Universal Beta Splatting
Authors: Rong Liu
Zhongpai Gao
Benjamin Planche
Meida Chen
Van Nguyen Nguyen
Meng Zheng
Anwesa Choudhuri
Terrence Chen
Yue Wang
Andrew Feng
Ziyan Wu
Venue: ICLR 2026 Poster (2026)
Keywords: Radiance Field
Splatting
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning

Abstract: We introduce Universal Beta Splatting (UBS), a unified framework that generalizes 3D Gaussian Splatting to N-dimensional anisotropic Beta kernels for explicit radiance field rendering. Unlike fixed Gaussian primitives, Beta kernels enable controllable dependency modeling across spatial, angular, and temporal dimensions within a single representation. Our unified approach captures complex light transport effects, handles anisotropic view-dependent appearance, and models scene dynamics without requiring auxiliary networks or specific color encodings. UBS maintains backward compatibility by approximating to Gaussian Splatting as a special case, guaranteeing plug-in usability and lower performance bounds. The learned Beta parameters naturally decompose scene properties into interpretable without explicit supervision: spatial (surface vs. texture), angular (diffuse vs. specular), and temporal (static vs. dynamic). Our CUDA-accelerated implementation achieves real-time rendering while consistently outperforming existing methods across static, view-dependent, and dynamic benchmarks, establishing Beta kernels as a scalable universal primitive for radiance field rendering.

Published as a conference paper at ICLR 2026
UNIVERSAL BETA SPLATTING
Rong Liu"?; Zhongpai G Benjamin Planche', Meida Chen', Van Nguyen Nguyen', Meng Zheng', Anwesa Choudhuri", Terrence Chen', Yue Wang', Andrew Feng', Ziyan Wu²
'University Southern California, Los Angeles, CA, USA United Imaging Intelligence, Boston, MA. USA
OJIS
3DGS
Otes
4DCS
Our
Curs
Figure 1: Visualization of UBS rendering quality. For static real-world scenes (left), UBS achieves superior rendering reflective and specular materials compared to 3DGS (Kerbl et al., 2023). For dynamic volumetric scenes (right), UBS maintains high visual fidelity in complex spatio-temporal scenarios where 4DGS (Yang et al., 2024a) produces blurring artifacts.
ABSTRACT
We introduce Universal Beta Splatting (UBS), unified framework that general- izes 3D Gaussian Splatting to N-dimensional anisotropic Beta kernels for explicit radiance field rendering. Unlike fixed Gaussian primitives, Beta kernels enable controllable dependency modeling across spatial, angular, and temporal dimen- sions within a single representation. Our unified approach captures complex light transport effects, handles anisotropic view view-dependent appearance, and models scene dynamics without requiring auxiliary networks or specific color encodings. UBS maintains backward compatibility by approximating to Gaussian Splatting as a special case, guarantecing plug-in in usability and lower performance bounds. The learned Beta parameters naturally decompose scene properties without ex- plicit supervision: spatial (surface vs. texture), angular (diffuse vs. specular), and temporal (static vs. dynamic). Our CUDA-accelerated implementation achieves real-time rendering while consistently outperforming existing methods across static, view-dependent, and dynamic benchmarks, establishing Beta kernels as a scalable universal primitive for radiance field rendering. Our project website is available at https://rongliu wwthtg. io/universal betasal---a-----tingge
INTRODUCTION
Real-time photorealistic rendering of complex remains fundamental challenge in computer vision and graphics. While Neural Radiance Fields (NeRF) (Mildenhall et al., 2020) achieve ex- ceptional visual quality through continuous volumetric representations, their reliance dense ray sampling imposes prohibitive computational costs. 3D Gaussian Splatting (3DGS) (Kerbl et al., 2023)
"This work was done during Rong Liu's internship at United Imaging Intelligence, Boston, MA, USA. Corresponning author.

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Published as a conference paper at ICLR 2026
addresses efficiency through explicit primitive-based rendering, achieving real-time performance with competitive fidelity. However, Gaussian kernels limit representational capacity: their fixed bell-shaped profiles struggle with sharp boundaries, while view-dependent effects require auxiliary spherical harmonic encodings that fragment the representation. Dynamic scene extensions (Yang et al., 2024b; Luiten al., 2024) further increase complexity through additional deformation networks.
Recent advances partially address these limitations through specialized designs. Deformable Beta Splatting (DBS) (Liu al., 2025a) improves geometric fidelity using spatially adaptive Beta kernels, but remains limited to 3D spatial dimensions and still requires separate Spherical Beta functions for view-dependent color. N -'ddimensioal Gaussian extensions like 6DGS (Gao et al., 2025a) and 7DGS (Gao et al., 2025b) incorporate view and temporal dimensions through conditional distributions, enabling them to capture scattering effects and dynamic changes. However, these approaches remain constrained by the Gaussian kernel's symmetric profile across dimensions- preventing independent control of spatial sharpness, angular specularity, and temporal dynamics.
The core insight is that different scene properties require different kernel behaviors. Spatial geometry benefits from adaptive sharpness-ffin for surfaces, peaked for textures. Angular appearance spans from diffuse to specular responses. Temporal dynamics range from static (constant support) to rapid motion (localized activation). While Gaussian kernels couple all dimensions with identical bell-shaped profiles, real scenes exhibit independent variations across space, angle, and time.
We introduce Universal Beta Splatting (UBS), unified framework that generalizes radiance field rendering to N-diimensional anisotropic Beta kernels. Unlike fixed Gaussian profiles, Beta kernels provide per-dimension shape control through learnable parameters, enabling each dimension to adopt its optimal form. This allows a single primitive type simultaneously model spatial geometry, view-dependent appearance, and temporal dynamics while maintaining computational efficiency.
Our approach addresses three technical challenges: (1) spatial-orthogonal Cholesky parameterization preserving orthonormal spatial structure while enabling flexible tross-dimensional correlations, (2) Beta-modulated conditional slicing transforming high-dimensional primitives into renderable 3D representations with dimension-specifii anisotropy, and (3) fully accelerated CUDA implementation ensuring real-time performance.
Crucially, UBS maintains backward compatibility: when Beta parameters equal zero, the kernel approximates to Gaussian, recovering 3DGS, 6DGS, or 7DGS depending on dimensionality. This ensures seamless drop-in replacement with guaranteed performance lower bounds. Beyond rendering improvements, learned Beta parameters naturally decompose scenes into interpretable components without explicit supervision: spatial parameters separate geometry from textures, angular parameters distinguish diffuse from specular, and temporal parameters isolate static from dynamic elements.
We validate UBS across static, view-dependent, and dynamic benchmarks, achieving state-of-the-art visual iumlity-improving PSNR by up to +8.27 dB on static scenes and +2.78 dB on dynamic scenes compared to corresponding Gaussian baselines. Our contributions are:
Universal Beta Splatting: A unified N --dimensional representation with per-dimension shape control, enabling simultaneous modeling of spatial, angular, and temporal properties through anisotropic Beta kernels with spatial-orthogone Cholesky parameterization. Efficient CUDA implementation: Fully accelerated differentiable rendering with custom kernlls achieving real-time performance.
Interpretable scene decomposition: Emergent separation of geometric, appearance, and motion components through learned Beta parameters without supervision.
Backward compatibility: Reduction to approximate existing methods as special cases, ensuring plug-in usability with performance lower bounds while enabling substantial improvements.
RELATED WORK
Neural Radiance Fields and Caussian Splatting, Neural Radiance Fields (NeRF) (Mildenhall et al., 2020) revolutionized novel view synthesis through continuous volumetric representations. While extensions like Instant-NGP (Mäller et al., 2022), Mip-NeRF (Barron al., 2022), and ZiR-NeRF (Barron et al., 2023) improved training speed and quality, NeRF methods remain limited by slow inference and inability model off-ray transport phenomena.
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3D Gaussian Splatting (3DGS) (Kerbl et al., 2023) achieved real-time rendering through explicit anisotropic Gaussian primitives. Extensions address anti-aliasing (Yu et al., 2024; Yan et al., 2024), compression (Lee al.. 2024a; Morgenstern et al., 2024), and semantic applications (Shi et al., 2024; Zhou et al., 2024; Peng et al., 2025; Xiong et al., 2026). However, 3DGS's spherical harmonic encoding struggles with complex view-dependent effects. To better capture view dependence, N- dimensional Gaussians (N-DG) (Diolatzis et al., 2024) extended to higher dimensions, refined by 6D Gaussian Splatting (6DGS) (Gao et al., 2025a) using conditional slicing. While elegant, Gaussian distributions cannot produce sharp angular cutoffs without many small primitives, limiting efficiency for specular materials.
Alternative Kernel Designs. Recent work explores kernels beyond standard 3D Gaussians. Several methods modify the kernel shape to better capture geometric structure: GES (Hamdi al., 2024), TNT-GS (Liu et al., 2025b), Quadratic Gaussian Splatting (Zhang et al., 2025), 3D Half-Gaussian Splatting (Li al., 2025), and Disc-GS (Qu et al., 2024) propose generalized exponentials, truncated, Gaussian splats. Other approaches expand the distribution
second-order, half, and discontinuity family, including Deformable Beta Splatting (DBS) (Liu et al., 2025a), 3D Student Splatting (Zhu al., 2025), and Gabor Splats (Zhou et al., 2025) for sharper edges heavier- -tille behavior. For appearance modeling. Textured Gaussians (Chao et al., 2025) enrich per-splat color detail. Beyond volumetric kernels, 3D Convex Splatting (Held al., 2025b), Triangle Splatting (Held al., 2025a), and Deformable Radial Kernel Splatting (Huang et al., 2025) introduce other geometric primitives as alternatives to ellipsoids. While these methods improve specific aspects of spatial reconstruction. they remain confined to 3D spatial kernels, without extending to unified nigh-dimensional splatting that jointly models spatial, angular, or temporal dimensions.
