Learning Unified Representation of 3D Gaussian Splatting
Authors: Yuelin Xin
Yuheng Liu
Xiaohui Xie
Xinke Li
Venue: ICLR 2026 Poster (2026)
Keywords: Representation Learning
3D Gaussian Splatting
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: Proposed a new representation of 3DGS based on submanifold field that is more suitable for learning.

Abstract: A well-designed vectorized representation is crucial for the learning systems natively based on 3D Gaussian Splatting. While 3DGS enables efficient and explicit 3D reconstruction, its parameter-based representation remains hard to learn as features, especially for neural-network-based models. Directly feeding raw Gaussian parameters into learning frameworks fails to address the non-unique and heterogeneous nature of the Gaussian parameterization, yielding highly data-dependent models. This challenge motivates us to explore a more principled approach to represent 3D Gaussian Splatting in neural networks that preserves the underlying color and geometric structure while enforcing unique mapping and channel homogeneity. In this paper, we propose an embedding representation of 3DGS based on continuous submanifold fields that encapsulate the intrinsic information of Gaussian primitives, thereby benefiting the learning of 3DGS.

Published as a conference paper at ICLR 2026
LEARNING UNIFIED REPRESENTATION OF 3D GAUSSIAN SPLATTING
Yuelin Xin
Yuheng Liu'
Xiaohui Xie
Xinke Li
UC Irvine
City University of Hong Kong
ABSTRACT
A well-designed vectorized representation is crucial for the learning systems na- tively based on 3D Gaussian Splatting. While 3DGS enables efficient and explicit 3D reconstruction, its parameter-basee representation remains hard to learn as fea- tures, especially for neural-network-basss models. Directly feeding raw Gaussian parameters into learning frameworks fails to address the non-unique and hetero- geneous nature of the Gaussian parameterization, yielding highly data-dependent models. This challenge motivates us to explore more principled approach to represent 3D Gaussian Splatting in neural networks that preserves the underly- ing color and geometric structure while enforcing unique mapping and channel homogeneity. this paper, we propose an embedding representation based on continuous submanifold fields that encapsulate the intrinsic information of Gaussian primitives, thereby benefiting the learning of 3DGS. Implementation available at htthtt::/
3DGS
com/ cilix- ai/ wee---iigg-eemddinnnn..
INTRODUCTION
Recent advances in 3D Gaussian Splatting (3DGS) (Kerbl et al., 2023) have established it as powerful technique for representing and rendering 3D scenes, enabling high-fidelity, real-time novel view synthesis through explicit parameterization of Gaussian primitives. This representation has cat- alyzed growing body of work exploring carning-bassed methods that operate directly on Gaussian primitives, supporting tasks such as compression (Shin et al., (025)) generation (Yi et al., (024) Xie et al. 2025), and understanding (Guo et al.) 2024). In these pipelines, the native parameterization [r, q. s, c, o) is often adopted as the input or output of neural architectures.
Despite its effectiveness in optimization- based reconstruction, we identify fundamental limitations when this parametric representation is employed as a learning space for neural networks. Specifi- cally, the native parameterization 0 conflicts with the inductive biases of standard neural architec- tures in three critical ways. First, the mapping from parameters to rendered output exhibits non- uniqueness. Ambiguities such quaternion sign duality and symmetry-ing induced variances create one-to-many mapping. This mapping creates "embedding collisions" where distinct parameter in- puts producing identical visual outputs generate conflicting supervision signals, leading to training instability and poor convergence (Bengio et al. 2013 Wang & Isola 2020). Second, the parameter components suffer from numerical heterogeneity. Spatial positions span large magnitudes while quaternions remain normalorma violating the homogeneous feature distribution assumption re- quired for effective gradient flow (loffe Szegedy 2015). Third, these parameters inherently reside on distinct mathematical manifolds, such as positions in IR3, rotations in SO(3), and appearance in spherical harmonic coefficients. Forcing these non-Euclidean variables into the Euclidean feature spaces of standard encoders breaks their intrinsic geometric structure, making the representation difficult to compress or regularize effectively.
These theoretical misalignments translate into substantial practical failures across various applica- tions. Our empirical analysis reveals that parametric encoders are critically unstable; for instance, due to the sign ambiguity, simply negating quaternion (a mathematically equivalent rotation) can cause complete decoding failure in parameter-traile trained autoencoders, see App. Fig. b. This instability extends to generative tasks involving latent manipulation, where we observe that linear interpolation
Corresponding author.

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Published as a conference paper at ICLR 2026
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Figure 1: A scene of N Gaussian primitives can be represented by NV sets of parameters 0 (shown in pink). Data this parametric space resides on different manifolds and is heterogeneous and non- Euclidean, introducing challenges for encoders fit disparate data manifolds implicitly. Shown in purple is the proposed representation, instead of relying on Gaussian parameterization, we introduce canonical submanifold field space (M, F)) that uniquely represents Gaussian primitive with an iso 5so-probability surface.
in the parametric latent space results in geometric "jitters" and unnatural transitions. Furthermore, parametric embeddings lack robustness noise, with minor perturbations causing disproportionate reconstruction errors. Crucially, for downstream tasks extensively explored in recent work includ- ing generative modeling YYi al., 2024 Zhou et 2024b). compression (Shin al., 2025 Girish et al. 2024), and editing (Chen et al. |l.] 2024a). these flaws manifest as discontinuous latent spaces and ineffciint encoding. Instead of capturing robust geometric semantics, models are forced to resolve parameter ambiguities with more efforts, leading to sub- sub-optimal performance in learning based re- construction (Charatan et al. 2024) and limited generalization across diverse domains.
