces memory usage and runtime, while PSNR first improves and then declines. This indicates that moderately finer partitions allow GMR compact regions more effectively, whereas overly fine splits may fragment primitives and degrade quality.
Loss Function Design. As shown in Tables and method exhibits slight underperformanee on the LPIPS metric. To address this, we investigate whether incorporating an LPIPS loss term can yield improvements.
Our baseline loss function follows the vanilla 3DGS formulation, using an We experiment with two new weighting schemes that include LPIPS: (1) LISSSIIMPPIPS 8:1:1, and (2) LI:SSIM:LPIPS = 6:2:2. The results of this ablation study on the T Temples dataset are shown in Fig. 7 (with additional results in the Appendix). Our analysis reveals a trade-off between perceptual and distortion metries: increasing the weight of the LPIPS loss improves LPIPS scores leads to slight deduction in PSNR and SSIM.
ratio of 8:2.

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3.8
16
Memory, (GB)
PSNR (d8)
3.4
Runtime
14
KD Tree Depth
KD Tree Depth
KD Tree Depth
Figure Hyperparameters. The impact of KD tree depth on runtime performance, memory
footprint, and rendering quality.
0.24
SSIM-
0.51
LPIPS
0.23
PSNR
0.22
LPIPS Weight (%)
LPIPS Weight (%)
LPIPS Weight (%)
Figure 7: Comparison of different loss weights. A trade-off between perceptual and distortion metrics.
GMR versus Random Subsampling. We conduct ablation study on the Temples dataset to evaluate the contribution of each stage in our pipeline: geometric compaction and appearance opti- mization. A random subsampling baseline is included to assess the effectiveness of our design choices.
The results, presented in Table 3, compare two compaction schemes followed by the same fine- tuning procedure. Our findings demonstrate that: first, our compaction procedure is significantly more effective than random subsampling and other pruning baselines (as shown in Tab, I). Second, the subsequent appearance optimization stage provides substantial quantitative improve- ments for both compaction approaches. The significant performance gain over the baseline validates the necessity and effectiveness of both stages in our proposed pipeline.
Table 3: Ablation study. Mean PSNR, SSIM, and LPIPS on the Tanks and Temples dataset at each stage of GHAP pipeline, with random subsampling as a control. Each operation is applied cumula- tively to all subsequent stages.
Temps&Temples
Method
SSIM
PSNR1
LPIPS
3DGS-15K
23.084
0.194
10% Compaction
0.502
14.015
0.483
10% Random Compaction
0.333
9.573
0,555
+15K Fine-tuning
0.818
23.312
0.242
+15K Fine-tuning
0.712
21,312
0.282
Conclusion and Discussion
We propose an optimal transport-basee based Gaussian mixture reduction framework for 3D Gaussian Splatting, achieving compact yet faithful representations. By minimizing composite transport diver- gence with appearance fine-tuning, our method preserves high visual fidelity while retaining only 10% of Gaussians, outperforming prior compaction techniques. The framework scales efficiently via block-wise KD-tree partitioning and integrates seamlessly with diverse 3DGS pipelines.
Future directions include enhancing robustness across challenging scene types, incorporating percep- tual objectives, developing multi-scale and overlap-aware partitioning. adopting auto-tuned schedules, and extending dynamic 3DGS for real-time temporal rendering.
Acknowledgments and Disclosure of Funding
Qiong Zhang is supported by the National Natural Science Foundation of China Grant 12301391. Cheng Meng and Qiong Zhang are supported by Big Data and Responsible Artificial Intelligence for National Governance, Renmin University of China. The authors would like to thank the anonymous reviewers for their constructive suggestions.
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Appendix
A
More Details about the Blockwise GMR Compaction
A.1 Introduction to Optimal Transport (OT)
The Optimal Transport (OT) problem dates back to 1781, when French mathematician Gaspard Monge (1746-1818) first formulated This "transport plan" minimizes the transportation cost, hence the term "optimal transport". The original formulation is known Monge's 's problem. In 1947, Russian economist Leonid Kantorovich (1912-1986) relaxed Monge's formulation, leading to the so-called Monge-Kand Kantorovich problem.
finding the minimal-cost way to move sand into hole.
Specifically, let P and Q be two distributions function, and let the coupling
metric space X, let X' X R4 be cost
I(P (,, v) / (dxx,)) -())), x(, = P())
be the set of joint distributions with marginals P and Q. For any cost function the total transportation cost induced by a plan क E 1I( Q) is defined
/ c(x,y) (o..))((ae (dd, dy)-
I.(TT) = Jxxx
Here, TΦ. y) indicates how much "mass" transported from x to y. The first constraint, T2x dy) P(x), ensures that the mass location is spread over X, while the second constraint, I (dx,, y) Qv)), ensures that the destination at y receives the required mass. The OT problem seeks the optimal plan
arg min (IT I(P,Q)))
minimizing the transportation cost. The corresponding minimal cost,
TP
induces a divergence between P and Q. This OT divergence provides a principled way to mea- sure distributional similarity, enabling applications in density matching, generative modeling, and dimensionality reduction.
The optimal plan x¹ and OT divergence typically lack closed-form solutions. Numerical algorithms [18] are used to compute the OT between discrete measures. When and Q are discrete, the OT divergence reduces to:
20