Dynamic Neural Radiance Fields and Gaussian Splatting, Early NeRF extensions modeled dynamics through deformation fields: D-NeRF (Pumarola et al., 2021) learns sanonical-to-defomee mappings, Nerfies (Park et al., 2021a) handles non-rigid deformations, HyperNeRF (Park et al., b021b) models topological changes, and HexPlane (Cao & Johnson, 2023) uses factorized representations.
For Gaussian Splatting, 4D Gaussian Splatting (4DGS) (Yang et al., 2024a) extends primitives temporally for real-time dynamic rendering. Dynamic 3D Gaussians (Luiten et al., 2024) and related works (Wu et al., 2024; Yang et al., 2024b) combine canonical representations with learned deformations. 7D Gaussian Splatting (7DGS) (Gao et al., 2025b) unifies spatial, temporal, and angular dimensions without separate deformation networks. Despite this unification, Gaussian's symmetric profile enforces smooth transitions a single primitive cannot simultaneously represent sharp spatial edges, abrupt temporal motion, and narrow specular highlights.
Our Universal Beta Splatting addresses these limitations through controllable Beta kernels with per-dimension parameters, enabling dimension-specifie adaptation and computational efficiency.
3 PRELIMINARY
3D Gaussian Splatting (3DGS). 3DGS (Kerbl et al., 2023) represents scenes as collections of anisotropic 3D Gaussians parameterized by position μt E R opacity o, rotation quaternion q, scale 8, and color features f. The covariance matrix Σ = RSS R is constructed from rotation and scale matrices. To render an image, each 3D Gaussian ellipsoid is projected into the 2D image plane using the viewing transformation W. This yields 2D mean jı' E IR and 2D covariance matrix E' = JWEW"S JT, where is the Jacobian of the projection. For a pixel location æ E R?, the Mahalanobis distance the projected splat is
rx(x)) = (x 14)
E (x m1))
The Gaussian contribution to the pixel is then computed using a Gaussian kernel multiplied by opacity:
o ( = ((x; μ, E) = = tr((2)
Finally, after sorting the primitives in front-to-back order, the pixel color is obtained using standard alpha compositing:
C(x) Σ
co
II (1 (z)))
(3)
1=1
/=1
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Stracture
Specular
Dynamic
Rendering
Sparial Decomposition
Angular Decompooition
Temporal Decomposition
Figure 2: Visualization decomposition. Our learned Beta parameters provide interpretable scene decomposition across spatial, angular, and temporal dimensions without explicit supervision.
Deformable Beta Splatting (DBS). DBS (Liu et al., 2025a) replaces Gaussian kernels with adaptive and bounded Beta kernels, defining density (xx) = (x;; ,a, 2, b) o, where
(x:; pc E. (1 r())))
B(b) cxp(b), b be I, r(x) E 10.
(4)
The Beta kernel closely approximates scaled Gaussian when initialization. As training progresses, the learnable parameter allows the kernel shape adapt: negative produces flatter profiles with sharper boundaries suitable for solid geometry, while positive b yields sharper peaks that better capture high-frequency details. This adaptability enables DBS model diverse geometric structures more effectively than fixed Gaussian kernels. Appendix D includes more background and motivation.
N-Dimensional Gaussian. 6DGS (Gao al., 2025a) and 7DGS (Gao et al., 2025b) extend 3DGS through conditional distributions. The key idea is to model a joint distribution over spatial and non- spatial dimensions (view/time), then condition on the query to obtain view-specific or time-specific 3D Gaussian for rendering. 6DGS incorporates viewing direction into a 6l) joint distribution:
~N((
(N((
14p 1ia
Σp Σ
Edd
(5)
Σn
Conditioning on view yields conditional covariance Econd = Σp E End
IT pdt conditional mean
Hoond 14p + Ey (.a), and modulated opacity Ocond expl (-ppp--dd--- paa) Σイ (d pa)).
7DGS adds temporal dimension X, for dynamic scenes, with conditioning on both time and view producing spatio-temporal-angg
modulation. These methods establish high-dimensional repre- sentations but remain limited by fixed Gaussian profiles that cannot independently control cross- dimensional dependencies-a limitation UBS addresses through adaptive Beta kernels.
METHOD
In this section, introduce Universal Beta Splatting (UBS), unified framework that extends explicit radiance field rendering to N-dimensional anisotropic Beta kernels.
4.1 N-DIMENSIONAL. BETA KERNEL
The core innovation of UBS the N-dimensional anisotropic Beta kernel that generalizes the DBS density function to higher dimensions:
oxx q) B(x qi u. E, b)
(6)
where a 3R3 represents spatial coordinates, the query q E N- encodes additional dimensions (view direction, time, etc.), la E R.V is the mean vector, Σ E N is the covariance matrix, E T controls the Beta shape parameters across dimensions (Betas are shared spatial domain), and [o, 1] is the opacity. The shape parameters b are transformed to ensure positive Beta exponents: Bi = exxp(b;), enabling each dimension to adopt optimal kernel forms-fiee from flat (negative b,) peaked (positive b;).
In addition to the primitive shape, each kernel primitive also carries an RGB color c E [0, 1 Crucially, because the Beta kernel natively incorporate view and temporal dimensions through q, no auxiliary color encoding is required to model view-dependent appearance or dynamics. This unified representation significantly reduces the per-primitive parameter count: for static scenes, UBS
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achieves 41% parameter reduction compared to 3DGS, and for dynamic scenes, UBS uses 73% fewer parameters than 4DGS, as detailed Table 7 in the Appendix B. In contrast, existing methods rely on auxiliary encodings: 3DGS, 6DGS, and 7DGS employ 48 3a-aammmeter spherical harmonics, 4DGS requires 144-parameter 4D spherical harmonics, and DBS needs additional spherical Beta functions for color modeling-
To maintain valid spatial geometry and guarantee backward compatibility while preserving high- dimensional expressiveness, we partition the N-dimensional space as:
Σ que
E mq Σg
E
where subscript denotes spatial components and denotes the additional dimensions. This parti- tioning allows conditioning on non-spatial dimensions to obtain renderable 3D representations.
4.2 SPATIAL-ORTHHOGONAL CHOLESKY PARAMETERIZATION
While mean vectors, beta shapes, and colors straightforward to parameterize, the covariance matrix requires additional care as it must be symmetric and positive semi-definite. In 3DGS, this achieved by factorizing the covariance parameterized by quaternions and s are per-axis scales. However, quaternions do not generalize to higher dimensions (N >> 3). N-dimensional Gaussian methods therefore resort to Cholesky factorization Σ = LL", which loses explicit orthogonality and often leads to suboptimal solutions the spatial subspace.
Σ R diag(s)² R where R is a rotation matrix
To address this limitation, we introduce the preserves rotation-seaa scale structure the spatial subspace while enabling flexible cios-diieeni correlations. For N 3 + C' dimensions, we construct:
Cholesky parameterization that
R.g diag sz)
0 L.)
(8)
Lgz
where Rg E SO(3) is the spatial rotation computed via first-order Taylor approximation Rg 2 I+A, of the matrix exponential (with A, skew kkew-symmetric s, are spatial scales, Sg are scales for other dimensions, and Lyz encodes cross-correlations. As shown in Table 10 (Appendix C), this first-order approximation provides nearly identical fidelity while greatly reducing computation. The covariance Σ LL maintains explicit 3D geometric structure while allowing expressive correlations across view and time. More theoretical and empirical analysis are available Appendix C.
4.3 BETA-MODULATED CONDITIONAL SLICING
In N dimeensioonl Gaussians, all dimensions coupled: when the input view or time changes, the primitive is inevitably dragged along- it shifts in space, stretches in shape, and fades in opacity. To break this constraint, we introduce Beta-Modulated Conditional Slicing.
To render at query condition we condition the N-dimensional Beta kernel to obtain 3D spatial representation. We introduce Beta modulation to the standard conditional Gaussian formulation:
Beta Parameters. We define per-dimension Beta exponents with spatial parameter sharing: / [Ba, B2, Ba, ......,, Bacl where 3, 4 exp(b.) shared across the first three spatial dimensions and Br = exp(b,.) are individual parameters for the C non-spatial dimensions. We share the spatial Beta parameter across all three spatial dimensions to ensure that splatted kernels maintain consistent spatial geometry. This design choice prevents inconsistent spatial artifacts while still allowing per-dimension control in angular and temporal domains.
Beda-Modulated Conditioning, The conditional mean and covariance incorporate dimension-specifee Beta modulation:
Mglg = Mrr + Eng Σ diag(S,) (q (g-1))),
(9)
Ex1g En- Ezg D, diag(B,) Eger
(10)
where diag(ā,) = diag(1B,,. ......... k1) applies simension-specifie Beta modulation to the non- spatial conditioning terms while preserving isotropic spatial structure.
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Beta-Modulated Opacity. We adopt factorized opacity with inherent multidimensional support:
d = ( mg));, d; tanh(d, E (0, o(q) - =0 II d
(11)
2=1
where É, denotes the Cholesky factor of En, such that L = Σ5 Then tanh maps per- dimension Mahalanobis distance di into bounded value d; E 1). satisfying the Beta kernel input definition. The modulated product form allows independent control over each non-spatial dimension's ss contribution.
Final Beta-Splat Density. The conditioned 3D Beta kernel becomes:
axx q) B(;;; Mxlg Exlgs bz) o(g),
(12)
where the spatial Beta b₂ controls the 3D) kernel shape uniformly across all spatial dimensions.
4.4 UNIVERSAL COMPATIBILITY AND INTERPRETABILITY
Adaptive Specialization. UBS naturally specializes for different applications through its query dimensions: (1) = d (view direction, N = 6) for view-dependent rendering with anisotropic BRDF modeling. denoted as UBS-6D; (2) q [t. d] (time + view, N 7) for dynamic scenes capturing ppatio-emporral-nnule correlations, denoted as UBS-7D; and (3) seamless extension to higher dimensions for additional modalities like lighting or material properties.