We thus propose principled alternative that represents each Gaussian primitive as continuous field defined on its iso-probability surface. This submanifold field representation establishes a unique correspondence between Gaussians and their geometric-photonen biguities of parametric representations. By discretizing this field as a colored point cloud sampled from the probability surface, we obtain numerically uniform and geometrically consistent repre- sentation. We further employ a variational autoencoder to learn embeddings from these discretized submanifold fields, together with Manifold Distance metric based optimal transport that bet- ter correlates with perceptual quality than parameter-space distances. Extensive experiments show higher reconstruction quality, stronger cross-domain generalization, and more robust latent represen- tations. The learned embedding space demonstrates potential for downstream applications through unsupervised segmentation and neural field decoding with the proposed embeddings. To summarize the contribution of this work, we:
properties, removing the am-
identify and formally characterize the fundamental limitations of parametric Gaussian rep- resentations for neural learning, including non-uniqueness and numerical heterogeneity.
propose submanifold field representation that provides provably unique and geometrically consistent encoding of Gaussian primitives.
develop variational autoencoder framework incorporating novel Manifold Distance met- ric based on optimal transport theory for effective learning in the submanifold field repre- sentation space with extensive experimental evidence.
2 RELATED WORKS
3D Gaussian Splatting, Since its re-introduction by Kerbl et al. (2023). 3DGS has rapidly become
core method for novel view synthesis and 3D representation. By placing explicit Gaussian primitives
in 3D space and employing efficient rasterization and accumulation, 3DGS achieves real-time
dering with high fidelity (Bao et al., 2025 Lin et al. 2025c). Several studies improve efficiency (Jo)
al.
2024
Lee et al. 2024). while others leverage large-scale datasets for generalization (Ma al.
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2025 Li et al. 2025a). ). At the application level. 3DGS has been adopted for digital human Li al.
2024 Kocabas et al., 2024; Wang et al. 2025), self-driving scene modeling (hhou et al. 2024a c
Yan et 20233, and physics-basee simulation (Jiagg et al., 2024: Xie et al. [024] Zhong al.
2024). Beyond fixed Gaussian parameters, another line augments primitives with latent embeddings
capture semantics, open-vocabulary understanding (Qin et al.
(024) and deformation model-
ing (Zhobro et al.) 2025). These efforts demonstrate that 3DGS provides high-fidelity appearance and serves as a versatile representation with broad potential (Sun et al. 2025).
3DGS Parameters Regression. To enable fast and flexible reconstruction without per-scene op- timization, recent works seek to directly obtain Gaussian splats through feedforward prediction networks. For example, CCharatan et al. (2024): Chen et al. (2024b) proposed to predict Gaussian parameters directly from multi-view input, while ZZeng et al. (2024) generate pixel-wise parame- ter maps and lift them to 3D via depth estimation. This paradigm has been extended to pose-free settings (Hong et al., 2024, Chen et al., 2024c Tian et al. 2025), and transformer-base
methods further improve scalability and generalization (Li et al. 2025b; Jiang et al. 2025 Lin et al. 2025b). Overall, these methods directly output Gaussian parameters from neural networks, showing that 3DGS serve an effective target for network- based prediction in efficient reconstruction.
Embedding Gaussian Primitives. Recent works move beyond reconstruction and encode Gaussian parameters into latent spaces for tasks such as generation, editing, and compression. Zhou et al. (2024b); Lin et al. (2025a); ; Wewer et al. (024) learn structured latent variables from 3D Gaussian space fulfill generation tasks. Editing and styessrrasfer methods diffusion style condition- ing to manipulate Gaussians primitives in latent or rendering spaces (Chen et al.) 2024a Vachha &
Haque 2024, Lee et al. 2025 Palandra
2024 Zhang et al. [0224] Kováes al. 2024 Yu al.,
2024). Other works improve rendering quality by optimizing Gaussian parameters under dillusion
priors Tang al.] [203 Yi et al., 2024] Chen et al., 2024d). while compression methods (Girish
et al. [024]] Yang et al., 2025) reduce storage and computation by quantizing and embedding Gaus- sian parameters. These approaches show potential of embedding Gaussians into neural latent spaces, but they assume Gaussian parameters are naturally compatible with neural learning, over- looking that these parameters were designed for optimization-bee based reconstruction. This oversight underlies our analysis in Section and our proposal of more suitable formulation.
3 METHOD
3.1 PRELIMINARIES: GAUSSIAN SPLATTING PARAMETERIZATION
scene under 3D Gaussian Splatting represented as set of N oriented, and view-dependently colored Gaussian primitives {G each contributing to the final rendered image via rasterization and alpha compositing. Each Gaussian primitive G; is usually represented by a parameter tuple 0, (14, q. S,, Ca, ), where:
#4 E R the center position of the Gaussian in world coordinates;
qi E S3(3): a unit quaternion representing the local rotation;
s (IR + )): scale parameters along the rotated axes;
* E R3*K, spherical harmonic (SH) coefficients for view-dependent color for K E Z;
o, E R: logit-tranfformed opacity value a, = o(o)), where or is a sigmoid function.
