Quartet of Diffusions: Structure-Aware Point Cloud Generation through Part and Symmetry Guidance
Authors: Chenliang Zhou
Fangcheng Zhong
Weihao Xia
Albert Miao
Canberk Baykal
Cengiz Oztireli
Venue: ICLR 2026 Poster (2026)
Keywords: computer vision
point cloud generation
structure-aware
part-based
symmetry-aware
Primary area: applications to computer vision, audio, language, and other modalities
TL;DR: We propose a structure-aware point cloud generation pipeline through part and symmetry guidance, guaranteeing symmetry and achieving SOTA results

Abstract: We introduce the *Quartet of Diffusions*, a structure-aware point cloud generation framework that explicitly models part composition and symmetry. Unlike prior methods that treat shape generation as a holistic process or only support part composition, our approach leverages four coordinated diffusion models to learn distributions of global shape latents, symmetries, semantic parts, and their spatial assembly. This structured pipeline ensures guaranteed symmetry, coherent part placement, and diverse, high-quality outputs. By disentangling the generative process into interpretable components, our method supports fine-grained control over shape attributes, enabling targeted manipulation of individual parts while preserving global consistency. A central global latent further reinforces structural coherence across assembled parts. Our experiments show that the Quartet achieves state-of-the-art performance. To our best knowledge, this is the first 3D point cloud generation framework that fully integrates and enforces both symmetry and part priors throughout the generative process.

Published as a conference paper at ICLR 2026
QUARTET OF DIFFUSIONS: POINT CLOUD GENERATION THROUGH PART AND SYMMETRY GUIDANCE
STRUCTURE-AWARE
Chenliang Zhou, Fangeheng Zhong, Weihao Xia, Albert Miao, Canberk Baykal, Cengiz Oztireli Department of Computer Science and Technology, University of Cambridge
ShapeGiF
DPC
LATA-CD
PVD
PointElow
LION
Quartet Diffusions
FrePelad
DCT-3D QQaattet
+400
680
Shape latent diffusion
Symmetry diffusion
Part diffusion
Assembler diffusion
1900
ShopeGit
Peinth]om
PVD DPC
100
DET33D
Quarnet
100
200
300
400
500
600
Training (bours)
Figure 1: Our method, the Quartet of Diffusions, harmoniously orchestrates interpretable and controllable pipeline for suucture-aware, high-quality, diverse, and efficient point cloud generation guided by part and symmetry clues.
ABSTRACT
We introduce the Quartet of Diffusions, structure-aware point cloud generation framework that explicitly models part composition and symmetry. Unlike prior methods that treat shape generation composition, our approach leverages four coordinated diffusion models to learn distributions of global shape latents. symmetries, semantic parts, and their spa- tial assembly. This structured pipeline ensures guaranteed symmetry, coherent part placement, and diverse, high-quality outputs. By disentangling the generative process into interpretable components, method supports fine-grained control over shape attributes, enabling targeted manipulation of individual parts while preserving global structural consistency. A central global latent further reinforces structural coherence across assembled parts. Our experiments show that the Quar- tet achieves ofate-of-the-art performance in generating high-quality and diverse point clouds while maintaining symmetry. To our best knowledge, this is the first 3D point cloud generation framework that fully integrates and enforces both sym- metry and part priors throughout the generative process.
holistic process or only support part
INTRODUCTION
Structure-aware 3D shape generation aims model not only the holistic surface geometry of ob- jects but also the underlying organizational principles that govern their composition and forma- tion (Chaudhuri al. 2020). A truly structure-aware representation must comprise two essential components (Chaudburi et al. 2020): the geometry of atomic structural elements (e.g., low-level
"Corresponding author