Universal Compatibility. UBS provides backward compatibility by design, reducing to existing methods under specific parameter settings, For N = (pure spatial), UBS reduces to Deformable Beta Splatting with a (a:) (ax; M4n E,, bz) and further approximates to standard 3DGS when by = 0 (Gaussian limit). For higher dimensions with all Beta parameters zero (b = 0), UBS approximates corresponding Gaussian methods: 6DGS for riew-dependent rendering (N = 6) and 7DGS for dynamic scenes (N = 7). This guaranteed compatibility ensures UBS serves as a seamless drop-in replacement while providing performance lower bounds.
Interpretable Decomposition. Beyond compatibility, learned Beta parameters provide interpretable scene decomposition without explicit supervision. Spatial parameters distinguish geometry types: negative by produces flat kernels for smooth surfaces, while positive b, creates peaked kernels for fine textures. Angular parameters separate appearance models: negative yields broad responses for diffuse surfaces, while positive by generates sharp peaks for specular reflections. Temporal parameters isolate motion patterns: negative 6, maintains broad support for static elements, while positive by provides localized activation for dynamic components, as illustrated in Figure 2. This decomposition enables applications like relighting and motion analysis without requiring separate encodings auxiliary networks.
4.5 OPTIMIZATION AND IMPLEMENTATION
For primitive management, we adopt the kernel-agmostic MCMC optimization from DBS (Liu et al., 2025a), which provides stable, cerribbution-rreeevvi updates independent of kernel form.
Loss Function. We optimize with reconstruction losses and regularization:
L = (1 ASsIM) L1 + ASSII + No Σ lo.1 + As Σ ||sil1a,
(13)
where opacity regularization A, ensures valid MCMC densification and scale penalty λ encourages primitive relocation. Spatial noise injection ftz M4 A V >e promotes exploration.
Densification, Cloned primitive opacity follows o' = - VI preserve distribution. Under opacity regularization, adjusting opacity maintains density distribution regardless of kernel type.
CUDA Implementation. We provide fused CUDA kernels for Beta evaluation, conditional slicing, and -othooona operations. The pipeline maintains compatibility with existing frameworks: N ---dimensional kernels are conditioned to 3D then rasterized using standard splatting. This modular design enables real-time performance while allowing future integration with acceleration techniques like FlashGS (Feng al., 2025) and extended camera models from 3DGUT (Wu al., 2025).

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5 EXPERIMENTS
5.1 EXPERIMENTAL SETUP
Datasets. We evaluate UBS across five diverse datasets covering both static and dynamic scenes:
Static: we use (1) NeRF Synthetic (Mildenhall et al., 2020) with 8 synthetic scenes under controlled lighting for geometric reconstruction assessment; (2) Mip-NeRF 360 (Barron et al., 2022) sharing unbounded real-world scenes challenging anti-aliasing and view synthesis; and (3) 6DGS-PBR (Gao et al., 2025a) featuring 7 physically-bast
scenes with complex view-dependent effects including volumetric scattering, translucent materials, subsurface scattering, and medical volumetrics.
Dynamic: we evaluate on: (4) D-NeRF (Pumarola et al., 2021) with 8 synthetic sequences featuring various deformation types; and (5) 7DGS-PBR (Gao et al., 2025b) containing 6 physically based correlations including cardiac motion, daylight transitions, animated volumetric effects, and translucent deformations.
dynamic scenes with complex
Evaluation Metrics. We assess reconstruction quality using standard metrics: Peak Signal-to- Noise Ratio (PSNR), Structural Similarity Index Measure (SSIM) (Wang al., 2004), and Learned Perceptual Image Patch Similarity (LPIPS) (Zhang et al., 2018). Additionally, we report rendering speed (FPS) and training time evaluate computational efficiency. All metrics are computed on held-out test views following standard evaluation protocols.
Implementation Details. UBS is implemented in PyTorch with custom CUDA kernels for Beta kernel evaluation and conditional slicing operations. We conduct on single NVIDIA RTX 4090 24GB GPU for static and ablation experiments and single NVIDIA Tesla V100 16GB GPU for dynamic experiments to be consistent 7DGS. Training employs the Adam optimizer for 30K iterations with learning rates of 1.6 x 10-4 for positions, x 10 for opacity, x 10 for scales, and 1 x 10-3 for other parameters. The regularization weights are set X 0.01, A 0.01, and X 1. For N-dimensional cases, we randomly initialize temporal and directional means (f4, ft)) within [0, 1]. Beta parameters are initialized to zero, corresponding the Gaussian limit for guaranteed convergence. For dynamic scenes, we use a batch size of to maintain consistency with 4DGS and 7DGS baselines. Unlike these methods which evaluate test sets every 500 iterations and report best results in their papers, we report at the final iteration (30K) to demonstrate convergence stability.
5.2 COMPARISON WITH THE STATE OF THE ART
Table 1 compares our UBS with 3DGS (Kerbl al., 2023), 4DGS (Yang et al., 2024a), 6DGS (Gao et al., 2025a), and 7DGS (Gao et al., 2025b) on both static and dynamic benchmarks.
Static Scenes. Our method UBS-6D consistently achieves superior reconstruction quality across all datasets while maintaining competitive training efficiency on three static scene benchmarks:
NeRF Synthetic: UBS-6D achieves 34.92 dB PSNR, improving by +1.13 dB over 3DGS and +1 +1.32 dB over 6DGS. Despite lacking strong view-dependent effects, the dataset benefits from Beta kernels' geometric adaptivity, with notable improvements on scenes requiring diverse surface representations (ficus: +1.41 dB over 3DGS, materials: +2.40 dB over 3DGS).
Mip-NeRF 360: UBS-6D reaches 28.66 dB PSNR, surpassing 3DGS by +1.23 dB and 6DGS by +2.31 dB. The real-world unbounded scenes showcase UBS's effectiveness on complex lighting conditions, with substantial gains on indoor environments (counter: +2.43 dB over 3DGS, bonsai: +3.00 dB over 3DGS).
6DGS-PBR: UBS-6D demonstrates the most significant improvements with 40.10 10 dB PSNR (+12.70 dB over 3DGS, +3.42 dB over 6DGS average). The physically-based scenes with volumet- ric scattering and translucency particularly benefit from per-dimension Beta control: volumetric effects like cloud improve by +5.81 dB over 6DGS, while complex materials like bunny achieve a remarkable +8.27 dB improvement over 6DGS.
Overall, UBS-6D demonstrates exceptional performance challenging view-dependent scenarios in 6DGSDDGPPAR In addition, UBS-6D establishes new of results on NeRF Syntheric and Mip-NeRF 360 shown in Table 4 in Appendix A.
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Table 1: Comparison with 3DGS and 6DGS static benchmarks of NeRF Synthetic (Mildenhall al., 2020),
Mip-NeRF (Barron al., 2022), 6DGS-PBR (Gao et al., 2025a), and with 4DGS and 7DGS on dynamic
benchmarks of 7DGS-PBR (Gao et al., 2025b) and D-NeRF (Pumarola et al., 2021). 'Train' means time in mins.
3DGS
6DGS
UBS-6D (Ours)
Dataset
Scene
PSNR1
SSIM
LPIPS|
Train
PSNR1
SSIM
LPIPS|
Truin
PSNR1
SSIM1 LPIPS.
Train
chair
35.60
0.988
0.010
4.8
35.55
0.986
0.011
19.6
36.72
0.990
0.009
8.7
NeRr Synthetic
drums
26.28
0,955
0.037
4.1
26.63
0.951
0.038
15.4
27.19
0.960
0.031
8.1
cicus
35.49
0,987
0.012
3.0
34,62
0,984
0.015
11.3
36,90
0.990
0.009
8.2
hottdo
38.07
0.985
0.020
3.5
37.96
0,983
0.021
10.0
38.67
0.988
0.016
9.1
legc
36.06
0,983
0.016
4.0
35.22
0.979
0..000
15.3
36.95
0.985
0.014
8.4
materiala
30.50
0,960
0.037
2.7
30.63
0,960
0.041
7.8
32.90
0.977
0.025
8.9
mic
36.67
0,992
0.006
3.0
37.10
0,992
0.007
9.4
37.86
0.005
9.5
shhi
31.68
0.906
0.106
4.1
31.09
0.899
0.118
12.7
32.13
0,914
0.100
8.8
STATIC SCENES
Javy
33.79
0.970
0.030
3.6
33.60
0.967
0.034
12.7
34.92
0.975
0,026
8.7
bicyele
25.19
0.764
0.212
29.2
22.17
0.535
0.425
114.9
25.64
0.793
0,186
30.2
flowers
21.39
0.604
0.338
19.6
18.99
0.417
0,461
88.8
21.96
0.635
0.313
19.6
Mip-NeREr NeRF 360
garden
27.27
0.862
0.109
30.4
26.71
0.843
0.142
164.9
28.10
0.883
0.094
36.8
stung:
26.61
0.772
0.215
22.7
23.83
0.665
0.338
60.2
26.75
0.788
0.199
18.9
Ereehill
22.47
0,631
0.329
20.1
21.51
0,556
0.425
94.0
23.57
0.684
0.280
21.9
room
31.49
0.917
0.221
18.2
31.34
0.904
0.256
51.4
32.81
0,940
0.173
21.9
counter
28.96
0.906
0,202
17.6
29.17
0.880
0.253
43.5
31.39
0.937
0.150
27.8
kitchen
31.40
0.926
0.127
21.5
30.48
0.907
0.1588
71.3
32,64
0.940
0.107
26.6
bonsai
32.05
0.940
0.206
15.7
32.91
0.935
0.228
48.4
35.05
0.962
0.154
22.7
lavg
27.43
0.814
0.218
21.7
26.35
0.738
0..99
81.9
28.66
0.840
0.184
25.2
bunny
29.19
0.985
0.074
22.8
37.41
0.991
0.052
24,1
45.68
0.996
0.023
32.1
6DGS PHR
cloud
34.03
0,985
0.058
23.1
41.00
0.991
0.050
21.6
46.81
0,995
0.027
30.6
explesion
27.06
0,953
0.097
11.3
41.61
0,990
0.031
18.3
44.54
0,994
0.018
32.4
smoke
26.82
0.964
0.088
24.2
41.41
0.993
0.041
31.1
44.02
0.995
0.028
33.5
suzanne
22.45
0,885
0.159
19.5
26.96
0.928
0.106
32.9
27.35
0.931
0.098
35.1
dragen
26.53
0,812
0.123
12.0
34.95
0,933
0.044
16.3
38.06
0.974
0.034
23.7
et-sean
25.69
0.917
0.099
4.5
33.46
0.964
0.058
11.0
34.24
0.969
0.046
6.5
avg
27.40
0.929
0.100
16.8
36.68
0.970
0.055
22.2
40.10
0.979
0.039
27.7
4DGS
7DGG
UBS-71 (Ours)
Dataset
Soene
PSNR1
SSIM?