---

Example (OT Divergence Between Discrete Distributions). For P = Ei=1 Un Under and Q Ejst Um Ymo the OT divergence is
m
m
Q) min
Tec(Z
Σ
Jaj
U/.
Taj
U4
(3)
==1
Exact solutions can be found via linear programming, while approximations are obtained using algorithms like Sinkhorn.
A.2
Relationship Between Composite Transportation Divergence and OT
The Composite Transportation Divergence (CTD) [34, 35] extends OT to Gaussian mixtures. For two mixtures On Σ a; u4. 2x) and d'm = Ei= af Mj), the CTD
m
Teldns )')) inf
To En), E,
Tij
03
Comparing this with the discrete OT divergence (3), the CTD treats Gaussian mixtures as discrete distributions over the space of Gaussians, defining the divergence as the OT between them.
To illustrate, consider warehouses and factories in the space of Gaussian distributions. F. The ith warehouse, at (0:5 15, E,), holds a; units of material, while the jth factory, at (t: p/, ky)), requires of units. The cost to transport material from to j is c(((; u4, E,), d(; 2,)), and TH 2 denotes the transported amount. The total cost under plan π is En, T. (i; Ex), ( ED) The coupling set 1l(a, a') {Ta Σj= Wij = 0ig >i=1 Trj a ensures: (a) correct material removal from warehouses, and (b) correct delivery to factories. The OT problem seeks the plan *" E [aa a') minimizing the total cost. The minimal cost corresponds to the CTD between the mixtures, quantifying the optimal transport cost between them.
Our compaction method leverages this interpretation, approximating a mixture with fewer Gaussians by minimizing their CTD-based dissimilarity.
A.3 Algorithm Intuition
At first glance, the optimization problem in (1) is bilevel: the OT plan must be found for any candidate {ā,, p.y. Ey}リ and the objective must be minimized. However, as shown by Zhang et al. [17], this simplifies with clear interpretation. The column-wise marginal constraints each Gaussian primitive (14, Σ,) transports its full mass to its "closest" counterpart. The optimal plan thus corresponds to clustering ((4, E,) }2_ into clusters (C, 1 with the original Gaussians forming cluster "barycenters". Mathematically, the simplified program is:
redundant, and
min
Σ :. E,)): [n] C1 L.... .C FEC,
where LJ denotes disjoint union. The optimal plan has a closed form:
az
if = arg ming e (d(:; Ex), o(: Pr. Ex))
iij
o
otherwise.
With this, and the reduced mixture parameters can be updated alternately, formalized in the following k-means-like algorithm:
Assignment Step: Each Gaussian 64, Ex) is assigned to cluster C, by minimizing
clo(; pr Ex) E,)), analogous to k-means" nearest-centroid assignment.
Update Step: For the cost function
c(O, ||I 111--11 E
21

---

the parameters (Fy, Ey) are updated as weighted averages:
pjj Eiec, aijl Eiec, ai
Σ Eiec, a;E,
Eiec, ai
This mirrors k-means' centroid update.
A.4 Algorithm Convergence
The following theorem guarantees convergence of the CTD sequence generated by Algorithm 1. In the worst case, the algorithm requires exponentially many steps, but typically converges in about iterations. A full proof is given in Zhang et al. [17].
Theorem (Convergence of the Algorithm). Suppose c(..) is continuous, and any > 0 and
o², the set {Ø c(0", Φ) I A} is compact under some Gaussian space metric. Let (() be the sequence generated by the update step with initial {a) (0) ( " iteration t. Then:
}. and let Te (1++1) be the CTD at
There exists T and {āg, F.. Ei} such that for all 2 T, {a m -. {aj. Pj. Σj} and Teth ((++1) T, where T is the CTD between the original mixture and {az. Σ7}.
2. The limit point (i Eij) is local minimum of Te-
3. An MM-based exhaustive algorithm with (m")) complexity solves (I).
B Experiment Steps and Complexity Analysis in More Detail
B.1 Detailed 3DGS Pipline
The advantages of 3DGS in rendering speed and image fidelity have made it applicable wide range of tasks, including human reconstruction, Al-generated content, autonomous driving, and beyond [36, 37, 38, 39]. Extensions to dynamic 3DGS, editable 3DGS, and surface representation have further broadened its utility [40, 41, 42]. 3D Gaussian Splatting (3DGS) represents a scene as a set of anisotropic 3D Gaussian primitives, each parameterized by its spatial location, shape, opacity, and radiance. The pipeline consists of the following key steps:
Initialization. From Structure-front fromum--orroo-oooioon (SfM), obtain calibrated camera poses and sparse point cloud. Each point is initialized as a 3D Gaussian with an opacity oç:
1 (x m4) x (x 14)
(x:; 14, Ea) |
exp
- 2
where μa E R the position (mean), Σ, E R333 3 is the covariance matrix (anisotropic shape), and a; E [0, Projection & Rasterization. Each 3D Gaussian is projected to 2D using the camera model,
Σ = JWE,JW"
where W is veiw transformation matrix and J is the Jacobian of the projective transform. Rasterization is done using a diffcrentiable splatting approach which makes optimization possible. The pixel color is computed via volumetric blending:
3. Image Formation (Alpha Blending).
N T, a. a as G, with T, = II (1 (1-aa):
i-1
C= -T.
j=l
where o; is the SH-predicted color of the i-th Gaussian in front-to-back order. 4. Optimization. The Gaussian parameters {M:, Σj. ai ci} are optimized to minimize a photometric loss:
f - (1 - x) ||c C + LSSIM:
where Ć is the rendered image, C the ground truth, and A [0, balances the two loss terms.
22

---

5. Adaptive Densification & Pruning. During training. Gaussians are cloned (under- reconstruction) or split (over-reconstruction) based on view-space gradient magnitude, and low-contribution Gaussians are pruned.
B.2 Algorithm Complexity
For Algorithm 1, we discuss its computational cost for reducing n Gaussians to m Gaussians. The assignment step requires computing pairwise distances between all imput Gaussians and cluster centers, resulting in a complexity of (nm). The update step involves computing the barycenters. Since the assignments
already known and the update for each cluster is linear in the number of assigned Gaussians, this step has complexity of (n). The overall time complexity of Algorithm is (nm)).
Our GHAP algorithm consists of two components: Geometric Compaction and Appearance Opti- mization (fine-tuning). In this analysis, we focus only on the time complexity of the geometric compaction stage. The KD-tree is constructed by recursively splitting the dataset along the me- dian of selected coordinate axis until maximum depth d input size be n, and the time complexity of building the tree be T(n). This follows the recurrence: T(n) 27' (n/2) + (n logg nt). The recursion terminates after dl levels, yiclding total KD-tree construction complexity of O (nd log2 (n)). In the second step of geometric compaction, Algorithm applied independently within each KD-tree block with Gaussians reduced ps Gaussians. Based on our analysis in the previous paragraph, the per-block cost is O(ps²). This along with the fact that there are 2d = n/s blocks, the total complexity becomes: O ((n/s) ps²) = O (pns) = O(ms). Combining with the KD-tree construction cost, the total time complexity of the geometric compaction step (og n + mT/2")).
(n/g((/s) is reached. Let the
B.3 Detailed Experiments Steps
To ensure fair comparison in our experiments, all methods undergo 30,000 total iterations under consistent training conditions. MCMC uses 30K iterations of its own update process in [24]. For other baselines, we summarize the backbone architecture, compaction methods, and fine-tuning steps in the table below.
Method
Backbone
Compaction
Fine-tune (150001-300 iterations)
(0- 15k iterations)
(15001th iteration)
3DGS
None
d
3DGS+GHAP
Our compaction
d
LightGaussian
LightGaussian compaction
d
PUP-3DGS Trimming the FAT MesonGS
PUP-3DGS compaction
d
Trimming the FAT compaction
d
MesonGS compaction MiniSplatting compaction
MiniSplatting
MiniSplatiing-D-GII D
Our compaction
LocoGi
LocoGS compaction
d
a Vanilla 3DGS densification and pruning in [3].
Mini-Splatting densification and pruning in [12].
с LocoGS 3DGS update in [33].
3DGS fine-tuning in [3].
Key implementation details:
For compaction methods applicable to pre-trained models (e.g., LightGaussian, PUP-3DGS, Trimming the FAT, MesonGS), we initialize from the same backbone model (trained using vanilla 3DGS) and apply their respective compaction directlyexcluing any compression modules.
23