The local geometry of the Gaussian is govemned by its covariance matrix, constructed as
Σ; = R(q;) d R(q.)
(1)
where R(q.) is the rotation matrix corresponding to the quaternion q. This defines ellipsoidal spatial density, centered at μ4 whose shape and orientation determine the contribution of G, to the rendered scene. The color at given view direction d e $ is computed per channel using SH basis functions denoted by
(d) [SH (d), (d), SH((a)
where (d) in c-channel calculated by Eter (C,)),(( (d) Y," is the real- valued spherical harmonic of degree I and order m. The final rendering aggregates contributions
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from all Gi via soft visibility- -eelhttd compositing process. This native parameterization is well- suited for gradient-bart based scene optimization. However, it introduces significant challenges when used representation for learning.
3.2 PARAMETERIZATION ILL-SUITED AS LEARNING SPACE
The parameter representation 0 poses fundamental challenges when used as a learning space for neural networks. We identify two critical issues: representation con-uniqueness and numerical het- erogeneity. Each undermines the stability and effectiveness of neural network training.
Representation Non-uniqueness. The parametric representation suffers from a many-to-one map- ping that violates basic requirements for stable learning. To understand this, we first formalize what rendering effect single Gaussian primitive produces.
Definition (Single Gaussian Radiance Field (SGRF)) A SGRF is radiance field Φ R R The field is defined by the local density at point x E R along direction d E S²:
Φx d) = PG cg(d),
where px(x) - exp (- T /]) is volume density function and og cg( is a color radiance field coupled with opacity. Specifically, given parameter set 0 {n, S. c, can be derived by equation! [and eg (d) (0) Color(d) can be derived by equation
The SGRF, derived from the multi-Gausse rendering framework by Kerbl et al. 2023), specifies how the final value at any pixel is rendered in scene containing only one Gaussian splat. Further- more, let & be the space of SGRFs, and θ IR be the paramater space of Gaussian primitives, each parameter set 0 E θ provides a complete representation that generates a correponding field @g E T. We indicate that single SGRF may correspond to multiple parameterizations of Gaussian primitives, as formalized in the following proposition.
Proposition 1 (Non-uniqueness of the SGRF Parametric Representation) The parametric rep- resentation SGRF is not unique. Formally, there exist at least two distinct parameter sets, 0, E θ and 02 E θ with 0, * Oz. that generate the exact same field og E 4
The non-uniqueness is from quaternion sign ambiguity, geometric symmetries, and rotation- spherical harmonic interactions producing equivalent parameter combinations (see App. A for proof). The non-uniqueness of will create "embedding collisions" where different parameter vectors produce identical rendered output (Wang & Isola 2020). This makes the learning objec- tive ||0 Ouagelll pl ambiguous, as multiple parameter configurations can achieve the same visual result. The resulting conflicting gradients lead to training instability and poor convergence indicated by [Bngii et al. (2013).
Numerical Heterogeneity The parameter components violate the homogeneous distribution as- sumption of standard neural architectures. Neural networks typically assume features share similar statistical properties for effective gradient flow (Toffe & Szegedy| (015).. However, 3D Gaussian parameters span vastly different ranges. For example, pre-activation scales range from -15 to 3, while quaternions stay unit-normalized. More fundamentally. these parameters follow differ- distributions and live different manifolds: positions fa E R rotations q E SO(3), scales s E (1R+),, and SH coefficients with exponential decay. Concatenating them ignores their het- erogeneous nature. Small noises quaternions can drastically alter geometry, while small noise in high-order SH coefficients is negligible, yet the network treats all dimensions equally.
The non-uniqueness and numerical heterogeneity of the native parameter space make unsuitable for neural network learning, which would generate unstable embeddings (see our experiments in Sec. and App. D). We therefore introduce a submanifold field representation that ensures unique mappings and respects the geometric structure of 3D Gaussians.
3.3 REPRESENTATION ON SUBMANIFOLD FIELD
To address this issue, we propose converting each Gaussian primitive G, to a novel geometric repre- sentation &,, which is color field defined on 2D submanifold in 3D Fuclidean space, illustrated
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Manifold Sampling
Submanifuld Field Variational Aute-Encuder
Parameter Fitting
Spiare
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To embed the proposed submanifold field representation into vector form suitable for neural networks, we devise a Submanifold Field Variational Auto-encoder (SF-VAE) that embeds any input submanifold field 32-D vector, then reconstructs the original parameter set O,. SF- VAE learns in our new representation space instead of the parametric space.
Figure 2:
in Fig. For Gaussian density N pi 2i,,, we define the iso-probability surface fixed radius as:
M (x (x - -x m4) =-]]
which forms an ellipsoid surface, namely, two-dimensional submanifold, centered μ4. On this submanifold, we define a field function:
(x = (o:) (olor;(dx):
(5)
where dx (x pb)||x pell denotes the unit direction vector for x E M,, and Color, represents the view-dependent color parameterization in equation 2 possible iso-probability submanifolds as defined equation space as:
Let M. be the space of all we define our unified representation
(6) The representation & 6 encodes both geometric properties (shape, orientation) via M, and ap- pearance attributes view-dependent color) via F, in continuous framework. We have the following proposition (proof is provided in App. B).