---

Published as a conference paper at ICLR 2026
parts) and the structural patterns that dictate how these elements are assembled into coherent shapes or scenes. These patterns may manifest as hierarchical decompositions, part-whole relationships, spatial composition graphs, or local and global symmetries.
In this work, we focus on 3D point cloud generation with explicit awareness of two fundamental structural priors commonly observed in natural and man-made objects: part awareness and sym- metry awareness. Part awareness captures the notion that objects are composed of semantically meaningful, interrelated components with geometrically distinct properties. Symmetry awareness, in contrast, models the intrinsic geometric regularities-such as reflective or rotational symme- often govern the spatial arrangement of parts. Importantly, part awareness serves as a prerequisite for identifying and enforcing local symmetries, which typically operate at the level of inter-part relationships. Without symmetry, shapes risk appearing disjointed or unnatural, under- mining their functional performance in practical applications (e.g. some prior work in Fig.
Compared to unstructured 3D data, structured representations offer substantial advantages. They not only enhance the visual plausibility and structural coherence of generated shapes, but also facilitate downstream tasks such as part segmentation, alignment, structure inference, and fine-grained shape editing (Schor et al. 2019 Li et al. 2020; Hertz et al. 2022, Koo et al., 2023). However, existing generative approaches largely remain constrained distribution learning problem. These methods often prioritize global visual fidelity and geometry-level details in the distribution, while neglecting the explicit modeling of inherent struc- ture (Vahdat al., [0222] Yang et al.] [019 Mo al., 2023 Zhou et al., 2021a). As result, they tend to produce shapes that suffer from poor part organization, broken or inconsistent symmetries, and limited generalization to unseen or complex object categories (Li et al. 2020). While few re- cent efforts support part-level generation, they rarely incorporate symmetry as a learned, relational prior. To date, there remains a notable gap: the absence of unified, principled frameworks that jointly encode part-level composition and symmetry structures to guide the generation of coherent, interpretable, and generalizable 3D shapes.
- treating point cloud synthesis as an un-
To address this, we propose the Quartet of Diffusions (the Quarter), a structure-aware point cloud generation framework that explicitly leverages part and symmetry guidance. Specifically, our pipeline employs four diffusion models (Ho et al. 2020) to learn the distributions of shape latents, symmetries, semantic parts, and assemblers that assemble the full point cloud from parts. By explic- itly modeling these structural distributions, the ensemble of four diffusions harmoniously orches- trates an effective pipeline for structure-aware point cloud generation through part and symmetry guidance: Our method enables the generation of high-quality, diverse point clouds with guaranteed symmetry. The disentangled structural representation makes the generation process interpretable and controllable, facilitating targeted modifications individual parts while preserving global con- sistency. Structural coherence is further reinforced by a global shape latent, which anchors part assembly to the overall geometry. This approach addresses key limitations in structure modeling and represents the first 3D shape generation method to guarantee symmetry in the generated shapes.
2 RELATED WORK
Symmetry in 3D shapes
Symmetry is fundamental geometric property observed across natural
and human-made objects. Extensive research exists on symmetry detection (Mitra et al.] 2013a
Atallah| 1985 Mingwwrth & Kittler 1988 Mitra et al. 20066; [Je et al. 2024 Zhou et al. 202 lb
Gao et al., 2020 Fukunaga & Hostetler 1975) Comaniciu & Meer 2002) (for more details, see
Appendices and B.I.. Symmetry clues are widely used in various 3D vision tasks including ac-
quisition and representation (Yang et al. [ et al. 2008 Li et al. 2010 Pauly et al. 2005
Thrun Weghreit 2005 Xu et al. [2099] Zheng et al. (010), classification (Kazhdan et al. 2004
Martinet et al., |0066 Podolak et al. 2006), perception (Reisfeld et al.) 1995). manipulation (Mi-
tra et al., 2007; Podolak et al. 2007 Panozzo et al. 2012 Gal et al. 2009) Mehra et al. 2009).
reconstruction (Phillips et al. 2016 Xu et al. 2024 Tulsiani et al. 2020), inverse rendering (Wu
ct al. 2020b), and refinement (Mitra et al. (2007). However, symmetry aware 3D shape generation
remans underexplored. Aside from our method, few approaches guarantee symmetry. PAGENet (Li ct al.) 2020) promotes symmetry via an MSE loss between a shape and its reflection, but it is limited
to reflectional symmetry and does not guarantee it.
2