---

All methods undergo identical fine-tuning (15k iterations) to achieve the target retention ratio.
This standardized protocol ensures that performance differences stem solely from the methods themselves, climinating variations due to training procedures. All experiments are conducted on a server with 256 GB RAM and 96-core Intel Xcon Platinum 8255C CPU, and on workstation equipped with five NVIDIA RTX 3090 GPUs, each with 24 GB of VRAM.
C An Additional Comparison Experiments
As documented in prior work, 3DGS-MCMC [24] reinterpretes the 3D Gaussian Splatting (3DGS) process through the lens of Markov Chain Monte Carlo (MCMC), thereby naturally exploring broader parameter space. In particular, it formulates pruning as a state transition within the MCMC framework and incorporates Li regularization, avoiding abrupt hard-threshold deletions and enabling the natural removal of unnecessary Gaussian elements. Given these properties, a further comparative analysis with 3DGS-MCMC is warranted.
To facilitate a fair comparison, we configured 3DGS-MCMC under two settings: one constrained to 300k Gaussians, and another initialized with 3000k Gaussians followed by compaction using method to reduce the count to 300k. Quantitative results on Tandt & Temples are summarized the table below, As evidenced by the results presented in the table, GHAP combined with 3DGS-MCMC
Method
SSIM
PSNRT
LPIPS
MMM-C-300K--
0.813
22.925
0.239
BDGS-MCMCC-3000--
0.827
22.786
0.209
yields superior performance compared to the native 3DGS-MCM approach constrained to 300k Gaussians. This demonstrates the effectiveness of our method in accurately approximating the surface geometry of 3DGS, highlighting the advantage of our compaction strategy.
D Ablation on Joint Geometry and Appearance Fine-tuning
In our configured compaction process, compaction was performed only once at the 15,001 iteration. We conducted an ablation study on the timing of compaction to examine the impact of varying compaction frequencies on the final outcome. Two alternative compaction strategies were considered:
the first involved compacting to 20% at the 15,00Ist iteration, followed by an additional 50% compaction at the 20,00lst iteration; the second strategy applied compaction of 40%, 50%, and 50% at the 15,00lst, 20,001st, st, and 25,001st iterations, respectively. All three strategies ultimately resulted in final retention rate of 10%. Experimental results, as presented in the table below, indicate that jointly refining geometry and appearance over multiple stages does not lead to evidently improved performance. Therefore, in practical applications, emphasis should be placed on executing appearance optimization for as long as possible, rather than pursuing multi-stage compaction.
Compacting Iteration
p
SSIM
PSNR1
LPIPS
15001
0.1
0.817
23.313
0.242
15001, 20001
0.2,0.5
0.817
23.44
0.247
15001, 20001, 25001
0.4.0.5,0.5
0.811
23.359
0.255
E
Additional Numerical Results and Scene Visualizations
We report more comprehensive set of results for the comparison experiments in Table 4, including one different retention ratio: 20%. As shown in the updated results, our compaction method significantly outperforms other methods either post-processing compression methods or end-to-end methods.
24

---

In addition to the previously shown Figure 5, we present more detailed qualitative comparisons across multiple scenes in Figure 8. Our method consistently preserves the visual quality of the original models. In particular, when applied to stronger 3DGS variants with improved densification strategies, such as Mini-Splatiin-DD, our compaction framework performs even better. This is reflected in the fact that, after compaction, Mini-Splatting-D D often achieves higher rendering quality than the original 3DGS baseline.
Table 4: We compared GHAP with four post-processing methods (LightGaussian, PUP-3DGS, Trimming the Fat, MesonGS) at 20% retention, as well as two end-to-end methods (LocoGS, 3DGS-MCMC) GHAP also replaces the pruning in Mini- Mini-Splatiin. Results show that method substantially outperforms both post-processing and end-to-end approaches.
Tanks& Temples
MipNeRF- 360
Deep Blending
Method
SSIM
PSNR1
LPISS
Gaussias
SSIM
PSNR
LPIPS]
k Gaussians
SSIM1
PSNR1
LPIPS
Coussians
original
0853
0.169
1577
29.816
0.238
LocoCS
0.843
23,655
0.191
571
0.798
27.049
0.257
674
0.903
29.977
0.261
529
GHAP (ours)
0,838
23.615
0.212
314
0.788
26.973
0.275
527
0.907
29.864
0.252
496
LightGaussia
0.315
PUP-3DGS-20S
0,509
22.613
0.228
315
0.790
26.671
0,257
525
0.905
29.719
0.248
495
Trimming the Fut-20%
0.819
22,198
0.232
315
0.781
26.191
0,280
524
0,500
29.082
0.272
191
MesonOS-200
314
3DGS-MCMC
0.779
22.141
0.282
315
0.763
25.957
0.300
263
0.885
28.9776
0.298
496
MiniSplatiing 20%
0.824
22.953
0.223
142
0.794
26.728
0.267
215
0.904
29.763
0.265
240
MiniSplatinggGGAAAP (ours)
25

---

3DGS
Compact (o 0.1)
Mini-Splatting
Compact (o 0.1)
Atom
Compact (o 0.1)
Truck
train
kitchen
bonsal
bicycle
counter
Rlwwer
Earda
stupp
treehill
Playroom
drjohnson
Figure 8: More scenes visualization, Visual comparisons across additional scenes before and after compaction.
26