LPIPS
Train|,
PSNRT
SSIM
LPIPS.)
Train.
PSNRT
SSIMI LPIPS.
Train]
heart1
27.23
0.949
0.046
103.0
34.66
0.983
0.023
114.2
37.44
0.990
0.013
38.7
7D0S- PBR
heart2
25.09
0.919
0.085
103.4
30.99
0.959
0.057
384.6
31.99
0.966
0.043
33.5
cloud
24.63
0.938
0.100
123.7
29.28
0.955
0.075
102.6
30.65
0.955
0.066
50.0
dust
35.81
0.954
0.037
97.0
36.85
0.955
0.038
69.8
39.17
0.985
0.028
46.5
DYNAMIC SCENES
29.25
0,927
113.7
31.64
0,937
74.1
31.70
0.057
53.7
suzanne
24.41
0.912
0.127
222.5
27.09
0,941
0.072
193.9
27.12
0.934
0,069
136.8
Javg
27.74
0.933
0.077
127.2
31.75
0.955
0.055
116.7
33.00
0.966
0.046
59.9
b.balls
32.45
0,979
0.028
50.7
33.61
0.980
0.025
86.6
33.39
0,982
0.022
49.6
A.warrior
33.69
0,944
0.069
35.3
32.30
0.931
0.088
31.3
33,81
0.934
0,086
36.3
D-NoRF
book
31.90
0.965
0.037
38.0
30.19
0.953
0.047
35.7
30.93
0,956
0.045
34.0
i. jacakks
28.43
0.962
0.041
66.9
31.37
0.967
0.038
34.1
30.96
0.966
0.040
43.2
lege
24.28
0,903
0.098
55.4
27.64
0.935
0.067
78.5
27.85
0.925
0.072
31.3
mutant
38.64
0.990
0.009
39.1
39.21
0,992
0.008
42.5
40.75
0,994
0.006
60.0
standup
39.00
0.990
0.009
34.4
38.42
0.987
0.014
33.5
39,08
0,991
0.009
41.1
trex
28.35
0.973
0.026
100.7
29.94
0.978
0.021
63.4
30.05
0.975
0.023
41.6
Javg
32.09
0.963
0.040
52.6
32.84
0.965
0.039
50.7
33.35
0.965
0.038
42.1
Dynamic Scenes. We compare our method UBS-7D with 4DGS and 7DGS on 2 dynamic benchmarks. Note that we consider the base 7DGS without MLP networks for adaptive Gaussian refinement (AGR) to ensure fair comparison and isolate the benefits of our primitive design from auxiliary networks:
7DGS-PBBL UBS-7D achieves 33.00 dB PSNR, outperforming 4DGS by +5.26 dB and 7DGS by +1.25 dB. The mhysically-based dynamic scenes with pptio--emmporalaangllar correlations show substantial improvements, particularly on complex volumetrie dynamics (dust: +2.32 dB over 7DGS) and cardiac motion sequences (heart +2.78 dB over 7DGS).
D-NeRF: UBS-7D reaches 33.35 dB PSNR (+1.26 dB over 4DGS, +0.51 dB over 7DGS). The synthetic monocular sequences benefit from Beta kemels' ability to model diverse temporal behaviors, with notable gains on scenes requiring both spatial detail and temporal precision (mutant: +1.54 dB over 7DGS, h. warrior: +1.51 dB over 7DGS).
The consistent improvements across PSNR. SSIM, and LPIPS metrics confirm that UBS's enhanced representational capacity translates to comprehensive quality gains. Figure 3 demonstrates that UBS
8

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Published as a conference paper at ICLR 2026
Ground Truth
3DGS
6DGS
UBS-6D (Ours)
>
Ground Truth
4DGS
7DGS
UBS-7D (Ours)
Figure 3: Qualitative comparison of methods for static and dynamic scenes.
achieves better visual quality in both static and dynamic scenes compared with baseline methods. Additional quantitative benchmark comparisons are provided in Appendix A.
Efficiency Analysis. Beyond reconstruction quality, UBS demonstrates superior computational efficiency through its streamlined primitive design and optimized CUDA implementation. Table summarizes the efficiency comparison across all benchmarks, with detailed per-scene results provided in Table in the Appendix B. For static scenes, UBS-6D achieves significantly faster training than 6DGS on NeRF Synthetic (8.7 vs 12.7 minutes. 31 15% reduction) and Mip-NeRF 360 (25.2 81.9 minutes, 69.2% reduction) with competitive memory usage. On 6DGS-PBR, UBS-6D requires longer training time (27.7 vs 22.2 minutes) due to our use of higher primitive counts (300K vs 68K) to fully exploit Beta kernels' representational capacity for complex volumetric effects. For dynamic scenes, UBS-7D delivers substantial training time reductions 48.7% on 7DGS JDGS-PBR (59.9 vs 116.7 minutes) and 17.0% on D-NeRF (42.1 50.7 minutes), both compared to 7DGS. These efficiency gains stem from our unified Beta kernel approach, which eliminates auxiliary spherical harmonic encodings while enabling efficient CUDA-accelerated conditional slicing operations.

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Published as a conference paper at ICLR 2026
Table 2: Efficiency comparison static and dynamic benchmarks (averaged across scenes). 'Train' indicates training time in minutes; 'Mem' indicates storage size in MB. Detailed per-scene results and per per-primitive parameter comparisons are provided in Appendix B.
3DGS/4DGS
6DGS/7DGS
UBS (Ours)
Dataset
FPSt Num) Mem Train.
FPSt
Num Mem!
Train]
FPSt
Num, Mem.
Train
NeRF Synthetic
695.6 289.3K
68.5
3.6
648.0
240.7K
69.8
12.7
345.7
300.0K
40.1
8.7
Mip-NeRF 360
178.7 3.3M
775.9
21.7
157.4
2.0M
581.2
81.9
100.4
3.1M
409.0
25.2
6DGS-PBR
305.9 147.7K
34.9
16.8
339.8
84,3K
24.4
22.2
290.2
300.0K
40.1
27.7
7DGS-PBR
192.6 642.0K
613.1
127.2
376.0
88.4K
29.7
116.7
359.6
234.3K
39.4
59.9
D-NeRF
296.4 255.3K
380.3
52.6
377.8
43.8K
16.0
50.7
368.2
168.8K
28.4
42.1
Table 3: Ablation study of UBS across representative static and dynamic scenes.
Static Scenes
Dynamic Scenes
Ficus
bonsai
cloud
lego
Mean
PSNRT
Train]
FPS)
PSNR
Train,
FPS
PSNRT
Train]
FPS
PSNR1
Train
FPS1
|PSNR
Train],
FPS?
skew-sym. Cholesky
36.34
207.2
34.24
142.5
241.9
31.87
29.5
Cholesky who spatial ortha.
35.76
22.8
119.2
34.21
75.0
40.4
28.41
16.3
162.2
28.02
15.0
175.4
31.60
32.1
124.1
w/o condition
217.2
53.9
31.63
22.0
wh Beta opacity
33.76
17.9
213.0
33.17
62.1
57.3
21.93
11.3
260.3
25.68
11.9
209.5
28.64
25.8
185.0
full
188.4
34.62
48.9
30.7
full w/ CUDA
36.96
9.3
262.4
34.65
24.1
78.5
28.58
10.2
184.8
28.55
7.8
259.1
32.19
12.9
196.2
5.3 ABLATION STUDY
We conduct ablations on representative samples static scenes in 6DGS-PBR (Gao et al., 2025a) dataset (ficus, bonsai) and of dynamic scenes in 7DGS-PBB (Gao al., 2025b) dataset (cloud, lego) to validate our design choices and efficient CUDA implementation (Table 3).
Oothhoo-Orthogona Parameterization. Pure rotation- scale parameterization and full Cholesky without spatial orthogonality reduce quality by 0.29 dB and 0.56 dB, respectively, confirming the importance of preserving the spatial SO(3) structure and maintaining flexibility of other dimensions. Beta-Modulated Conditioning, Removing Beta modulation decreases performance by 0.53 dB, with larger drops on dynamic scenes (lego: 1.25 dB), validating iintension-spccfffe shape control for responses. Beta-Modulated Opacity. Disabling Beta-modulated opacity causes the most significant degradation (-3.52 dB average), particularly on volumetric scenes (cloud: -..68 dB), proving the gated product form opacity crucial for anisotropic falloff. CUDA Acceleration. Custom CUDA kernels reduce training time by 58% (12.9 vs 30.7 minutes) and improve rendering speed by 26% (196.2 vs 155.8 FPS), enhancing real-time performance through fused Beta evaluation and conditional slicing operations.
decoupling
These studies confirm each component's 's contribution to UBS's performance, with the complete model showing consistent gains across diverse scenarios while maintaining computational efficiency.
6 CONCLUSION
We presented Universal Beta Splatting (UBS), a unified framework extending radiance field rendering to N --dimensional anisotropic Beta kernels. By replacing fixed Gaussian profiles with adaptive per-dimension shape control, UBS simultaneously models spatial geometry, view-dependent appear- ance, and temporal dynamics within single primitive. Our spatial-orthogonal parameterization and meda-modultted conditional slicing enable independent dimensional control while preserving geometric consistency. Experiments demonstrate significant improvements up +8.27 dB PSNR on volumetric scenes and 48.7% faster training on dynamic sequences while maintaining backward compatibility by approximating Gaussian methods. The learned Beta parameters provide interpretable scene decomposition without explicit supervision, and our CUDA implementation achieves real-time rendering, establishing Beta kernels practical universal primitives for radiance fields.
