Direct Conditional Parameterization for N-Dimensional Splatting

Zhongpai Gao; Benjamin Planche; Van Nguyen Nguyen; Meng Zheng; Anwesa Choudhuri; Arun Innanje; Terrence Chen; Ziyan Wu

cs.CVOct 16, 2026

Direct Conditional Parameterization for N-Dimensional Splatting

Authors: Zhongpai Gao, Benjamin Planche, Van Nguyen Nguyen, Meng Zheng, Anwesa Choudhuri, Arun Innanje, Terrence Chen, Ziyan Wu

Organizations: United Imaging Intelligence, Boston, MA, USA

Abstract

N-dimensional splatting extends 3D Gaussian Splatting with view direction and time to model view-dependent and dynamic effects. Each primitive is lifted into a joint distribution over position and conditioning variables, then conditionally sliced at each rendering query. N-DGS and UBS share this step across Gaussian and Beta kernels: recovering the conditional mean, opacity, and covariance requires per-primitive matrix inversions and regression products in every forward pass. We introduce direct conditional parameterization, replacing this covariance-derived readout with explicit per-effect parameters: a Cholesky precision factor for opacity and a factorized query-to-position regression operator with learnable per-dimension coupling. Unrestricted conditional coordinates recover a valid joint covariance; our factorization adds spatial-scale and coupling priors. Applied to both kernels, the parameterization yields dGS (Gaussian) and dBS (Beta). Across five static and dynamic benchmarks, they match or exceed their covariance-derived counterparts on quality, with gains up to +1.26 dB in the main MCMC evaluation. Production slicing is 6.9-7.7 times faster than on-the-fly covariance-derived conditioning; complete-render profiling gives 1.25-1.91 times speedups on four trained scenes. Direct conditioning provides a structured, kernel-agnostic alternative for N-dimensional splatting. Project page: https://gaozhongpai.github.io/dgs/.

Figures & tables

Figure 2: Overview of the d GS pipeline. Given query deviation $\boldsymbol{\delta}=\mathbf{q}-\boldsymbol{\mu}_{q}$ , opacity uses Cholesky factor $\mathbf{L}$ , while position uses the scale-normalized coupling factor $\mathbf{V}_{pq}$ and scaled precision $\tilde{\mathbf{V}}_{qq}$ with per-dimension coupling $\boldsymbol{\Lambda}$ . The factor norm follows the primitive scale; the complete regression and final shift also depend on $\tilde{\mathbf{V}}_{qq}$ and $\boldsymbol{\delta}$ . The same pipeline extends to d BS ( Section 4.4 ).

	Implicit		Direct (ours)
Operation	N-DGS	UBS	d GS	d BS
Invert $\boldsymbol{\Sigma}_{qq}$	$\boldsymbol{\Sigma}_{qq}^{-1}$	$\boldsymbol{\Sigma}_{qq}^{-1}$	—	—
Regr. matrix $\mathbf{P}$	$\boldsymbol{\Sigma}_{pq}\boldsymbol{\Sigma}_{qq}^{-1}$	$\boldsymbol{\Sigma}_{pq}\boldsymbol{\Sigma}_{qq}^{-1}\!\mathrm{diag}(\boldsymbol{\Lambda}_{\beta})$	—	—
Cov. correction	$\mathbf{P}\,\boldsymbol{\Sigma}_{pq}^{\top}$	$\mathbf{P}\,\boldsymbol{\Sigma}_{pq}^{\top}$	—	—
Position shift	$\mathbf{P}\,\boldsymbol{\delta}$	$\mathbf{P}\,\boldsymbol{\delta}$	$\mathbf{V}_{pq}\mathrm{diag}(\boldsymbol{\Lambda})\mathbf{V}_{qq}\boldsymbol{\delta}$	$\mathbf{V}_{pq}\mathrm{diag}(\boldsymbol{\Lambda}_{\beta})\mathbf{V}_{qq}\boldsymbol{\delta}$
Opacity	$\mathcal{K}_{G}(\boldsymbol{\delta}^{\top}\!\boldsymbol{\Sigma}_{qq}^{-1}\!\boldsymbol{\delta})$	$\mathcal{K}_{B}(\hat{\mathbf{L}}_{q}^{-1}\!\boldsymbol{\delta};\boldsymbol{\beta})$	$\mathcal{K}_{G}(\\|\mathbf{L}^{\top}\!\boldsymbol{\delta}\\|^{2})$	$\mathcal{K}_{B}(\mathbf{L}^{\top}\!\boldsymbol{\delta};\boldsymbol{\beta})$