Reviews:
[{"review_id": "QTENlsNLup", "reviewer": "Reviewer_1EBm", "summary": "This paper presents Gaussian Herding Across Pens (GHAP), a Gaussian compaction method that integrates similar Gaussians, instead of pruning less-important Gaussians. The compaction pipeline consists of two stages, (a) geometric compaction via Gaussian Mixture Reduction (GMR) and (b) appearance finetuning. To enhance the efficiency, these compaction steps are conducted in a block-wise manner divided by KD-tree partition. Experimental results show that GHAP reduces the number of Gaussians of existing representations as a plug-and-play module. Also, it achieves comparable rendering quality with the same compaction setting compared to existing Gaussian compaction methods such as LightGaussian and PUP-3DGS.", "questions": "- In Table 2, the rendering quality of LightGaussian outperforms the rendering quality of this method for a 10% retention rate on Mip-NeRF 360 dataset. Mip-NeRF 360 dataset contains both indoor and outdoor scenes which have been employed as a standard real-world benchmark for novel-view synthesis. Can you explain why this method shows worse performance on Mip-NeRF 360 dataset?\n\n- For better understanding, adding RD curves for Table 1 and Table 2 would be helpful.\n\n- Although several compaction methods are mentioned in Section 2, the authors provide a comparison only with LightGaussian and PUP-3DGS. It is recommended to conduct an additional comparison with RadSplat, Mini-Splatting, and Trimming-the-Fat.", "limitations": "- Despite the effectiveness of the proposed compaction strategy, this paper contains only limited technical contributions to be accepted at this conference. \n\n- Although the experimental results show comparable performance, they are insufficient to clearly demonstrate that the proposed method outperforms existing approaches.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 4, "originality": 4, "strengths_and_weaknesses": "### Strengths\n\n- While the existing compaction methods reduce the number of Gaussians by pruning, this method integrates similar Gaussians.\n- This method can be adapted to existing Gaussian representations as a plug-and-play module. \n- Experimental results indicate that this method can maintain high fidelity despite a small retention rate.\n\n---\n\n### Weaknesses\n\n- To validate the effectiveness of the KD-tree partitioning strategy, it would be recommended to provide an ablation study on it.\n\n- The technical contribution of this paper is a simple compaction method for 3D Gaussians, which may be insufficient to be accepted at this conference.\n\n- The experimental results show less effective results according to the experimental setting. To be specific, it cannot outperform all existing compaction methods for all datasets. Despite the high fidelity, comparative methods show better scores for some experimental settings, as shown in Table 2.", "was_revised": true, "final_justification": "During the rebuttal period, the authors have addressed all of my concerns regarding the limited technical advancements and performance. Notably, they have demonstrated that the proposed method outperforms various approaches through extensive experiments. Therefore, I raise my final rating to positive.", "rebuttal": "Thanks for giving us excellent scores on originality and significance. We respond to your comments below and will \nincorporate all of your suggestions into the final version.\n\n- **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:\n    - **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.\n    - **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.\n    - **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.\n- **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.\n\n| Method-$\\rho$/Dataset       | **TNT**     |              |             |             | **360** |              |          |             | **DB**      |              |             |             |\n| - |-| - | -| - | - | - |-| - |-------------| - |-| - |\n|                             | SSIM↑       | PSNR↑        | LPIPS↓      | k Guassians | SSIM↑           | PSNR↑        | LPIPS↓   | k Gaussians | SSIM↑       | PSNR↑        | LPIPS↓      | k Gaussians |\n| **GHAP (ours)-10%**         | ***0.818*** | ***23.312*** | ***0.242*** | 157         | ***0.764***     | ***26.404*** | ***0.314*** | 263         | ***0.905*** | ***29.647*** | ***0.264*** | 248         |\n| LightGaussian-10%           | 0.756       | 22.113       | 0.306       | 158         | 0.735           | 25.674       | 0.331    | 263         | 0.869       | 28.01        | 0.327       | 248         |\n| **GHAP (ours)-20%**         | ***0.835*** | ***23.615*** | ***0.212*** | 314         | ***0.788***     | ***26.973*** | ***0.275*** | 527         | ***0.907*** | ***29.864*** | ***0.252*** | 496         |\n| LightGaussian-20%           | 0.779       | 22.486       | 0.271       | 315         | 0.765           | 26.353       | 0.288    | 526         | 0.873       | 28.011       | 0.315       | 495         |\n- **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**.\n\n| Method-$\\rho$/Dataset              | **TNT**     |              |             |             | **360**     |              |             |             | **DB**      |              |             |             |\n|-|-|-|-|-|-|-|-|-|-|-|-|-|\n|                                    | SSIM↑       | PSNR↑        | LPIPS↓      | k Guassians | SSIM↑       | PSNR↑        | LPIPS↓      | k Gaussians | SSIM↑       | PSNR↑        | LPIPS↓      | k Gaussians |\n| **3DGS+GHAP (ours)-10%**           | ***0.818*** | ***23.312*** | 0.242       | 157         | ***0.764*** | ***26.404*** | 0.314       | 263         | ***0.905*** | ***29.647*** | ***0.264*** | 248         |\n| Trimming the Fat-10%               | 0.776       | 21.535       | 0.293       | 156         | 0.731       | 25.255       | 0.343       | 263         | 0.887       | 28.056       | 0.302       | 247         |\n| MesonGS-10%                        | 0.811       | 20.714       | ***0.208*** | 157         | 0.773       | 24.924       | ***0.264*** | 263         | 0.896       | 28.693       | 0.264       | 248         |\n| 3DGS-MCMC-10%                      | 0.779       | 22.141       | 0.282       | 157         | 0.763       | 25.957       | 0.309       | 263         | 0.885       | 28.976       | 0.298       | 248         |\n| **3DGS+GHAP (ours)-20%**           | ***0.835*** | ***23.615*** | 0.212       | 314         | 0.788       | ***26.973*** | 0.275       | 527         | ***0.907*** | ***29.864*** | ***0.252*** | 