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Published as a conference paper at ICLR 2026
ACKNOWLEDGMENTS
This work was supported in part by the U.S. Army Combat Capabilities Development Command under contract numbers W91 INF-14---000 and 1-22C-24-D-0001. The authors thank Mr. Clayton Burford of the Battlespace Content Creation (BCC) team at Simulation and Training Technology Center (STTC). The content of the information does not necessarily reflect the position or the policy of the Government, and no official endorsement should be inferred.
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Universal Beta Splatting Supplementary Material
A MORE COMPARISONS
A.1 COMPARISON WITH STATE-OF-THE-ART METHODS
Table 4 presents comprehensive comparison of UBS-6D with state-of-the-art neural rendering methods on the Mip-NeRF360 (Barron et al., 2022) and NeRF Synthetic (Mildenhall et al., 2020) benchmarks. Our method achieves the highest PSNR on both datasets, demonstrating the effectiveness of adaptive Beta kernels over fixed Gaussian profiles. The consistent improvements across both synthetic and real-world scenes validate the universal applicability of Beta kernel framework.
Table 4: Comparisonh with SOTA methods on Mip-NeRF360 (Barron et al., 2022) and NeRF Synthetic (Mildenhall et al., 2020).
Mip-NeRF360
NeRF Synthetic
Methods
PSNR
SSIM
LPIPS!
PSNR1
SSIM
LPIPS
implicit
Instant-NGP (Müller et al., 2022)
25,51
0.684
0.398
33.18
0.959
0.055
Mip-NeRF360 (Barron et al., 2022)
27.69
0.792
0.237
33.25
0.962
0,039
Zip-NeRF (Barron et al., 2023)
28.54
0.828
0.189
33.10
0.971
0,031
3DGS (Kerbl al., 2023)
27.20
0.815
0.214
33.31
0.969
0.037
GES (Hamdi al., 2024)
26.91
0.794
0.250
2DGS (Huang et al., 2024)
27.04
0.805
0.297
33.07
Mip-Splatting (Yu et al., 2024)
27.79
0.827
0.203
33.33
0.969
0.039
explicit
3DGS-MCMC (Kheradmand et al., 2024)
28.29
0.840
0.210
33.80
0.970
0.040
DRKS (Huang al., 2025)
26.76
0.787
0.236
33.82
Textured GS (Chao al., 2025)
27.35
0.827
0.186
33.24
0.967
0.043
Quadratic GS (Zhang al., 2025)
27.39
0.813
0.213
Dise-GS (Qu al., 2024)
28.01
0.833
0.189
DBS (Liu et al., 2025a)
28.60
0.844
0.182
34.64
0.973
0.028
UBS-6D (Ours)
28.66
0.840
0.184
34.92
0.975
0.026
Table reports quantitative comparison of UBS-7D against state-of-the-art kernel-based methods on the real-world N3V dataset (Li et al., 2022). UBS-7D achieves the best overall average PSNR of 32.22 dB, outperforming Ex4DGS (32.11 dB) and 7DGS (31.59 dB), while winning in 4 out of 6 scenes. This demonstrates the unified Beta primitive's effectiveness in capturing complex temporal dynamics and appearance variations in real-world dynamic content. Earlier methods 4DGS (30.19 dB) and 4DGaussians (28.63 dB) show significantly larger performance gaps, validating the advantage of our N-D anisotropic kernel design for challenging 4D reconstruction tasks.
Table 5: Comparison with other kernel methods on the real-world N3V dataset (Li et al., 2022).
Cut Roasted Beef
Coffee Martini
Cook Spinach
Flame Salmon
Flame Steak
Sear Steak
Model
Average
4DGS (Yang et al., 2024a)
26.51
32.11
31.74
26.93
31.44
32.42
30.19
4DGaussians (Wu al. 2024)
26.69
31.89
25.88
27.54
28.07
31.73
28.63
Ex4DGS (Lee et al., 2024b)
28.79
33.23
33.73
29.29
33.91
33.69
32.11
7DGS (Gan et al., 2025b)
29.07
32.23
32.93
28.83
33.07
33.41
31.59
UBS-7D (Ours)
29.73
33.30
33.88
28.77
33.83
33.83
32.22
A.2 COMPARISONS ON THE 6DGS- PBR DATASET
Table compares UBS-6D (Ours) with 3DGS (Kerbl et al., 2023), DBS (Liu et al., 2025a), and 6DGS (Gao et al., 2025a) the 6DGS-PBR dataset. UBS-6D consistently achieves the best performance across all seven scenes, surpassing all baselines in reconstruction accuracy and perceptual quality. On average, UBS-6D achieves 40.10 dB PSNR, 0.979 SSIM, and 0.039 IPIPS, outperforming
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Table 6: Results on the 6DGS-PBR (Gao et al., 2025a) dataset. 'Train' indicates time minutes.
3DGS
DBS
6DGS
UBS-6D (Ours)
Scene
PSNRT
SSIMT
Train]
PSNRT
SSIM
Train
PSNR1
SSIMf
Train
PSNR
SSIM?
Train]
buany
29.19
0.985
22.8
30. 10
0.982
23.5
37.41
0.991
24.1
45.68
0.996
32.1
cloud
34.03
0.985
23.1
43.88
0.993
12.7
0.991
0.995
explosion
27.06
0.953
11.3
32.42
0.965
17.9
41.61
0.990
18.3
44.54
0.994
32.4
smoke
suzarne
22.45
0.885
19.5
26.58
0.923
30.3
26.96
0.928
32.9
27.35
0.931
35.1
dragon
26.53
12.0
29.71
et-scan
25.69
0.917
4.5
27.52
0.923
4.3
33.46
0.964
11.0
34.24
0.969
6.5
avg
27.40
0.929
16.8
31,29
0.942
22.41
36.68
0.970
22.2
40.10
0.979
27.7
Table 7: Per-primitive parameter count comparison across static and dynamic splatting methods. UBS significantly reduces parameter counts compared to existing NDGS methods: UBS-6D is 59% of 3DGS and UBS-7D is 27% of 4DGS.
Static Solutions
Dynamic Solutions
Comparison
3DGS
6DGS
UBS-6D (Ours)
4DGS
7DGS
UBS-7D (Ours)
(Geom. Color)
Parameter
11
48
59
28
48 76
32 + 35
17
144
161
36 4 48
84
41 3
Ratio
1x
1.29x
0.59x
1x
0.52x
0.27x
6DGS by +3,4 dB PSNR and DBS by +8.8 dB, with comparable training efficiency (27.7 min vs. 22.2/22.4 min). The clear gap between 6DGS/UBS-6D and 3DGS/DBS arises from the presence of conditional slicing, which allows the mean and covariance to adapt with viewing conditions. In contrast, 3DGS and DBS rely on static spatial kernels, limiting their ability to capture these view- dependent effects. By combining high-dimensional Beta kernels with conditional slicing, UBS-6D attains complex visual effects and photorealism while maintaining competitive runtime.
B EFFICIENCY ANALYSIS
B. PER-PRIMITIVE PARAMETER ANALYSIS
Table 7 provides detailed breakdown of parameter counts per primitive across different splatting methods. UBS achieves significant parameter efficiency by eliminating the need for auxiliary color encodings through its native high-dimensional representation.
For static scenes, 3DGS requires 59 parameters per primitive: 11 for geometry (pos: 3, scale: 3, rotations 4, opacity: 1) and 48 for spherical harmonic color encoding (16 coefficients x RGB channels). 6DGS increases this 76 parameters by expanding the covariance representation to 28 parameters while maintaining the same 48-parameter SH encoding. In contrast, UBS-6D requires only 35 parameters: 32 for N-dimensional geometry (pos: dí r: spatia i -ort hogonal covariance: 21, beta shapes: 4, opacity: 1) and just for direct RGB color, achieving 41% parameter reduction compared 3DGS.
For dynamic scenes, the parameter efficiency becomes even more pronounced. 4DGS requires 161 parameters per primitive: 17 for 4D geometry (pos: 3, time: 1, 4D scale: 4D rotation: opacity: 1) and 144 for temporal spherical harmonics (48 coefficients x RGB channels). 7DGS reduces this 84 parameters: 36 for geometry (pos: 3, time: dir: 7D covariance: 28, opacity: 1) and 48 for standard SH encoding. UBS-7D achieves the most compact rep- resentation with only 44 parameters: 41 for unified geometry (pos: 3, t time: 1, dir: r: apatial-orthogome
covariance: 28, beta shapes: 5, opacity: and 3 for direct RGB color, representing 73% parameter reduction compared to 4DGS.
This dramatic parameter efficiency stems from UBS's ability to natively encode view-dependent and temporal variations within the kernel itself, eliminating the need for separate spherical harmonic or temporal color encodings. The Beta kernel's inherent anisotropy across dimensions captures complex appearance variations that would otherwise require dozens of harmonic coefficients, making UBS both more expressive and more parameter-efficiet than existing methods.
15

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Published as a conference paper at ICLR 2026
Table 8: Efficiency comparison with 3DGS and 6DGS static benchmarks of NeRF Synthetic (Mildenhall al., 2020), Mip-NeRI (Barron et al., 2022), 6DGS-PBR (Gao al., 2025a), and with 4DGS 7DGS on dynamic benchmarks of 7DGS-PBR (Gao et al., 2025b) and D-NeRF (Pumarola et al., 2021). 'Truin' means training time in minutes; 'Mem' means storage size MB.
3DGS
6DGS
UBS-6D (Ours)
Dataset
Scene
FPSt
Num,
Mem,
Train
FPSt
Num
Mem
Train.