8 (E, (M, (M,, M, E M, F. :;
Proposition 2 (Uniqueness of Submanifold Field Representation) For every SGRF óg E T 6. This establishes one-to-one corre-
there exists unique corresponding representation spondencee between the elements of de and & Formally, for any two distinct fields Φg,1, Øg.2 E b their corresponding representations E,,&, &g E & are also distinct.
The submanifold field &, thus provides numerically stable and provably unique representation space on which we can safely build learning objectives and neural architectures.
3.4 ENCODE SUBMANIFOLD FIELDS AS EMBEDDINGS
We design a variational auto-encoder to encode submanifold field representation, shown in Fig. The network architecture, learning objectives and dataset introduced.
Encoder-decoder Architecture. We employ a point-cloud-bent -assd network to encode and decode one sub-manifold field. Particularly, we uniformly sample P points from the submanifold field &
as a colored point cloud p = { ((xm. F(dx... })} m-1- We then employ a PointNet (Qi et al. 2017) encoder / to obtain latent embedding by dimension D. The decoder g consists of two neural networks, namely, the coordinates transform
f(z IP) where z RP is the embedding with
network ge R Rº + g and color field 9s R RD - R3. The decoded point cloud from
decoder is given by,
P g(z,Up:)) ((e, z)). gr ()).gr(((e((())) ([ae((en, z)), 2
where Up = P= is a set of coordinates sampled from a unit sphere surface. Such canonical set works as the initial input for two implicit functions ge and gy, and queries coordinates and
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color field. Furthermore, recover the original Gaussian parameters 0, for rendering purposes, estimate the covariance matrix Σ, by principal component analysis (PCA), and SH coefficients c; by fitting the spherical harmonics to PD
Learning Objectives. We introduce Manifold Distance (M-Dist) for the reconstruction objective in encoder-decoder training. Given two submanifold fields & (M, F) and & = (M, F), propose to measure their similarity based on the Wasserstein-2 distance from optimal transport defined as
S df d cx), (x,))))
inf W2
"EI(o,0")
where 62 = F(dx), e F(dy), [(ā oj) is the set of all joint probability measures (transport plans) with marginals and og, and the ground distance is defined as
((x,, (y, = ||x - y Al|es cyl12
balancing spatial and color terms. In practice, both M and M are discretized as colored point clouds P and P. The empirical Wasserstein-2 distance W is then computed between these point clouds by
with RT
(P,P))=
min rET(a.))
Σ
Σ
Tú (,,
(10)
(x0, EP (rs
Finally, the learning objective for variational auto-encoder is
VAVE = EP-VAR(P) (P) dRa (f(z (((aP))IW(O..))
(11)
where VAE(P) = g(f(z P)) and the second term is the KL divergence loss for variational auto-encoder implementation, and is a balance factor.
Dataset Preparation. Since this embedding model only encodes single Gaussian primitives, which have no semantic meaning out of scene's global context, we can use a randomly generated dataset of Gaussian primitives to train this model, thus making it domain-invariant to data. The implemen- tation details of this generated dataset can found App. C.
4 EXPERIMENTS
4.1 EVALUATION SETUP
Baseline Implementation Details. To isolate the effect of representation choice, we adopt a self- implemented encoder-decoder framework for both the parametric representation 0 and the proposed submanifold field representation E. While comparisons with existing 3DGS learning methods are possible, they typically involve task-specific architectures that confound the role of representation itself. Direct reuse of prior pipelines would not yield a controlled comparison, so we implement both baselines in the same VAE-style framework to attribute differences solely to the representation.
We implement and train three matched embedding models: our submanifold field VAE (Sec. [.4). and two baseline parametric VAEs operating directly on 0. For the parametric mod- els, each Gaussian primitive is represented as 56-D vector (3+44-3K for I=ma==3), omitting global coordinates match the SE-VAE setting. three-layer MIP encodes this input to 32-D) latent 156-5512+5122---
x 2), and the decoder, either uses a MLP map the latent back 0., or uses the same decoder of SF-VAE to map to P. This setting further decouples evaluation results with the training objective functions. Apart from input dimension. all models share identical depth, width, latent size (32), and optimizer settings (all using Adam), ensuring a matched capacity.
We evaluate the proposed representation and compare it with the baseline primarily using two datasets. For oeject-level tasks, we utilize ShapeSplat (Ma et al. 2025). a large-scale 3DGS dataset derived from ShapeNet (Chang et al. 2015), comprising 52K objects across 55 categories. For scene-level experiments, we employ Mip-NeRF 360 (Barron et al. (022). which contains medium-scale scenes with abundant high-freyuency details. Additionally, unless stated otherwise,
Datasets.

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Reconstruction quality comparison for object-level (ShapeSplat) and scene-level (Mip- NeRF 360) datasets. All models trained on the randomly generated dataset. The three models have a parameter count of 0.62M, 0.66M and 0.62M respectively. The relatively extreme perceptual metrics values in ShapeSplat come from the use of background during measurement.