496         |\n| Trimming the Fat-20%               | 0.819       | 22.498       | 0.232       | 315         | 0.781       | 26.494       | 0.28        | 524         | 0.9         | 29.082       | 0.272       | 494         |\n| MesonGS-20%                        | 0.822       | 20.699       | ***0.207*** | 314         | 0.776       | 25.006       | 0.262       | 527         | 0.897       | 28.696       | 0.262       | 496         |\n| 3DGS-MCMC-20%                      | 0.813       | 22.925       | 0.237       | 314         | ***0.801*** | 26.850       | ***0.257*** | 527         | 0.893       | 29.393       | 0.277       | 496         |\n\n| Method-$\\rho$/Dataset              | **TNT**     |              |             |             | **360** |              |             |             | **DB**     |              |             |             |\n|-|-|-|-|-|-|-|-|-|-|-|-|-|\n|                                    | SSIM↑       | PSNR↑        | LPIPS↓      | k Guassians | SSIM↑           | PSNR↑        | LPIPS↓      | k Gaussians | SSIM↑      | PSNR↑        | LPIPS↓      | k Gaussians |\n| MiniSplatting-10%                  | 0.799       | 22.661       | 0.265       | 78          | 0.759           | 26.022       | 0.318       | 111         | 0.895      | 29.395       | 0.289       | 125         |\n| **MiniSplatting-D+GHAP-10% (ours)** | ***0.835*** | ***23.232*** | ***0.198*** | 79          | ***0.802***     | ***27.09***  | ***0.25***  | 112         | ***0.909*** | ***30.042*** | ***0.254*** | 127         |\n| MiniSplatting-20%                  | 0.824       | 22.953       | 0.223       | 142         | 0.794           | 26.728       | 0.267       | 215         | 0.904      | 29.763       | 0.265       | 240         |\n| **MiniSplatting-D+GHAP-20% (ours)** | ***0.855*** | ***23.403*** | ***0.171*** | 155         | ***0.821***     | ***27.31***  | ***0.214*** | 219         | ***0.912*** | ***30.17***  | ***0.238*** | 250         |\n| LocoGS                             | 0.843       | 23.655       | 0.191       | 571         | 0.798           | 27.049       | 0.257       | 673.8       | 0.903      | 29.972       | 0.261       | 529         |\n\n\n- **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.\n\n| Depth | 0      |            |            | 1       |           |            | 2       |            |            | 3      |            |            | 4       |            |            | 5          |            |            | 6      |            |            | 7       |            |            | 8      |            |            |\n| - |-|-| - |-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|\n|              | PSNR↑  | Runtime(s) | Memery(MB) | PSNR↑   | Runtime(s) | Memery(MB) | PSNR↑   | Runtime(s) | Memery(MB) | PSNR↑  | Runtime(s) | Memery(MB) | PSNR↑   | Runtime(s) | Memery(MB) | PSNR↑      | Runtime(s) | Memery(MB) | PSNR↑  | Runtime(s) | Memery(MB) | PSNR↑   | Runtime(s) | Memery(MB) | PSNR↑  | Runtime(s) | Memery(MB) |\n| 3DGS+GHAP    | 13.610 | 2.7        | 3706       | 14.427  | 5.0       | 3247       | 14.908  | 6.5        | 3134       | 14.756 | 7.0        | 3106       | 16.138  | 5.8        | 3098       | 16.376     | 5.1        | 3097       | 15.440 | 4.5        | 3096       | 15.7963 | 3.7        | 3095       | 15.488 | 1.0        | 3095       |\n    \n- **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.\n\n| Method/$\\rho$ | GHAP    | LightGaussian| PUP-3DGS  | Trimming the Fat| MesonGS |\n| - |-|-|-|-|-|\n| 10%                   | **23.312**  | 22.113       | 21.535    | 21.535          | 20.714  |\n| 20%                   | **23.615**  | 22.486       | 22.603    | 22.498          | 20.699  |\n| 30%                   | **23.740**  | 22.582       | 23.102    | 22.802          | 20.891  |\n| 40%                   | **23.656**  | 22.588       | 23.394    | 22.892          | 21.142  |\n| 50%                   | **23.791**  | 22.609       | 23.582    | 22.911          | 21.432  |"}, {"review_id": "wICsUrRHaU", "reviewer": "Reviewer_D3B2", "summary": "The paper addresses the problem of reducing the number of Gaussians needed to represent a scene, using an optimal transport perspective. The authors first train a 3DGS model. The resulting Gaussians are then partitioned into blocks based on camera centers. In the first stage, the geometric parameters (i.e., centers and covariances) are iteratively adjusted using K-means to minimize the divergence between the original Gaussian mixture and the changed mixture. After this geometric optimization, the appearance parameters (color and opacity) are refined. The method is evaluated on three datasets, and the results show that, even when retaining only 10% of the Gaussians, the final rendering quality is largely preserved.", "questions": "As provided in the weakness section:\n\nPlease include timing results for your approach to help assess its computational cost.\n\nPlease add a comparison to [24] in Table 2, as it is also a relevant probabilistic method that performs well with fewer Gaussians.\n\nPlease revise Figures 3 and 4 for clarity and accuracy:\n– In Figure 3, increase the font size or enlarge the image for better readability.\n– In Figure 4, correct the \"PNSR\" typo (should be \"PSNR\"), verify the PSNR values (they appear to be switched), and consider adding a difference map to make the visual impact of your method more evident.", "limitations": "- Please provide the limitations of your technique and possible future directions.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strength:\n\nMinimizing the number of Gaussians required to represent a scene is a relevant and timely research problem. The paper presents a novel approach that effectively reduces the number of Gaussians to 10% of the original set, while maintaining only marginal degradation in rendering quality.\n\nWeakness:\n\nThe paper does not provide any information on the computational cost of applying the proposed method. Please include the runtime to help assess the practical overhead of integrating your method.\n\nParts of Figure 3 are difficult to read due to small font sizes. Consider enlarging the fonts or the entire figure for better clarity.\n\nYou cite [24] as another probabilistic approach that, while different in methodology, also performs well with fewer Gaussians compared to 3DGS. Please include this method in the comparisons (in Table 2), as it would be valuable to see how your approach performs compared to that.