---

Published as a conference paper at ICLR 2026
Part-based 3D shape generation Part-based 3D shape generation leverages the modular struc-
ture of objects (Mitra et al.] [2014] 2013b) and emerges as a vital paradigm for modeling complex
geometries Zerroug & Nevatial 1999; Kim et al. 2013), enhancing diversity (Schor et al.] 2019
Chen et al. 2024) and facilitating downstream tasks including recognition (Hoffman & Richards
1984). retrieval (Chang et al., 2015 Mitra et al. 2014), and manipulation (Huang al. [2014). Early
methods focus on geometric template fitting or hierarchical part assembly (Funkhouser et al.
2004
Bokeloh al., 2010 Kalogerakis et al., 2012 Berthelot al.] 2017 Cohen-Or & Zhang
2016
Fish et al.; 2014). More recent work harnesses implicit shape representations (Koo et al.,
2023
Hertz et al. 2022; Talabot et al.) 2025 Hui et al. [022] Genova et al. 2019 2020). For example,
PartSDF (Talboot et al. 2025) models parts using implicit neural fields, enabling both continuous
interpolation and discrete composition.
Deep neural networks are widely adopted for part-based shape modeling (Schor et al.
Li et al. 2019
2020 Dubrovina et al., 2019; Li et al.] 2024| Wu et al., 2020a Huang et al. 2015 et al. 2017
Mo et al. 2019 Zou et al. [017 Nash & Williams 2017 7, Wu et al. 2019 Wang et al. 2018 2019
Gao et al. 2019). ). For instance, CompoNet (Schor et al. 2019) enhances diversity by varying both parts and their compositions, while PASTA (Li et al. 2024) generates shapes conditioned on part arrangement for fine-graing une-grained control. Similarly, our Quartet performs structure- awarre 3D generation by modeling both parts and their compositions, with the added benefit of symmetry enforcement.
Diffusion models Diffusion models (Ho al., 2020) are generative models based on Markovian
diffusion processes (see Appendix [B33 for more details). They have demonstrated remarkable per-
formance various domains (Croitoru al., [2023 Cao al. [2024] Yang et al. [023), particularly
in image (Dhariwal & Nichol, 2021; Ho et al. 22200 Rombach et al., [022 Ramesh et al.) [022]
Brooks et al. 2023 Saharia et al. (002)) speech (Chen et al. 2020, leong et al. |021] Liu et al, 2022). video (Ho et al.
2022
Xing et al. 2024 Luo et al.) 2023 Yang et al. 2023). 3D scene (Wei
ct al. 2023 Zhai et al., 2023
Tang et al. 2024), and 3D object generation (Zhou et al. 202 la Luo
& Hu 2021 I Vahdat et al. 2022; Nakayama et al. 2023 Wu et al. 2023, Mo et al. 2023 Liu al.)
2019 Zhou et al. 2024; Koo et al. 2023). Accordingly, our Quartet recruits four diftusion models
to learn the distributions of shape latents, symmetries, parts, and assemblers.
3 METHOD
We aim to model the distribution of point clouds x E X' S. R3xN as that over a collection of semantic parts {D through part and symmetry guidance. Specifically, we view each point cloud x U T,P, as a composition of its M semantic parts and their corresponding assemblers. Each assembler T3 is a transformation comprising translation, rotation, and scaling in 3D Euclidean space, mapping the part P, to the correct position, orientation, and scale in the original point cloud.
The Quartet models four distributions using parameterized diffusion models: 1. point cloud shape point cloud parts Pe (D, Sj, (Sec. 3.3 , and 4. assemblers p* (T) w Dj. S,, z). additionally conditioned on
latents Pe (z) (Sec. 1.1. 2. part-wise symmetries Pc (S, z) (Sec. (.2).
part latent wj (Sec. 3.4 These components collectively define the Quartet of Diffusiooss architec-
ture. Pipeline overviews are shown in Figs. and
Modeling point cloud distributions in this way offers several key advantages: 1. It introduces vari- ability at both the shape and assembly levels, enabling combinatorially greater diversity; 2. By explicitly generating and enforcing symmetry early in the pipeline, the model produces outputs with more realistic and consistent symmetric properties; 3. The separate modeling of symmetries, parts, and assembly allows for better interpretability and fine-grained control individual parts can be manipulated independently without compromising global structure; and 4. A central shape latent z ensures structural coherence across all components. These benefits are demonstrated through exten- sive experiments in Sec.
The Quartet is trained by sequentially optimizing the parameters of four distributions to fit the given dataset X. Point cloud generation is performed in two phases: part generation and part assem- bly. In part generation, a shape latent z is first sampled from the latent distribution pe (z). Con- ditioned on z, symmetry groups S; are sampled from Pç (Sj z), and parts Pj are drawn from Pe (p.) S) for j = ..., M; In part assembly, parts P, encoded into latent representa-
3