Table 1:
Input Representation
Encoder
Decoder
PSNR f
SSIM 1
LPIPS
M-Dist
Li-Dist
ShapeSplat
Parametric
MLP
MLP
37.512
0.888
0.152
0.184
0.040
Parametric
MLP
SF-VAE
44.725
0.896
0.136
0.051
0.097
Submanifold Field
SF-VAE
SF-VAE
63,408
0.990
0.010
0.041
0.098
Mip-NeRF
360
Parametric
MLP
MLP
18.818
0.564
0.452
0.510
0.034
Parametric
MLP
SF-VAE
20.923
0.730
0.359
0.055
0.173
Submanifold Field
SF-VAE
SF-VAE
29.833
0.953
0.079
0.048
0.179
Table 2: Reconstruction quality comparison under cross-domain setting. All models trained either SpapeSplat or Mip-NeRF 360 dataset are tested on another dataset. We show that the generalization ability of SF Embedding framework is inherently domain-agnosti even without random data.
Train set
Test set
Input Represent.
Encoder / Decoder
PSNR
SSIM
LPIPS
Parametric
MLP MLP
9,753
0.356
0.615
ShapeSplat
Mip-NeRF.
Parametric
MLP / SF-VAE
14.845
0,675
0.336
Submanifold Field
SF-VAE SF-VAE
19.194
0.821
0.309
Parametric
MLP MLP
55.624
0.957
0.067
Mip-NeRF.
ShapeSplat
Parametric
ML.P SE-VAE
60.777
0.987
0.013
Submanifold Field
SE-VAR SP-VAR
62.576
0.990
0.014
we train the embedding models using the randomly generated Gaussian primitive dataset, with 500K randomly generated data samples; implementation details are provided App. C.
Evaluation Metrics. To comprehensively assess both perceptual fidelity and representation qual- ity, we report PSNR, SSIM, and LPIPS on rasterized reconstructions against ground truth Gaussian splats, well as L2 distance in the Gaussian parameter space. Crucially, also include our proposed Manifold Distance (M-Dist) as an evaluation criterion. By compariogg M-Dist with parameter-space distances (L), we demonstrate that M-Dist aligns more closely with per- ceptual "gold standard" metrics such as PSNR and LPIPS, validating our claims Sec.
4.2 EVALUATION ON REPRESENTATION LEARNING FRAMEWORK
Zero-shot Reconstruction. We present a comprehensive quantitative and qualitative analysis of reconstruction quality for both object-level and scene-level data, summarized in Tab. Tab.[[and Fig. All models are trained on the same randomly generated 3D Gaussian primitives dataset and evaluated on ShapeSplat and Mip-NeRF 360, using three matched encoder-decoder configurations control for bias. Across all perceptual metrics (PSNR, SSIM, LPIPS), the submanifold field representation consistently outperforms parametric baselines. For example, on SpapeSplat, SF- SF-VAE achieves sub- stantially higher PSNR and SSIM and a much lower LPIPS, indicating both improved fidelity and perceptual quality. Similar performance gains are observed scene-level reconstruction, where the submanifold field model demonstrates better performance across diverse spatial contexts.
Importantly, the Manifold Distance (M-Dist) metric shows stronger empirical correlation with quality metrics like PSNR and L.IPS than traditional LA parameter distances, supporting our claim that M-Dist more robust and meaningful similarity measure for 3D Gaussian representations, truthfully reflecting perceptual differences rather than merely parameter discrepancies. The consis- tent improvement margin across both datasets highlights the advantage learning in the submani- fold field space, which better preserves intrinsic structure and view-de wew-deedddent appearance, confirm- ing the efficacy of our representation for high-fidelity 3D Gaussian modeling.
Cross-domain Reconstruction. We also evaluate cross- roms-domain generalization by training real-world dataset and testing on the other (object-level ++ scene-level) under an identical training protocol and capacity budget in the reconstruction study. Concretely, train either the proposed

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Ground Truth
MLP
MLP SF-VAE
SE-VAE
Ground Truth
MLP//MLP
MLP/SF-AA
SE-VAE
Figure 3: Qualitative results for rasterized reconstruction. Samples selected arbitrarily from Mip- NeRF 360 and SpapeSplat. Parametric models can induce confusion in parameter space, failing to embed and restore the correct Gaussian parameters.
Noine Level
Yoire Level -11
Nein Leval
Neire Lenel 0.5
Add Naise Embedding
Fananoats Eark. SF Evals
M-Dist
0.05
0
0.1
0.2
0.
0.1
0.5
Noise
Figure 4: Reconstruction results using embeddings with noise. Left: Visualization of reconstructed scene from noisy embeddings of Gaussian parameters (MLP) and SF-VAE. Right: Comparison on M-Dist for different noise levels added to embedding space, tested on Mip-NeRF 360. Noise level is defined as the ratio between the noise magnitude and the embedding variance.
SF-VAE or the parametric MLP baselines on a source set and evaluate on target set, rendering novel views and reporting PSNR, SSIM, and LPIPS averaged over test samples (see Tab. 2 Across both transfer directions, the SF-based embedding consistently achieves higher reconstruction quality than the parametric baseline, indicating reduced sensitivity to domain-specific statistics (e.g.. scale, lighting, complexity). Particularly, comparing these transfer results with Tab. T we find that the model trained synthetic random data actually outperforms the models transferred from real- world domains. This indicates that our random learning strategy effectively strips away domain priors, which establishes approach unified representation, where single model trained synthetic data can fundamentally generalize to real-world contexts.