\n\nFigure 4 needs correction. It contains a typo (\"PNSR\" instead of \"PSNR\") and the PSNR values appear to be mislabeled or inconsistent. Moreover, the visual differences between the “before” and “after” images are not clearly visible, even when zoomed in. It would be helpful to include a difference map to better highlight the changes introduced by your method, as they are not significant.", "was_revised": true, "final_justification": "I have reviewed all of the reviewers’ comments, and overall, I believe that since the method outperforms all the other reported compression baselines, it represents a valuable contribution. I strongly encourage you to revise and correct the figures carefully in the final version of the paper. Additionally, please ensure that the new experiments and timing reports are incorporated, as they strengthen the evidence for your contribution.", "rebuttal": "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.\n\n- **Computational cost of the method.**\n    - **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 $O(n\\log n + \\rho s^2\\log n)$.\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.\n    - **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.\n\n| Method/Retention Rate | GHAP      |        |        | LightGaussian |        |        | PUP-3DGS  |        |        | Trimming the Fat |        |        | MesonGS   |        |        |\n| :-------------------- | --------- | ------ | ------ | ------------- | ------ | ------ | --------- | ------ | ------ | ---------------- | ------ | ------ | --------- | ------ | ------ |\n|                       | time（s） | memory | PSNR↑  | time（s）      | memory   | PSNR↑  | time（s） | memory | PSNR↑  | time（s）        | memory | PSNR↑  | time（s） | memory | PSNR↑  |\n| 10%                   | 5.6       | 4857   | 23.312 | 10.8          | 3908   | 22.113 | 26.3      | 4001   | 21.535 | <0.1             | 3740   | 21.535 | 11.0      | 2830   | 20.714 |\n| 20%                   | 4.4       | 4895   | 23.615 | 11.2          | 3908   | 22.486 | 26.3      | 4001   | 22.603 | <0.1             | 3731   | 22.498 | 11.1      | 2830   | 20.699 |\n| 30%                   | 3.8       | 4976   | 23.740 | 9.0           | 3908   | 22.582 | 26.3      | 4001   | 23.102 | <0.1             | 3722   | 22.802 | 10.8      | 2830   | 20.891 |\n| 40%                   | 3.3       | 5060   | 23.656 | 10.7          | 3908   | 22.588 | 26.3      | 4001   | 23.394 | <0.1             | 3760   | 22.892 | 10.9      | 2830   | 21.142 |\n| 50%                   | 3.0       | 5141   | 23.791 | 10.8          | 3908   | 22.609 | 26.3      | 4001   | 23.582 | <0.1             | 3800   | 22.911 | 11.1      | 2830   | 21.432 |\n\n- **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.\n\n\n| Method-$\\rho$/Dataset    | **TNT**      |                |              |             | **360**     |              |               |             | **DB**      |              |             |             |\n|--------------------------|--------------|----------------|--------------|-------------|-------------| ------------ |---------------| ----------- |-------------| ------------ | ----------- | ----------- |\n|                          | SSIM↑        | PSNR↑          | LPIPS↓       | k Guassians | SSIM↑       | PSNR↑        | LPIPS↓        | k Gaussians | SSIM↑       | PSNR↑        | LPIPS↓      | k Gaussians |\n| **3DGS+GHAP (ours)-10%** | ***0.818***  | ***23.312***   | ***0.242***  | 157         | ***0.764*** | ***26.404*** | 0.314         | 263         | ***0.905*** | ***29.647*** | ***0.264*** | 248         |\n| 3DGS-MCMC-10%            | 0.779        | 22.141         | 0.282        | 157         | 0.763       | 25.957       | ***0.309***   | 263         | 0.885       | 28.976       | 0.298       | 248         |\n| **3DGS+GHAP (ours)-20%** | ***0.835***  | ***23.615***   | ***0.212***  | 314         | 0.788       | ***26.973*** | 0.275         | 527         | ***0.907*** | ***29.864*** | ***0.252*** | 496         |\n| 3DGS-MCMC-20%            | 0.813        | 22.925         | 0.237        | 314         | ***0.801*** | 26.850       | ***0.257***   | 527         | 0.893       | 29.393       | 0.277       | 496         |\n\n\n- **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.\n- **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. \n****"}, {"review_id": "XSmEOvbdd7", "reviewer": "Reviewer_Cgrv", "summary": "The paper proposes a method that makes 3D Gaussian Splatting representations compact.\n\nThe main pipeline is\n- Start with an optimized 3D representation.\n- Partition the 3D space using a KD-tree.\n- Within each partition, run Gaussian Mixture Reduction (GMR) to decrease the number of Gaussians.\n- Refine the appearance (colors) to obtain the final result.\n\nThe authors demonstrate that this pipeline can be applied on top of several existing methods, including the original 3DGS, Mini-Splatting, and AtomGS.", "questions": "- Would it be possible to include comparisons with a broader set of methods? This would highlight how effective the proposed GMR-based approach is relative to other techniques. Given this intriguing idea, I’d like to raise my initial score once those comparisons are more solid.\n- The limitation section seems to be missing. Could you discuss the limitations during the rebuttal?", "limitations": "Neither the main paper nor the supplementary material contains a Limitations section, which conflicts with the required checklist.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 4, "strengths_and_weaknesses": "**Strengths**\n- Employing GMR is an interesting idea, and combining it with spatial partitioning via a KD-Tree effectively reduces computational cost.\n- The storyline, from the optimal-transport to engineering details such as the KD-Tree, is easy to follow.\n\n\n**Weaknesses**\n\n- The experimental comparison is not thorough. Given the rapid advancements in this field, the paper should highlight its advantages more clearly. If the focus is post-training compactness, it should be compared with additional baselines such as MesonGS (ECCV 2024) and Fast Compression of 3D Gaussian Splatting (ICLR 2025). It would also be helpful to compare against end-to-end compact 3DGS variants such as ContextGS (NeurIPS 2024) and LocoGS (ICLR 2025).\n- Rate–distortion (RD) curves are essential for comparison, yet Fig. 5 shows only the proposed method. Comparing RD curves to those of other methods would be helpful in demonstrating the effectiveness of the proposed method.", "was_revised": true, "final_justification": "Common concerns about limited experimental results (including other baselines) and computational costs are well addressed.", "rebuttal": "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.