FPSt
Num. Mem. Train.
chair
412.8 489.0K
115.7
4.8
420.4 410.7K
119.1
19.6
356.3
300.0K
40.1
8.7
drums
562.3 390.9K
92.5
4.1
555.8 317.2K
92.0
349.1
300.0K
40.1
8.1
NeRF Synthetic
ficus
728.5 264.9K
62.7
3.0
757.4 226.4K
65.6
11.3
356.6
300.0K
40.1
8.2
hotdog
831.5 187,8K
44.4
3.5
700.5 161.0K
46.7
10.0
352.5
300.0K
40.1
9.1
Lego
582.4 347.0K
82.1
4.0
247.7 273.3K
79.2
15.3
340.3
300.0K
40.1
8.4
ma materials
1286.1 160.1K
37.9
2.7
1098.2 125.0K
36.2
7.8
360.7
300.0K
40.1
8.9
mia
614.9 195.7K
46.3
3.0
756.9
174.3K
50.5
9.4
326.1
300.0K
9.5
ship
546.4 279.2K
66.0
4.1
646.8
237.9K
69.0
12.7
330.3
300.0K
40.1
8.8
STATIC SCENES
avg
695.6
289.3K
68.5
3.6
648.0
240.7K
69.8
12.7
345.7
300.0K
40.1
8.7
bicysle
105.2
6.0M
1421.5
29.2
105.2
3.3M
957.2
114.9
43.9
6.0M
801.1
30.2
flowers
150.9
3.6M
850.0
19.6
150.9
2.4M
707.2
88.8
84.3
3.0M
400.5
19.6
Mip-NeRF: 360
garden
84.2
5.7M
1358.3
30.4
84,2
4.2M
1219.4
164.9
52.9
5.0M
667.6
36.8
stump
151.2
4.6M
1095.1
22.7
190.5
1.7M
488.2
60.2
64.9
4.0M
536.3
18.9
treehill
161.6
3.8M
886.5
20.1
130.9
2.6M
755.8
94.0
72.4
3.5M
467.3
21.9
room
222.0
369.1
1.IM
314.8
147.2
1.5M
200.3
233.5
1.2M
281.4
17.6
203.7
0.8M
239.8
43.5
143.8
1.5M
200.3
27.8
kitchen
183,0
1.8M
424.7
21.5
147.4
1.0M
285.9
71.3
144.8
1.5M
200.3
26.6
bonsai
317.1
1.3M
296.6
15.7
233.4
0.9M
262.7
48.4
147.1
1.5M
200.3
22.7
avg
178.7
3.3M
775.9
21.7
157.4
2.0M
581.2
81.9
100.4
3.IM
409.0
25.2
bunny
123.8
79.8K
18.87
22.8
349.5
9.7K
2.8
24.1
257.9
300.0K
40.1
32.1
6DGS-PBR P
cloud
370.5
60.0K
14.19
23.1
429.8
17.9K
5.2
21.6
247.7
300.0K
40.1
30.6
explosion
400.3
51.2K
12.11
11.3
302.8
25.9K
7.5
18.3
275.6
300.0K
40.1
32.4
176.6
55.8K
13.20
24.2
207.7
21.0K
6.1
31.1
225.0
300.0K
40.1
33.5
suzanne
186.5 316.2K
74.78
19.5
94.9
208.3K
60.4
32.9
325.9
300.0K
40.1
35.1
dragon
275.9 241.4K
57.09
12.0
266.9
127.8K
37.1
16.3
334.9
300.0K
40.1
23.7
ct-scan
607.2 229.2K
54.20
4.5
726.8
179.3K
52.0
11.0
370.2
300.0K
40.1
6.5
avg
305.9
147.7K
34.9
16.8
339.8
84.3K
24.4
22.2
290.2
300.0K
40.1
27.7
4DGS
7DGS
UBS-7D (Ours)
Dataset
Scene
FPSt
Num,
Mem.
Train.
FPS1
Num.
Mem. Train
FPSt
Num.. Mem. Train.
heart1
186.9 694.0K
444.6
103.0
401.0
82.4K
27.7
114.2
244.6
328.0K
55.1
38.7
7DGS- PBS-
heart2
160.4 869.2K
556.9
103.4
384.6
101.5K
384.6
297.1
328.0K
55.1
cloud
219.0 216.9K
138.9
123.7
386.2
44.2K
14.8
102.6
473.6
150.0K
25.2
50.0
DYNAMIC SCENES
dust
296.1 357.7K
229.2
97.0
394.8
10.9K
3.7
69.8
406.0
150.0K
25.2
46.5
Lame
151.2 947.8K
607.2
113.7
371.6
15.1K
5.1
74.1
461.4
150.0K
25.2
53.7
suzanne
141.8 766.1K
1,701.6
222.5
317.9
276.3K
92.8
193.9
275.0
300.0K
50.4
136.8
avg
192.6 642.0K
613.1
127.2
376.0
88.4K
29.7
116.7
359.6
234.3K
39.4
59.9
b, b.balls
219.9 276.
277.7
50.7
213.1
127.4K
43.5
86.6
366.6
150.0K
25.2
49.6
I.. h.warrior
299.4 298.4K
212.6
35.3
431.5
8.7K
2.9
31.3
395.2
150.0K
25.2
36.3
D-NoRF
book
325.1 174.7K
183.3
432.8
21.7K
7.3
35.7
452.2 150.0K
25.2
34.0
j.jacks
366.0 143.7K
534.8
66.9
432.8
21.7K
5.3
34.1
376.4 150.0K
25.2
43.2
Tego
320.0 186.2K
355.0
55.4
365.0
68.6K
31.4
78.5
347.4 150.0K
25.2
31.3
mutant
341.6 138.7K
88.9
39.1
395.8
33.9K
11.5
42.5
258.7
300.KK
50.4
60.0
standup
330.4 142.5K
107.1
34.4
399.2
12.7K
4.3
33.5
389.4 4 150.0K
25.2
41.1
trex
169.2 682.4K
1,085.2
100.7
352.4
62.0K
21.4
63.4
359.4
150.0K
25.2
41.6
avg
296.4 255.3K 380.3
52.6
377.8
43.8K
16.0
50.7
368.2
168.8K
28.4
42.1
Table 9: Impact of primitive capacity on NeRF Synthetic dataset. 'Mem" means storage size in MB.
100K Primitives
200K Primitives
300K Primitives
Scene
PSNRT
SSIM
Mem]
FPSt
PSNRT
SSIM
Mem,
FPS1
PSNR
SSIM
Mem.
FPSt
chair
36.16
0.989
13.4
543.4
36.62
0.990
26.7
341.3
36.62
0.990
40.1
356.3
drums
26.88
0.959
13.4
529.3
27.01
0.959
26.7
376.3
27.03
0.959
40.1
349.
ficus
36.17
0.988
13.4
574.4
36.44
0.989
26.7
431.9
36.57
0.989
40.1
356.6
hotddg
38.68
0,988
13.4
534.7
38.81
0.988
26.7
366.9
38.94
0.988
40.1
352.5
lego
35.66
0.981
13.4
572.1
36.67
0.984
26.7
370.7
36.97
0.985
40.1
340.3
matorials
32.49
0,974
13.4
531.7
32.85
0.976
26.7
367.7
32.93
0.977
40.1
360.7
mic
37.48
0.994
13.4
493.1
37.79
0.994
26.7
346.7
37.92
0.994
40.1
326.1
ship
31.88
13.4
32.10
26.7
344.8
32.09
330.3
avq
34.42
0.973
13.4
540.3
34.79
0.974
26.7
368.3
34,89
0.974
40.1
345.7
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Published as a conference paper at ICLR 2026
Table 10: Comparison of rotation parameterizations for covariance matrix construction the bicycle scene for UBS-3D. Our spatial -othiiont approach balances performance, efficiency, and parameter count.
Covariance Matrix Parameterization
Training Time
PSNR
# Parameters (N>3)
Parameters (N=3)
Quaternion (w/ CUDA)
3h26m
25.50
Not Applicable
7
Cholesky
1h14m
25.28
N(I+NY2
6
Skew-Symmetric w/ Matrix Exp
3h6m
25.49
N(1+N)/2
6
Skew-Symmetric w/ Cayley Transform
1h16m
25.52
N(1+Ny/2
6
Skew- "kew-Symmetric w/ 2nd order Taylor
lh27m
25.54
N(1+NV2
Syw-memmetiic w/ 1st order Taylor
1h
25.54
N(L+Ny2
B.2 COMPUTATIONAL EFFICIENCY
Table 8 presents a comprehensive efficiency comparison between UBS and baseline methods. For static scenes, UBS-6D achieves 31% % faster training than 6DGS NeRF Synthetic (8.7 7 vs 12.7 minutes) with 43% memory reduction (40.1 vs 69.8 MB) and 345.7 FPS rendering. On Mip-NeRF 360, UBS achieves the lowest memory usage (409.0 MB, 47% and 30% reduction vs 3DGS and 6DGS) with 100.4 FPS. On 6DGS-PBR, UBS achieves comparable training times (27.7 vs 22.2 minutes for 6DGS) with 290.2 FPS. For dynamic scenes, UBS-7D reduces training time by 53% and 49% compared to 4DGS and 7DGS on 7DGS-PBR (59.9 vs 127.2 and 1 116.7 minutes) with 359.6 FPS, and by 20% and 17% on D-NeRF (42.1 52.6 and 50.7 minutes) with 368.2 FPS. The efficiency gains stem from UBS's unified Beta kernel design, which eliminates auxiliary spherical harmonic encodings. Moreover, unlike previous Gaussian methods that rely on heuristic densification and pruning, UBS employs MCMC-based optimization that allows direct control over primitive count, enabling flexible speed-quality trade-offs as demonstrated in Table 9.
Table 9 demonstrates this flexibility by analyzing the speed-quality trade-off across varying primitive capacities (100K, 200K, 300K) on NeRF Synthetic. Results reveal that reducing primitives from 300K to 100K decreases quality by only 0.47 dB PSNR (34.89 to 34.42 dB) while significantly improving rendering speed by 56% (345.7 to 540.3 FPS) and reducing memory by 67% (40.1 1 to 13.4 MB). This demonstrates that UBS can be optimized for speed-critical applications by adaptively adjusting primitive counts to balance quality and efficiency.