4.3 SENSITIVITY STUDY OF REPRESENTATION
Robustness to Noise. To evaluate the submanifold field embedding's robustness noise, we grad- ually add higher levels of gaussian noise to the embedding space of the parametric model and the submanifold field model and test their reconstruction quality and M-Dist. To ensure fair compar- ison, noise level
ratio to variance instead of absolute noise magnitude. As shown in Fig. 4 the embedding space of submanifold field model is more robust to random perturbation, this makes submanifold field embeddings better learning target since it is less sensitive to potential noise introduced by downstream regression.
Latent Space Interpolation. To evaluate the regularity of the latent space of the proposed repre- sentation, we randomly sample pairs of source and target Gaussian primitives G, and G, and linearly interpolate each pair for fixed number of steps Compared with parametric space, the inter-
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Refermnce Image
MLP/MLP
MLLP/SEVVE
SE-VAE
Figure 5: Unsupervised graph clustering based on raw Gaussian parameters and various embed- dings. Submanifold field embeddings show better preservation of detailed semantics, showing its downstream applicability.
H
Table 3: Comparison between Gaussian Neural Fields trained
using submanifold field embeddings and raw Gaussian parame- ters. Top: ShapeSplat, bottom: Mip-NeRF 360.
(.)))
Emb :S Pammeters
Target
PSNR 1
SSIM f
LPIPS
Params
Figure 6: Setting of a Gaussian Neural Field, we compare be- tween the prediction target SF embedding and raw GS param- eters.
Raw GS Parameter
51.660
0.925
0.141
0.21M
SF Embedding
58.619
0.980
0.043
0.20M
Raw GS Parameter
19.922
0.648
0.410
1.87M
Embedding
24.395
0,804
0.261
1.85M
polation in submanifold field embedding space shows smooth transition path, while interpolation in parametric space shows undesired jitter in rotation and scale, indicating space irregularities, see App. D. This highlights the motivation to learn in the unified submanifold field embedding space.
4.4 REPRESENTATION APPLICABILITY
Unsupervised Clustering, To further probe the semantic structure of the learned embedding spaces, perform unsupervised graph clustering on both the raw Gaussian parameter space and the em- bedding outputs of each model. As visualized Fig. 5] clusters formed in the submanifold field embedding space exhibit more detailed semantic separation against the reference images compared to those formed using normalized parameters or parametric embeddings. For example, SF-VAE's embedding clustering in the first line of Fig. F outlines clearer separation of foreground objects with the background. The clusters appear smoother, less noisy, and with clearer boundaries, showing an ability distinguish between different entities. This indicates that the submanifold field embedding captures more dense semantics and discriminative features, validating its usefulness.
Gaussian Neural Fields. To validate the potential of our representation for advanced downstream tasks, we introduce the Gaussian Neural Field (GNF). Drawing inspiration from the decoding struc- tures in generative diffusion models (c.g., DiffGS by Zhou et al. (2024b)) and neural compression frameworks (Wu & Tuytelaars) 2024), the GNF functions as a coordinate- baste neural implicit field as illustrated Fig. [6] Specifically, employs a lightweight MLP (architecture detailed in App- D.4) to learn a continuous mapping from spatial coordinates x; to per-primitive descriptors. This setup allows us to evaluate the "learnability" of our representation: while regressing heterogencous raw parameters 0, often leads to optimization difficulties, our unified SF embeddings provide smooth and well-conditioned target for the neural field. As evidenced in Tab. B and visualization in App. D.4, the SF- SF-guided GNF outperforms the parameter-basse baseline in visual fidelity with equivalent training effort. This indicates that our representation more friendly to neural networks, hinting its utility for potential downstream generative and compression tasks.

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Published as a conference paper at ICLR 2026
(
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Figure 7: Behavior studies tested on Mip-NeRI 360. From left to right: (a) embedding dimension,
(b) generated training dataset size, (c) Submanifold Field discretized (i.e., point sample) grid size.
4.5 MORE STUDIES ON IMPLEMENTATION DETAILS
Ablation Study on SF- VAE Designs. We provide performance comparison with different frame- work designs based on Mip-NeRF 360. (1) For encoder f. we tested DGCNN encoder (Wang al.) 2019) beyond our implementation, where the DGCNN encoder achieves comparable reconstruction fidelity while it roughly x.75x slower in encoding and uses roughly 2x more GPU memory during inference; (2) For the decoder's 's unit sphere grid, we tested a 2D grid implementation with matching grid size, 2D grid achieves similar reconstruction results but takes more iterations to converge: (3) We implemented two versions of the fitting module: GPU- version using FP32 with batching and Cholesky decomposition, and a CPU-based one using FP64 without batching and Least Squares solver. Experiments show that the GPU version introduces only negligible quality degradation (0.4 PSNR and 0.01 SSIM), while achieving an average speedup of 85 x with a batch size of 4096.
Behavior Study of Latent Space Dimension. We evaluated different embedding space dimensions for the SF-VAE model to meet the best trade-off between compression and reconstruction quality. All models are trained on the generated dataset with L 3 order Spherical Harmonics. Results shown in Fig. [ (a). 32 is the optimal balance point between reconstruction quality and latent space compression. All values are tested with baseline input/output of P 12a. While this work does not specifically focus on compression effectiveness,, embedding space robustness shown in Sec. 4.3 suggests potential further latent tokenization and quantization.
Behavior Study of Training Set Size. To determine the number of random training samples re- quired to achieve the best reconstruction results, we vary the trainiing sample size from 5K to 500K (baseline), see Fig. (b). The results indicate the proposed representation is data efficient. When only using 2% of the baseline training sample, our model can achieve close-to-baseline performance.