Table 1: Per-primitive operations and production slicing-only forward time on A100 (1M primitives). Direct methods avoid query-block inversion, regression construction, and covariance correction. Parentheses compare optimized on-the-fly conditioning; Appendix H reports separate cached and factorized inference controls.

Figure 3: Qualitative comparison on static and dynamic scenes. Insets use identical crop coordinates and magnification across methods; boxes mark the source regions.

	Dataset	Method	#Gauss	Time $\downarrow$	FPS $\uparrow$	FPS ${}_{\text{CUDA}}$ $\uparrow$	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$
STATIC	NeRF Synthetic	N-DGS	300	10.43	183.58	348.95	34.04	0.973	0.027
		d GS ( Ours )	300	10.26	245.18	593.94	34.34	0.973	0.026
		UBS	300	6.80	155.96	306.33	34.48	0.975	0.026
		d BS ( Ours )	300	5.48	239.09	415.66	34.53	0.975	0.025
	Mip-NeRF 360	N-DGS	2,852	51.23	35.69	75.31	26.88	0.775	0.263
		d GS ( Ours )	2,574	49.85	42.49	200.56	27.67	0.794	0.227

Table 2: MCMC results: scene means from one run per scene (#Gauss in K, training time in minutes; dynamic batch size 1). Pairs share per-scene primitive caps; realized counts are reported. FPS and FPS

{}_{\text{CUDA}}

are arithmetic scene means using PyTorch and fused CUDA slicing.

	NeRF Synthetic			6DGS-PBR
Config	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$
$\Lambda{=}0$ ( d GS-O)	33.84	0.971	0.030	36.54	0.965	0.059
$\Lambda{=}1$ (fixed)	33.78	0.971	0.029	37.31	0.972	0.053
d GS w/o $\bar{s}$	31.26	0.957	0.046	34.78	0.951	0.075
$\Lambda$ learned ( d GS)	33.91	0.972	0.029	37.75	0.972	0.052

Table 3: Ablation study on NeRF Synthetic and 6DGS-PBR.

\Lambda{=}0

gives d GS-O (opacity only),

\Lambda{=}1

fixes full coupling, and learned

\Lambda

gives full d GS. “w/o

\bar{s}

” removes spatial scaling.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

	Methods	Mip-NeRF360			NeRF Synthetic
	Methods	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$
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
explicit	3DGS [ Kerbl et al., 2023 ]	27.20	0.815	0.214	33.31	0.969	0.037
	GES [ Hamdi et al., 2024 ]	26.91	0.794	0.250	-	-	-

Appendix

Table 4: Comparison with SOTA methods on Mip-NeRF360 [ Barron et al., 2022 ] and NeRF Synthetic [ Mildenhall et al., 2020 ] . Literature baselines retain their reported protocols; UBS and d BS use the same 3DGS evaluation protocol.

	Dataset	Method	#Gauss	Time $\downarrow$	FPS ${}_{\text{CUDA}}$ $\uparrow$	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$
STATIC	NeRF Synthetic	N-DGS	171	6.35	615.57	33.38	0.970	0.032
		d GS-O ( Ours )	201	5.83	754.98	33.84	0.971	0.030
		d GS ( Ours )	201	6.84	704.84	33.91	0.972	0.029
	Mip-NeRF 360	N-DGS	1273	27.09	147.46	26.40	0.739	0.316
		d GS-O ( Ours )	2363	25.58	279.30	27.18	0.779	0.252
		d GS ( Ours )	2463	28.21	237.82	27.37	0.781	0.251

Appendix

Table 5: Standard 3DGS densification: scene means from one run per scene (#Gauss in K, training time in minutes; dynamic batch size 4). FPS

{}_{\text{CUDA}}

is an arithmetic scene mean. For 3DGS [ Kerbl et al., 2023 ] and 4DGS [ Yang et al., 2024 ] comparisons, see the original N-DGS [ Gao et al., 2025a , Gao et al., 2025b ] .