C SPATIAL-ORTHOGONAL CHOLESKY PARAMETERIZATION
Our vatiial-rttogonal Cholesky parameterization (Equation in the main paper) addresses a fun- damental challenge in N-dimensional radiance field rendering: how to parameterize covariance matrices that simultaneously preserve interpretable 3D spatial structure while enabling flexible cross- dimensional correlations. This section provides theoretical justification for our design and empirical validation of our rotation parameterization choice.
C.I THEORETICAL ANALYSIS
Design Requirements. For the covariance matrix Σ E NxN in N-dimensional primitives, we require a parameterization that: (1) guarantees positive semi-definiteness (PSD), (2) preserves 3D spatial interpretability (rotation + scale structure), (3) enables cross-dimensional correlations with non-spatial dimensions (view, time), (4) generalizes to arbitrary N dimensions, and (5) computes efficiently during optimization.
Existing approaches fall short of these requirements. Quaternion rotation parameterization (used in 3DGS) provides excellent spatial interpretability but does not generalize beyond 3D. Pure Cholesky factorization generalizes to arbitrary dimensions but loses explicit spatial rotation structure, leading to suboptimal geometric representation (Table 10 shows 0.22 dB PSNR drop). Naive matrix exponential approaches preserve orthogonality and generalize to N-D but incur prohibitive computational costs (3h+ training time).
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Published as a conference paper at ICLR 2026
Construction. We construct the covariance through hybrid Cholesky factorization Σ LL with structured lower- triantrra matrix:
R.,diag(ss) Laz
-=
(14)
L
where Rs E (O(33 is the spatial rotation matrix, 8, IR are spatial scales, 8, E RCC are non-spatial scales for C' additional dimensions, and Lyz IR yields:
Cx3 encodes cross- vross-dimensional correlations. This
diad (8.) )² R ding 8,) R
(ia Lgz L a Lg
Σ
(15)
Skew-Symmetrie Rotation Parameterization. employ skew-symmetric matrix parameterization. A skew-symmetric matrix A₂ E RR3x3 satisfies A, -A J which in 3D has the explicit form:
For the spatial rotation matrix R. E SO(3),
W/
W2
A
W3
W1
(w wa) E RR
(16)
W2
W1
where w represents the learnable rotation parameters. The rotation matrix is obtained via the matrix exponential map:
A
A
RS =xp((A
+ A
(17)
h k=
2!
31
which mathematically guarantees orthogonality (R. R = D) and unit determinant (det = 1), ensuring R2 SO(3). For computational efficiency, use the first-order Taylor approximation:
R2 R I Ag:
(18)
which maintains approximate orthogonality for small ||ار|| while avoiding the expensive matrix expo- nential computation. This approximation initializes with 0 (identity rotation) and progressively learns rotation during optimization.
Mathematical Properties. Our parameterization satisfies all design requirements: (1) PSD: Guar- anteed by Cholesky properties (E = (2) Spatial Interpretability: The spatial block Σ, = diaggg ( R exactly recovers 3DGS parameterization. (3) Cross-Correlation:: Off- diagonal block Eng dia (s., L enables view-dependent spatial deformation. (4) Gen- eralization: Holds for arbitrary N = + C dimensions. (5) Efficiency: Spatial rotation R₂
uses meemttyy first-order Taylor approximation R2 I Az, avoiding expensive matrix exponential while maintaining approximate orthogonality.
C.2 EMPIRICAL ANALYSIS
Having established the structure, we now validate our choice spatial rotation parameterization through systematic comparison. We evaluate six strategies on the bi cycle scene from Mip-NeRF 360 using UBS-3D (pure spatial Beta splatting), focusing on the 3D spatial rotation subspace.
Table 10 presents our findings across different parameterization strategies:
Quaternion. The traditional 3DGS approach using quaternion rotation and diagonal scaling achieves competitive PSNR (25.50 dB) and the lowest parameter count (7 parameters: 3 position + scale quaternion magnitude after normalization). However, this approach does not generalize to higher dimensions (N > 3), making it unsuitable for our N Y-dimensional framework.
Cholesky Decomposition. Pure Cholesky factorization generalizes to arbitrary dimensions with N(N+1)/2 parameters. However, it suffers from 0.22 dB PSNR drop compared to quaternions, as it lacks explicit orthogonality constraints that help maintain geometric consistency in the spatial subspace.
LL offers faster training (lh14m) and
mymm-Syymettii Matrix Methods. Constructing spatial rotation matrices through R = exp(A) where A is skew-symmetric preserves orthogonality and generalizes to higher dimensions. The naive
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Published as a conference paper at ICLR 2026
implementation requiring full matrix exponentiation is prohibitively slow (3h6m), but approximation
methods offer significant improvements. The Cayley transform R (I - A) I + A) reduces training to 1hl6m with 25.52 dB PSNR. Taylor approximations provide the best balance: the 1st order Taylor approximation R = I + A achieves both the fastest training time (1h) and the highest PSNR (25.54 dB), making it optimal for spatial rotation parameterization.
Spatial-Orthogonal Cholesky. Based on these results, we select the skew-symmetric approach with 1st order Taylor approximation for spatial rotation parameterization, as it provides the best combi- nation of training speed (Ih) and reconstruction quality (25.54 dB). For our full UBS framework, we adopt hybrid
approach: semmmeymttiiic 1st order Taylor for the 3D spatial rotation subspace (preserving geometric interpretability and optimal performance) combined with Cholesky factors for modeling cross- coos-dimensional correlations between spatial and non-spatial dimen- sions (view, time). This design achieves the optimal trade-off between performance, generalizability. and computational efficiency across all UBS variants.
The analysis confirms that the skew-symmetric 1st order Taylor approximation provides the best spatial rotation parameterization, enabling our spatial-orthogonal design to maintain geometric consistency in the spatial subspace while allowing flexible correlations with additional dimensions in the full N-dimensional UBS framework.
D THEORETICAL FOUNDATION AND ANALYSIS
Fundamental L Limitations of Gaussian Kernels N-dimensional Gaussian kernels suffer from inherent dimensional coupling that prevents modeling of independent scene properties. In methods like 6DGS and 7DGS, all dimensions input view direction or time changes, the primitive is inevitably affected across all dimensions shifts in space, changes shape, and modulates opacity simultaneously. This coupling stems from two fundamental constraints the Gaussian kernel:
coupled through the joint Gaussian distribution: when the
1. Fixed Bell-Shaped Profile:
The Gaussian kernel maintains its characteristic bell-shaped expo- nential decay (000-r2/200 across all dimensions. This fixed functional form cannot produce flat responses needed for view-independent or temporally-static elements. For example, in 6DGS, primitive cannot remain view- new-independeet because the Gaussian's 's bell-shaped profile in the angular dimensions will always modulate opacity based on viewing direction, even when the scene element should appear constant across views.
The multivariate Gaussian distribution enforces symmetric relationships between dimensions through its covariance structure. When conditioned on viewing direction or time, the resulting 3D Gaussian inevitably shifts its spatial mean, changes its spatial covariance, and modulates its opacity according to the angular or temporal distance from the mean. This prevents primitives from exhibiting independence- fixed while only its angular response varies, static object cannot maintain constant appearance while dynamic elements change around it.
2. Symmetric Dimensional Coupling:
ppttiall--loallize texture cannot remain
These limitations force inefficient representations where many coupled primitives must approximate what should be expressible by fewer independent primitives.
In contrast, UBS's Beta kernels enable per-dimension shape control: spatial dimensions adopt flat profiles for view-independent geometry while angular dimensions use peaked responses for specular highlights, and temporal dimensions remain nearly constant for static elements while varying sharply for dynamic components. This dimensional independence allows each primitive to model complex anisotropic phenomena efficiently-a single UBS primitive can simultaneously represent static spatial surface with view-dependent reflectance, whereas Gaussian methods require multiple coupled primitives that cannot cleanly separate these properties.
Universal Beta Kernels: Mathematical Properties. N V-dimensional Beta kemne::
UBS addresses these limitations through the
N
B(x; μ, E, = II (1 ')) d, E 10,
(19)
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Published as a conference paper at ICLR 2026
Gaussian kemel: exp(-4.5d) Beta kernel: (1 d) mcplo)
b
Figure 4: Adaptive Beta kernels defined on E [0, 1). The curves are extended symmetrically to (-1, 1) for illustrative visualization only.
where (<(=)) represents the bounded distance. For the first three dimensions (i = 2, 3), de is defined by r; in Equation 1, while for the remaining dimensions (i 4,..., The shape parameter B, = 4 exp(6;) controls the kernel profile.
......,,,,, N), d; is defined by Equation 11.
The Beta kernel exhibits three key properties:
Property (Shape Adaptivity): The shape parameter by directly controls kernel behavior through 3; exp((::):
by < O: Flatter kernel profile (3, < 4, broader support)
b 0: Gaussian-like profile (8, 4, standard bell curve)
b, 0: More peaked kernel profile (3: 4, sharper localization)
When 6; = 5, we have 3; 4e-5 0.027 and the kernel becomes nearly constant (1-d,)0.022 ss 1. When b; 5, 3r 4e As 594 and the kernel becomes highly peaked. These practical ranges
(b, E [-5, 5)) are sufficient to span from flat to sharp kernel profiles, shown in Figure 4.
Property (Per-Dimension Shape Control): The gated product form enables independent shape control for each dimension through its own by:
( II (1 di eexp(b:)
(20)
Unlike Gaussian kernels where all dimensions share the same bell-shaped functional form (only varying in scale), Beta kernels allow each dimension to adopt fundamentally different shapes: dimension can be nearly flat (b, -3) while dimension j is highly peaked (b, 3). This per- dimension shape control, combined with cross- crossddimensional correlations through E, enables efficient representation of complex anisotropic phenomena.
Property (Backward Compatibility); When b, = 0, we have Br = 40xp(0) 4, and the Beta kernel approximates Gaussian:
(1 d)
for d [0,
(21)
guarantecing that UBS subsumes Gaussian methods as special cases when = 0. This ensures seamless compatibility with existing Caussian-baser methods.