Behavior Study of SF Discretized Size. To ensure the submanifold fields are truthfully represented in a discrete manner, we evaluate different sample sizes P, see Fig. 7 (c). Going from the lowest tested P 6² to the baseline P = 12a, we observe a gradual improvement in reconstruction quality, while going above P = 12a yields very little improvement. Since P directly correlates to the computational efficiency of the submanifold field model (see below), we keep P P= 12?.
Computational Efficiency. Increasing the point sample size P increases computation and memory. Encoding time remains low since a lightweight PointNet-based encoder shares weights with all input points, giving an inference speed of 1.72s per million Gaussians for P = 122 with a batch size of 4096 on an RTX 5090. Decoding time is 4.20s per million Gaussians, from emhedding Gaussian parameters. The complexity is O(P) or O(n²) w.r.t. the grid size We utilize the advantage of GPU parallel computation boost the caleulation for parameter fitting module. In the 4.20s of decoding time, the fitting module (PCA + SH fitting) consumes only about 0.48s which is negligible for large Gaussian scenes.
CONCLUSION AND LIMITATIONS
We introduced a soometry-awarr submanifold field representation for 3D Gaussian Splatting that maps each primitive to a color field on an iso probbabbbiity ellipsoid and proved the mapping is in- jective over core attributes. Built on this representation, our SF-VAE learns semantically mean- ingful latents and yields higher-fidelity reconstructions and stronger zero-shot generalization than capacity-matcher raw -ww-aaraeterr baselines. Limitations and Outlook. Our current setup operates at the modeling for more complex representation learning.
level, while this ensures data invariance, it omits explicit inter-splat structure
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Published as a conference paper at ICLR 2026
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Reviews:
[{"review_id": "5FHdslVhXt", "reviewer": "Reviewer_kCGA", "summary": "This paper argues that the standard parameter-based representation for 3D Gaussian splatting is not a great fit for neural networks. The limitations in regard to non-uniqueness, numerical inconsistency, and mismatches with learning spaces. This leads to instability, poor generalization, and noise sensitivity. To address this, they propose to model each Gaussian as a continuous field on its iso-probability surface and discretized colored point cloud. This preserves geometry and appearance while ensuring uniqueness and consistency, leading to better learning performance.", "questions": "Please, conduct the ablation study, behavior study, and sensitivity study in a structured way. \nThe ablation study must ablate (remove) the components of the design choices (like the network architecture). \nThe behavior study must showcase how the method behaves over the variables (like the embedding size, training data).\nThe sensitivity study must showcase how the input fluctuations impact the output (like the noise study).\nAuthors are recommended to follow a more formal definition and comment on these modifications during the rebuttal. \nPlease also address the weaknesses reported above.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* The paper convincingly explains why Gaussian parameters are a poor representation for learning. The arguments about non-uniqueness and manifold mismatch are backed by valid theoretical analysis. The contents in sections 3.2 and 3.3 are the major strengths, which are generally convincing.\n\n* The qualitative and quantitative results support the claims about robustness and generalization.\n\n* The paper is generally well written and easy to follow. The graphical illustrations are particularly helpful. \n\n* The unsupervised graph clustering (shown in Figure 5) convincingly communicates how the submanifold field embeddings preserve detailed semantics.", "weaknesses": "The ablation study in regard to the proposed design (of the architecture of Figure 2) is missing. Currently, the Behavior/sensitivity studies of Figure 6 are presented as the ablation. The choice radar plot of Figure 6 (c) does not really help. The sensitivity comparison of Figure 4 is unfair because the embedding noise and parametric noise do not mean the same thing. It needs to be either justified in an interpretable way or otherwise must be removed to avoid misleading. The confusion on rotation space claimed with the example of Figure 3 is not very convincing, as the source of the artifacts can be anything. For example, among many arbitrarily oriented splats (from initialization), it could be due to not being able to optimize the opacity parameter. Since there could be many splats (which may effectively not contribute) with undesired rotations, their effect can be minimized by minimizing the opacity only.", "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": "pNFUYfUpdS", "reviewer": "Reviewer_WJ16", "summary": "Gaussian splats have been used for 3D scene reconstruction with a lot\nof success. Recent works explore mapping the GS parameters to another\nembedding space using neural networks to allow applications, such as\ngeneration, editing, compression, and even predicting GS parameters\nmulti-view image sets directly. Authors here focus on such mappings\nand note that the common GS parameterization is not optimal. The main\nreasons are non-uniqueness, different parameterization lead to the\nradiance fields, and numerical heterogeneity, different parameters\nhave different magnitudes and such magnitude heterogeneity makes\ngradient-descent based learning challenging. They propose another\nparameterization as well as an auto-encoder model that maps this\nparameterization to an embedding space and back. The representation in\nthe embedding space is claimed to be more suitable for neural network\ntraining. The model is trained using synthetically generated Gaussians\nand applied then to the ShapeSplat and Mip-NeRF 360 datasets. Both\nobject-level and scene-level experiments are conducted. Comparisons\nbased on similarity of rasterized images and Gaussian parameters are\npresented.", "questions": "+ The issue of heterogeneity of parameter magnitude is a problem of\n  optimization. I wonder whether optimizers that approximates second\n  order optimization, such as Adam, would be less susceptible to this\n  issue.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "+ Authors identify an interesting and important problem.\n+ The explanation of the problem is very well done.\n+ GS is a very relevant topic and the paper is timely.