\n\n- **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**.\n\n| Method-$\\rho$/Dataset              | **TNT**     |              |             |             | **360**     |              |             |             | **DB**      |              |             |             |\n|-|-|-|-|-|-|-|-|-|-|-|-|-|\n|                                    | SSIM↑       | PSNR↑        | LPIPS↓      | k Guassians | SSIM↑       | PSNR↑        | LPIPS↓      | k Gaussians | SSIM↑       | PSNR↑        | LPIPS↓      | k Gaussians |\n| **3DGS+GHAP (ours)-10%**           | ***0.818*** | ***23.312*** | 0.242       | 157         | ***0.764*** | ***26.404*** | 0.314       | 263         | ***0.905*** | ***29.647*** | ***0.264*** | 248         |\n| Trimming the Fat-10%               | 0.776       | 21.535       | 0.293       | 156         | 0.731       | 25.255       | 0.343       | 263         | 0.887       | 28.056       | 0.302       | 247         |\n| MesonGS-10%                        | 0.811       | 20.714       | ***0.208*** | 157         | 0.773       | 24.924       | ***0.264*** | 263         | 0.896       | 28.693       | 0.264       | 248         |\n| 3DGS-MCMC-10%                      | 0.779       | 22.141       | 0.282       | 157         | 0.763       | 25.957       | 0.309       | 263         | 0.885       | 28.976       | 0.298       | 248         |\n| **3DGS+GHAP (ours)-20%**           | ***0.835*** | ***23.615*** | 0.212       | 314         | 0.788       | ***26.973*** | 0.275       | 527         | ***0.907*** | ***29.864*** | ***0.252*** | 496         |\n| Trimming the Fat-20%               | 0.819       | 22.498       | 0.232       | 315         | 0.781       | 26.494       | 0.28        | 524         | 0.9         | 29.082       | 0.272       | 494         |\n| MesonGS-20%                        | 0.822       | 20.699       | ***0.207*** | 314         | 0.776       | 25.006       | 0.262       | 527         | 0.897       | 28.696       | 0.262       | 496         |\n| 3DGS-MCMC-20%                      | 0.813       | 22.925       | 0.237       | 314         | ***0.801*** | 26.850       | ***0.257*** | 527         | 0.893       | 29.393       | 0.277       | 496         |\n\n| Method-$\\rho$/Dataset              | **TNT**     |              |             |             | **360** |              |             |             | **DB**     |              |             |             |\n|-|-|-|-|-|-|-|-|-|-|-|-|-|\n|                                    | SSIM↑       | PSNR↑        | LPIPS↓      | k Guassians | SSIM↑           | PSNR↑        | LPIPS↓      | k Gaussians | SSIM↑      | PSNR↑        | LPIPS↓      | k Gaussians |\n| MiniSplatting-10%                  | 0.799       | 22.661       | 0.265       | 78          | 0.759           | 26.022       | 0.318       | 111         | 0.895      | 29.395       | 0.289       | 125         |\n| **MiniSplatting-D+GHAP-10% (ours)** | ***0.835*** | ***23.232*** | ***0.198*** | 79          | ***0.802***     | ***27.09***  | ***0.25***  | 112         | ***0.909*** | ***30.042*** | ***0.254*** | 127         |\n| MiniSplatting-20%                  | 0.824       | 22.953       | 0.223       | 142         | 0.794           | 26.728       | 0.267       | 215         | 0.904      | 29.763       | 0.265       | 240         |\n| **MiniSplatting-D+GHAP-20% (ours)** | ***0.855*** | ***23.403*** | ***0.171*** | 155         | ***0.821***     | ***27.31***  | ***0.214*** | 219         | ***0.912*** | ***30.17***  | ***0.238*** | 250         |\n| LocoGS                             | 0.843       | 23.655       | 0.191       | 571         | 0.798           | 27.049       | 0.257       | 673.8       | 0.903      | 29.972       | 0.261       | 529         |\n\n- **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.\n\n| $\\rho$/Method | GHAP    | LightGaussian| PUP-3DGS  | Trimming the Fat| MesonGS |\n| :-------------------- |---------|--------------|-----------|-----------------|---------|\n| 10%                   | **23.312**  | 22.113       | 21.535    | 21.535          | 20.714  |\n| 20%                   | **23.615**  | 22.486       | 22.603    | 22.498          | 20.699  |\n| 30%                   | **23.740**  | 22.582       | 23.102    | 22.802          | 20.891  |\n| 40%                   | **23.656**  | 22.588       | 23.394    | 22.892          | 21.142  |\n| 50%                   | **23.791**  | 22.609       | 23.582    | 22.911          | 21.432  |\n\n- **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."}, {"review_id": "5R7LdfkF8E", "reviewer": "Reviewer_28Ys", "summary": "The authors propose a compaction method for pre-trained 3DGS scenes that first optimizes a Gaussian mixture model to represent the scene with fewer primitives, followed by a color finetuning that re establishes high quality textures. The method can be applied on top of existing state-of-the-art 3DGS variants for even better rendering quality. As a result, the proposed method achieves high rendering quality, measured with PSNR, SSIM and LPIPS, while using only 10% of the primitive count. To reduce the overall computing time and memory consumption, the scene is first split into blocks using a KD-tree and the GMR is performed on each block separately.", "questions": "What is the computing time and memory consumpty of the algorithm?\n\nHow does the method compare to Mini-splatting set to a 10% or 20% primitive count?\n\nWhat can be done to improve the LPIPS quality?\n\nHow can you ensure that the settings / loss functions are optimal? Have you made further ablations for different cost functions / fine tuning settings that are not included in the paper?", "limitations": "The limitations are barely discussed. The authors acknowledge a decrease in LPIPS performance but leave out important information about the runtime and memory consumption.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- The paper is written very well\n- The idea is clear and the explanations are easy to follow\n- The results are impressive\n- The visualizations, algorithms and formulas support the understanding\n\nWeaknesses:\n- The additional computing time / memory consumption is not reported. This is the biggest flaw, as it gives no intuition if the proposed method takes minutes, hours or days to compute.\n- Comparison to Mini-splatting (set to produce 10% or 20% primitives) is missing. Only Mini-splatting-D is reported\n- Line 289: “as demonstrated in the ‘Bicycle’ scene, where the poor performance of the Atom-GS baseline prevents our approach from maintaining”. This does not align with my observation in Figure 4. The grass is clearly better in the “before“ image.