20


Reviews:
[{"review_id": "olZNGj902b", "reviewer": "Reviewer_5UF1", "summary": "This paper proposes Universal Beta Splatting (UBS), a unified framework that generalizes 3D Gaussian Splatting to N-dimensional anisotropic Beta kernels for explicit radiance field rendering. Unlike fixed Gaussian primitives, UBS models spatial, angular, and temporal dependencies within a single representation, enabling controllable anisotropy and dynamic scene rendering without auxiliary networks. The method remains backward-compatible with 3DGS, ensuring usability and performance stability. Experiments demonstrate real-time rendering and consistent quality improvements across static, view-dependent, and dynamic scenes, establishing Beta kernels as a versatile and interpretable primitive for radiance field modeling.", "questions": "See Weaknesses", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "This paper presents a technically solid and conceptually original generalization of Gaussian Splatting via N-dimensional anisotropic Beta kernels, enabling unified modeling of spatial, angular, and temporal properties. The proposed spatial-orthogonal Cholesky parameterization and Beta-modulated conditional slicing are elegant design choices that balance flexibility and efficiency. The paper demonstrates clear motivation from prior limitations (e.g., 3DGS, 6DGS, DBS) and provides strong empirical validation with consistent improvements across benchmarks. The backward compatibility with existing Gaussian-based methods is particularly noteworthy, as it enhances practical impact and ease of adoption. Overall, the work is novel, well-executed, and clearly presented, offering both theoretical insight and real-world significance for radiance field rendering.", "weaknesses": "While the paper is technically sophisticated, a few aspects could be strengthened. First, the novelty relative to prior high-dimensional splatting methods (e.g., 6DGS, DBS) is somewhat incremental — UBS’s core Beta kernel formulation mainly generalizes existing ideas with additional per-dimension flexibility. A deeper theoretical justification of why Beta kernels are fundamentally better suited for radiance modeling than Gaussians would improve clarity. Second, the empirical evaluation could be more comprehensive: ablations isolating the effects of Beta modulation, conditional slicing, and spatial-orthogonal Cholesky parameterization are limited. Additionally, it would be helpful to discuss training stability and computational overhead, since the Beta formulation introduces additional parameters and nonlinearities. Finally, while interpretability claims are appealing, the qualitative decomposition results (e.g., spatial vs. angular vs. temporal) would benefit from more rigorous quantitative validation.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "9GF4REXJIp", "reviewer": "Reviewer_4uqJ", "summary": "The paper integrates the \"Deformable Beta Splatting\" (DBS) method with existing works, \"6DGS & 7DGS.\" It adopts the beta kernel function, (1 - distance_to_the_center)^(e^b), proposed in DBS, to replace the exponential decay term in the Gaussian function. This achieves similar effects to GES, which uses \\beta to control the decay rate and kernel shape through \\exp(-distance_to_the_center^\\beta). Experimental results demonstrate that the proposed combined method delivers good performance. However, the paper lacks a comparison of Gaussian quantities and model sizes. Additionally, the evaluation of dynamic reconstruction omits several state-of-the-art methods.", "questions": "1. It is essential to specify the number and size of Gaussians used in each table to ensure a fair and transparent comparison.  \n\n2. Comparisons or discussions with state-of-the-art primitive-enhanced GS methods and dynamic GS methods should be included. While achieving worse results is acceptable, providing a discussion would help the community better understand the reasons from a novel perspective.  \n\n3. The paper should introduce more challenges associated with the existing deformable beta splatting and ND-GS methods. Currently, the methodology contribution is insufficiently detailed, and the experimental section is also lacking in depth.", "limitations": "", "rating": 4, "confidence": 5, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The combination of beta splatting with ND-GS is worth exploring; however, its novelty is questionable. I would appreciate a rebuttal from the authors to clarify this aspect.\n\n2. Overall, the paper is well-structured and easy to follow, with clear and precise equations and method descriptions.\n\n3. The demonstrated advantage over 4DGS is satisfactory, highlighting the method's practical significance compared to existing solutions.", "weaknesses": "1. The concept of beta control on the truncated unit circle region (x < 1) was originally introduced in the \"Deformable Beta Splatting\" paper. ND-GS, 6D-GS, and 7D-GS are established methods with their own innovations in representation. This paper simply applies beta control to 6D- and 7D-GS to achieve improved results, building on the strengths of these prior methods rather than introducing significant novelty.  \n\n2. The number of Gaussians is a critical factor influencing the performance of GS-based methods, yet this aspect is omitted in the paper. Furthermore, file size, an equally important metric, is not demonstrated, leaving a gap in the evaluation of model efficiency.\n\n3. The dynamic reconstruction section fails to include many state-of-the-art methods, limiting the practical relevance of the proposed combination method. Notably, 4DGS is not the leading approach in this domain. Advanced methods such as Ex4DGS, SC-GS, and Deformable-3DGS are recognized as state-of-the-art but are absent from the comparison.\n\n4. The paper overlooks a significant body of work focused on improving primitive representations. The comparisons or discussions with notable methods should be included, such as:\n- Deformable Radial Kernel Splatting\n- TNT-GS: Truncated and Tailored Gaussian Splatting\n- Triangle Splatting for Real-Time Radiance Field Rendering\n- Textured Gaussians for Enhanced 3D Scene Appearance Modeling\n- Quadratic Gaussian Splatting: High Quality Surface Reconstruction with Second-order Geometric Primitives", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "sG1Feoywnk", "reviewer": "Reviewer_zazd", "summary": "The paper proposes Universal Beta Splatting, which generalizes 3D Gaussian Splatting into N-dimensional anisotropic Beta kernels for unified modeling of spatial, angular, and temporal dimensions. Unlike fixed Gaussian kernels, UBS enables per-dimension shape control for adaptive geometry, reflection, and motion representation. It introduces spatial-orthogonal Cholesky parameterization, Beta-modulated conditional slicing, and CUDA-accelerated rendering. UBS remains backward-compatible with 3DGS/6DGS/7DGS and automatically decomposes scenes into geometric, material, and dynamic components. Experiments show up to +8.27 dB PSNR gain and ~50% faster training, establishing Beta kernels as efficient universal primitives for radiance field rendering.", "questions": "See weakness.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "1.The proposed UBS framework can simultaneously model diverse scene types, including static surfaces, view-dependent effects, surfaces, and dynamic scenes, within a single unified representation.\n\n2.Each primitive in UBS is more parameter efficient than prior methods while remaining fully compatible with the 3DGS rendering pipeline, ensuring easy integration and deployment.\n\n3.Extensive experiments across multiple benchmarks demonstrate UBS’s superior performance and strong generalization, validating the effectiveness of the proposed approach.", "weaknesses": "1. The paper claims that UBS is backward compatible with previous methods such as 3DGS, treating 3DGS as a special case. However, 3DGS models color using spherical harmonics, while UBS adopts a direct three dimensional color representation, which raises concerns about true compatibility between the two formulations.\n\n2. The formulation in Equation (6) lacks theoretical justification or derivation. It is unclear how this design was obtained and whether it has any mathematical guarantees.\n\n3. The paper does not provide analysis on the stability or convergence of the optimization process, particularly when extending UBS to high-dimensional Beta kernels, where parameter coupling may become complex.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "pKgFdqHN1Q", "reviewer": "Reviewer_bWeR", "summary": "**Universal Beta Splatting (UBS)** targets the fragmentation in splatting pipelines, where geometry, view dependence, and dynamics are handled by separate components. The authors proposes a single primitive that models all three together. Specifically, it replaces fixed Gaussians with an N-dimensional anisotropic Beta kernel and uses conditional slicing to obtain a renderable 3D primitive for a given view/time, avoiding SH or auxiliary color networks while remaining backward-compatible (recovering Gaussian splatting as a special case). Experiments across static, view-dependent, and dynamic benchmarks show consistent gains with real-time performance, and the authors emphasize that learned Beta parameters yield interpretable spatial/angle/time factors.", "questions": "The demo appears to visualize surface normals. Did the authors report any quantitative normal-estimation results, and how are normals computed from the Beta kernel parameters?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- The UBS kernel unifies spatial geometry, view-dependent appearance, and temporal dynamics within a single N-dimensional anisotropic Beta primitive. This design is well-motivated and, in my view, poised to make a substantial contribution to the community.\n- The experimental results are outstanding, and the visualizations align well with the quantitative metrics.\n- UBS features a pluggable architecture, allowing for seamless integration with other downstream tasks.", "weaknesses": "- Missing references for other works about alternative kernel design:\n  - 3D-HGS: 3D Half-Gaussian Splatting, by Haolin Li et al.\n  - [NeurIPS 2024] DisC-GS: Discontinuity-aware Gaussian Splatting, by Haoxuan Qu et al.\n  - [CVPR 2025] Deformable Radial Kernel Splatting, by Yi-Hua Huang et al.\n- The paper assumes substantial familiarity with Deformable Beta Splatting and therefore isn’t self-contained for readers new to Beta-based splatting. Several core ideas (e.g., how a Beta kernel parameterization induces opacity and color) are only sketched at a high level, with key mechanics deferred to prior work. As a result, understanding the contribution of UBS as a unified N-D generalization requires first reconstructing DBS’s basics. I recommend adding a focused preliminaries section that (i) recaps the Beta kernel definition and its bounded support; (ii) derives the mapping from Beta parameters to per-primitive opacity and radiance/color; and (iii) clarifies what is re-used vs. newly introduced in UBS (especially around conditional slicing and the removal of SH). This would make the paper accessible without cross-referencing DBS and would sharpen the statement of novelty.\n- Minor typo error:\n  - L171: `across both spatial, angular, and temporal dimension` -> `across spatial, angular, and temporal dimensions`\n- Missing comparison on FPS and primitive counts. It would help to include a direct comparison of primitive counts (Gaussian vs. UBS) and the corresponding FPS to clarify the runtime trade-offs.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)