\n+ The proposed approach is sound and based on solid\n  theory. Proposition 2 is well appreciated.", "weaknesses": "+ The experiments on domain-shifts are not well explained. It is not\n  clear what is trained on the datasets and what is tested. I thought\n  the proposed method was trained using synthetically generated\n  Gaussians. This part needs to be explained better I believe.", "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": "X1lf1A2QT5", "reviewer": "Reviewer_Jmon", "summary": "This paper addresses key challenges of using parameter-based Gaussian representations as network inputs: non-uniqueness, numerical heterogeneity, and distinct mathematical manifolds. To overcome these issues while preserving the underlying geometric structure, the authors propose representing each Gaussian as a continuous field defined on its iso-probability surface. Experimental results demonstrate that the proposed representation achieves superior reconstruction quality and exhibits strong robustness to domain shifts compared to traditional parametric representations.", "questions": "- The decoding process involves batched PCA, eigen decomposition, and spherical harmonics fitting for every reconstructed Gaussian primitive. Could the authors clarify the computational cost of this step and whether it poses a bottleneck when decoding a large number of Gaussians?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper clearly explains the problem formulation and the motivations behind introducing a new representation for Gaussian primitives.\n- The idea of learning a discretized iso-probability surface as a geometric representation is interesting, and I believe it is also novel.\n- The writing is clear and effectively conveys the core ideas; the main figure is particularly helpful in illustrating the concept.\n- The proposed model can be trained using a randomly generated dataset, enabling domain-agnostic learning that can later be applied to embed optimized Gaussian primitives.", "weaknesses": "- While the paper aims to validate the idea of using iso-probability surfaces as an alternative representation of parameter-based Gaussian representations to make them more amenable to learning-based systems, it remains unclear how well this representation generalizes to various downstream tasks. From my understanding, one motivation for learning such representations is to enable applications such as compressing large sets of Gaussians or training generative models to sample new Gaussian primitives. However, the experimental results primarily demonstrate that the proposed representation can reconstruct Gaussian parameters more accurately in a controlled setting. This alone does not necessarily prove that the learned representation will be effective in downstream or scene-level applications.\n- Each iso-probability surface is independently encoded and decoded by the network, meaning that each latent embedding is learned in isolation without awareness of global scene-level context. This independence could limit the model’s ability to generate spatially consistent or semantically coherent Gaussian primitives when generating full scenes, which is an important consideration for practical use cases (e.g., scene generation).", "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": "GZIZpyB0MV", "reviewer": "Reviewer_jRea", "summary": "This paper addresses challenges in 3D Gaussian Splatting (3DGS) representations, where individual Gaussians have multiple parameter sets. The authors note that some parameters are non-unique (e.g., different rotation representations can yield the same state) and that parameter groups are from different distributions or manifolds. The authors claim this non-uniqueness and heterogeneity make the standard 3DGS parameter space ill-suited for neural networks tasked with generating or estimating Gaussians. To resolve this, the paper introduces \"submanifold field\" representations, a novel parameter space for these Gaussian features. The authors demonstrate the uniqueness of this new representation and propose a corresponding manifold distance (M-Dist).\nThrough experiments, the paper demonstrated superior performance against baselines.", "questions": "- To aid in understanding the concept, could the authors elaborate on the definitions and intentions behind the terms \"sub\" and \"field\" in \"submanifold fields\"?\n\n- While the paper discusses the ill-posedness of standard 3DGS features, is there a specific reason to formalize the non-uniqueness and heterogeneity so strongly through definitions and propositions?\n\n- In Section 4.2 (lines 362-364), it is stated that M-Dist has a higher correlation with the perceptual metric than L1. Could the authors also provide the correlation with L2? Furthermore, in Figure 4 (right table), the M-Dist for the SF embedding appears to increase significantly with noise. This seems to contradict the claim that M-Dist has a better correlation with perceived image quality than L1. Could the authors clarify this apparent discrepancy?\n\n- Regarding Figure 4 (robustness to noise): Since the different representations have different scales and properties, is it certain that adding uniform noise affects them in a comparable way?\n\n- For Figure 5 (interpolation): Would it be possible to include more straightforward or intuitive examples? From the current figure, it is unclear precisely how the SF-VAE interpolation is superior to the other methods.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- The paper presents an interesting and timely idea, given the growing interest in large-scale reconstruction models which could benefit from improved 3D Gaussian Splatting representations.\n- The proposed method demonstrates strong performance improvements over the baseline.\n- The authors provide a broad analysis, including evaluations of robustness, clustering, and interpolation.", "weaknesses": "**Motivation**: The motivation presented in the Introduction could be strengthened. The importance of resolving the identified issues should be emphasized more clearly, particularly by highlighting specific downstream applications and detailing why and when existing representations fail.\n\n**Experiments**: The current experiments focus on small MLP and VAE-style networks, which may not reflect real-world applications. Even if the goal is to isolate the effect of the new representation, it is unclear whether results from these small networks will generalize to the much larger, more complex models used in practice. Without testing or adapting existing large-scale reconstruction models, the broader applicability of the proposed representation remains uncertain.", "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)