Dataset		Scene	N-DGS			d GS-O ( Ours )			d GS ( Ours )
Dataset		Scene	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$
STATIC SCENES	NeRF Synthetic	chair	35.75	0.986	0.013	36.36	0.988	0.012	36.26	0.988	0.012
		drums	26.75	0.956	0.036	26.80	0.957	0.035	26.92	0.958	0.034
		ficus	33.54	0.984	0.014	35.30	0.988	0.011	35.03	0.988	0.011
		hotdog	38.10	0.985	0.022	38.13	0.986	0.020	38.15	0.986	0.020
		lego	35.63	0.981	0.019	35.87	0.982	0.017	35.79	0.982	0.017

Appendix

Table 6: Detailed quantitative comparison across all datasets at 30K iterations. N-DGS, dGS-O, and dGS are compared on static benchmarks (NeRF Synthetic, Mip-NeRF 360, 6DGS-PBR) and dynamic benchmarks (7DGS-PBR, D-NeRF).

Dataset	N-DGS	DU	d GS
NeRF Synthetic	33.38	32.91	33.91
6DGS-PBR	37.11	36.44	37.75

Appendix

Table 7: Direct-regression control under standard densification. Dataset-mean PSNR (dB), one training run per scene.

Protocol	Method	PSNR
MCMC	UBS	$34.50\pm 0.02$
	d BS	$34.53\pm 0.02$
Standard	$\boldsymbol{\Lambda}=\mathbf{0}$	$33.83\pm 0.02$
	$\boldsymbol{\Lambda}=\mathbf{1}$	$33.78\pm 0.01$
	Learned $\boldsymbol{\Lambda}$	$33.92\pm 0.02$

Appendix

Table 8: NeRF Synthetic PSNR (dB): mean and sample standard deviation of dataset means across three training seeds.

Parameterization	$C=3$	$C=4$
N-DGS, on-the-fly	0.224	0.269
N-DGS, cached	0.178	0.197
N-DGS, factorized query	0.141	0.180
d GS, direct	0.079	0.100

Appendix

Table 9: Slicing time (ms) for one million primitives on an A100. Medians of 100 trials after 20 warmups; inputs are materialized before timing.

Scene	Slicing	Remainder	Total	Speedup
lego	1.56 / 0.22	2.19 / 2.42	3.75 / 2.65	1.42 $\times$
counter	7.38 / 0.57	8.77 / 12.18	16.15 / 12.75	1.27 $\times$
garden	24.83 / 1.69	13.96 / 18.61	38.79 / 20.30	1.91 $\times$
bicycle	29.78 / 2.02	24.71 / 41.59	54.49 / 43.61	1.25 $\times$

Appendix

Table 10: Frame-time decomposition (ms) on trained checkpoints using matched training caps. Each pair lists N-DGS / d GS; the remainder includes all post-slicing stages.

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Three-dimensional Gaussian Splatting (3DGS) combines explicit primitives with efficient rasterization, yet recent systems increasingly use neural networks to generate or share Gaussian parameters. We characterize this trend along five axes: attribute decoding, spatial sharing, view-conditioned decoding, topology generation, and amortized inference. An analysis of 19 representative methods shows that these choices address different limitations and cannot be reduced to a binary neural label. We also isolate three forms of neural parameterization in a controlled mip-NeRF 360 study. Sharing appearance and opacity improves reconstruction quality, while decoding geometric structure offers no further gain. The evidence favors selective neuralization: shared functions help when they capture reusable correlations without sacrificing the local geometric freedom of explicit splats.

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