\n- The random subsampling is barely described and possibly not the best baseline\n- The ablation study is very limited. Different cost functions could be compared (for instance: only position, only covariance, additional opacity term)\n- Absolute number of Gaussian primitives should be mentioned in the main paper at least once\n- Algorithm 2 is not well formatted \n\n\nNice to have:\n- Experiments that use LPIPS as a loss during appearance optimization\n- Source code or example .ply files in the supplementary materials\n- Experiments that finetune the geometry as well during appearance optimization\n- large Gaussian primitives that overlap with neighboring blocks are not regarded for GMR", "was_revised": true, "final_justification": "Given that the authors have addressed all of my concerns (runtime, further comparisons, ablations, code release and formatting) and are willing to apply those changes in the final version, I increase my rating to an accept.", "rebuttal": "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.\n\n- **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**.\n\n| Method/$\\rho$  | GHAP      |        |        | LightGaussian |        |        | PUP-3DGS  |        |        | Trimming the Fat |        |        | MesonGS   |        |        |\n| :------------- | --------- | ------ | ------ | ------------- | ------ | ------ | --------- | ------ | ------ | ---------------- | ------ | ------ | --------- | ------ | ------ |\n|                | time（s） | memory | PSNR↑  | time（s）      | memory   | PSNR↑  | time（s） | memory | PSNR↑  | time（s）        | memory | PSNR↑  | time（s） | memory | PSNR↑  |\n| 10%            | 5.6       | 4857   | 23.312 | 10.8          | 3908   | 22.113 | 26.3      | 4001   | 21.535 | <0.1             | 3740   | 21.535 | 11.0      | 2830   | 20.714 |\n| 20%            | 4.4       | 4895   | 23.615 | 11.2          | 3908   | 22.486 | 26.3      | 4001   | 22.603 | <0.1             | 3731   | 22.498 | 11.1      | 2830   | 20.699 |\n| 30%            | 3.8       | 4976   | 23.740 | 9.0           | 3908   | 22.582 | 26.3      | 4001   | 23.102 | <0.1             | 3722   | 22.802 | 10.8      | 2830   | 20.891 |\n| 40%            | 3.3       | 5060   | 23.656 | 10.7          | 3908   | 22.588 | 26.3      | 4001   | 23.394 | <0.1             | 3760   | 22.892 | 10.9      | 2830   | 21.142 |\n| 50%            | 3.0       | 5141   | 23.791 | 10.8          | 3908   | 22.609 | 26.3      | 4001   | 23.582 | <0.1             | 3800   | 22.911 | 11.1      | 2830   | 21.432 |\n\n- **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**.\n\n| Method-$\\rho$/Dataset         | **TNT**     |              |             |             | **360**     |              |             |             | **DB**      |              |             |             |\n| ----------------------------- |-------------| ------------ | ----------- | ----------- |-------------| ------------ | ----------- | ----------- |-------------| ------------ | ----------- | ----------- |\n|                               | SSIM↑       | PSNR↑        | LPIPS↓      | k Guassians | SSIM↑       | PSNR↑        | LPIPS↓      | k Gaussians | SSIM↑       | PSNR↑        | LPIPS↓      | k Gaussians |\n| MiniSplatting-10%             | 0.799       | 22.661       | 0.265       | 78          | 0.759       | 26.022       | 0.318       | 111         | 0.895       | 29.395       | 0.289       | 125         |\n| MiniSplatting-D+GHAP-10%      | ***0.835*** | ***23.232*** | ***0.198*** | 79          | ***0.802*** | ***27.09***  | ***0.25***  | 112         | ***0.909*** | ***30.042*** | ***0.254*** | 127         |\n| MiniSplatting-20%             | 0.824       | 22.953       | 0.223       | 142         | 0.794       | 26.728       | 0.267       | 215         | 0.904       | 29.763       | 0.265       | 240         |\n| MiniSplatting-D+GHAP-20%      | ***0.855*** | ***23.403*** | ***0.171*** | 155         | ***0.821*** | ***27.31***  | ***0.214*** | 219         | ***0.912*** | ***30.17***  | ***0.238*** | 250         |\n\n- **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.\n    - **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:\n\n    | Metric/Loss                          | SSIM↑ | PSNR↑   | LPIPS↓ |\n    | ------------------------------------ | ----- | ------- | ------ |\n    | original (0.8 * L1 + 0.2 * ssim)     | 0.818 | 23.312  | 0.242  |\n    | 0.8 * L1 + 0.1 * ssim +  0.1 * lpips | 0.793 | 23.0572 | 0.239  |\n    | 0.6 * L1 + 0.2 * ssim + 0.2 * lpips  | 0.805 | 22.993  | 0.207  |\n    \n    - **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:\n\n    | Metric/Cost   | SSIM↑         | PSNR↑    | LPIPS↓   |\n    |---------------|---------------|----------|----------|\n    | original cost | 0.818         | 23.312   | 0.242    | \n    | + opacity     | 0.814         | 23.266   | 0.250    |\n\n    \n- **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:\n\n| Sampling Iteration/$\\rho$.         | SSIM↑ | PSNR↑  | LPIPS↓ |\n| ---------------------------------- | ----- | ------ | ------ |\n| [15k] / [0.1]                      | 0.817 | 23.313 | 0.242  |\n| [15k, 20k] / [0.2, 0.5]            | 0.817 | 23.44  | 0.247  |\n| [15k, 20k, 25k] / [0.4, 0.5, 0.5]  | 0.811 | 23.359 | 0.255  |\n\n\n- **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.\n- **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.\n- **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.\n- **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.\n- **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."}]

Author rebuttal:

Meta-review: This paper proposes to view 3DGS representations as a Gaussian mixture and perform compaction from the perspective of Gaussian mixture reduction via optimal transport.   
Strength:  
Framework is novel (1EBm, D3B2, Cgrv), plug-and-play (1EBm), high performance (1EBm, D3B2, 28Ys).       
Weakness:  
Ablation study (1EBm, 28Ys), effectiveness (1EBm), technical novelty (1EBm),  computation (D3B2, Cgrv, 28Ys), baseline comparison (D3B2, Cgrv, 28Ys)

After rebuttal, this paper receives all positive ratings (5444). Most reviewers think the proposed method is novel and the performance is high. Although some common concerns regarding computation cost, ablation study anb baseline comparisons are raised, the authors have done a good job during the rebuttal process, and all the reviewers are satisfied and vote for acception. Thus, the AC recommend to accept this paper. Please do include those new results and discussions in later versions.
Decision: Accept (spotlight)