---

Published as a conference paper at ICLR 2026
part latent
SVAE decoder
symmetry
<
part diffusion
assembler Y diffusion
diffusion
assembler
shape latent diffusion
fundamental domaain
part encoder
Shape latent
translation rotation scaling
one wheel
half hood
half body
SVAE
fourlshhels full hood
full body
encoder
symmetty group
full car
Figure 2: Overview the Quartet's architecture. Four diffusions are employed to learn the distri- butions of shape latents, symmetries, parts, and assemblers. By explicitly modeling different distri- butions, the Quartet provides an interpretable and controllable framework to generate high-quality, diverse 3D shapes with guaranteed symmetry. Beige blocks denote learnable modules; gray blocks indicate outputs directly generated from the models.
tions w, using an encoder ls (w, Pj. z). and the corresponding assemblers T₃ are sampled from Pv (T, wy, "y. Sj. 2). These assemblers are then applied to the parts to reconstruct the full point cloud
M
UTP,
1=0
Mathematically, the generation process thus be formulated as
p (1(p,, T,, w;) |-=== = po (z) ((( S, w;) w;)) x) po (z) II I(,, T, [z) (2)
M
po (z) II P< (S, 2-d
P) (p, Sy, z) as (w, P,. Pe (T' W.
where the second equality in (2) follows from the conditional independence parts given the shape latent Therefore, the overall point cloud distribution can be modeled as
(x) (1 T))) ff... [(,
A6 [I ds,
M II
dw;
dz
(4)
1=1
[ Pe (D, 16, do (w, |P,, ps (Tr W,,,,, S,, dS, dS,dw, dz.
po (z) II
(5)
3wD
3.1 SHAPE LATENT DIFFUSION
Although we model each distribution separately, they must remain aware of the underlying point cloud to coordinate coherently. Directly conditioning on the full point cloud is computationally expensive due its high dimensionality. Instead, we condition them on shape latents obtained from
variational autoencoder (VAE) (Kingma & Welling, 2013 Zhou et al. 2024 Vahdat et al. 2022) (see Appendix B.2 for background).
To effectively encode point clouds while preserving semantic information, we propose novel sparse variational autoencoder (SVAE) with latent diffusion modeling its latent distribution. SVAE builds on VAE architecture implemented with point-voxel convolutional neural networks (PVCNNs) (Liu et al., 2019, Zhou et al. (021a)). drawing inspiration from sparse autoencoders (Ng et al. 201T). Prior work suggests that enforcing sparsity in activation layers improves interpretabil-
ity (Cunningham et al. 2023, Makelov et al. 2024 Marks et al. 2024) and semantic disentangle- ment (Bricken et al.) 2023] O' Neill et al. 2024), which is critical for our downstream tasks. To this
end, we introduce a sparsity constraint on the final activation layer a., (x) of the encoder (z x)
while maximizing the VAE evidence lower bound (ELBO) CELBO (Eq. (24) in Appendix B.2
max Ex [.EBB (7) x)] subject to Exx (x)]l,
(6)
4

---

Published as a conference paper at ICLR 2026
where 8 >> 0 controls the sparsity strength. Leveraging Karush-Kuhn Kuhn-Tuckke (KKT) methods (Kuhn
& Tucker 1951] Karush| [139), we reformulate the constrained optimization problem into a La-
grangian F F(7, x: whose optimal point is global maximum the domain of n and obtain its
lower bound:
F(n xx = LELBO (7: - X (x)a, - 2 CELOO (7 x) - A (x) =: (1;
(7)
(8)
where the KKT multiplier. The SVAE is trained by maximizing this lower bound LsVAE- The hyperparameter À controls the trade-off between reconstruction quality and sparsity, with À recovering the standard ELBO.
While a simple Gaussian prior is commonly used for VAE latent distributions, evidence suggests that such restricted prior cannot accurately capture complex latent distributions (i.e. the prior hole problem (Vahdat et al., 2021: Zhou et al., 2023)) and degrade VAE performance (Chen et al. 2016). To address this, we follow prior work (Zhou et al.) 2024) and employ a diffusion model learn the latent distribution. Once trained, the shape latents can be directly sampled from it. This design enables effective and interpretable feature extraction via SVAE while enhancing latent expressiveness and maintaining runtime efficiency.
po (z), implemented with U-Net backbone,
3.2 SYMMETRY ENFORCEMENT
The Quartet guarantees symmetry by explicitly learning its distribution. Once learned, the model
can generate and enforce appropriate symmetries during point cloud generation. For a 3D shape P. finite group of rigid transformations S (T) S E(3), generated by { S2 ..... S J. said to be a symmetry group over a subset d C p if the full shape can be reconstructed by Sd, the application of S d, which is defined be the union of all images under the transformations in S:
U Sd P. SES
(9)
sd
where the group generation is defined as
I. S. E T.ET.nET
(10)
The minimal such subset d is called the fundamental domain for S. Figure domains for various parts in point cloud airplanes, cars, and chairs.
3 Fillustrates fundamental
As translational symmetry implies an infinite shape, for tractability we restrict symmetry groups to those generated by at most two transformations - reflections or rotations with angles 2 a IV such
that 2x E Z. By a special case of the Cartan-Dieudooné theorem GGallier & Gallier 2011 ) any 3D rotation be expressed
composition of two reflections across planes intersecting along the axis of rotation, where the rotation angle is twice the angle between the two planes. Consequently, we just need to search for the symmetry groups generated by at most three reflections. Each reflection represented using the Hesse normal form (Böcher 1915 Duda & Hart 1972). enabling efficient and parallelizable symmetry search (Je et al., 2024). When multiple symmetries exist, we select the one corresponding to the fundamental domain with the smallest cardinality. In practice, if a symmetry group is generated by fewer than three reflections, we pad the remaining slots with a special symbol to maintain consistent data dimensionality.
Since each part generated separately, symmetry enforeed at the part level. During part gener- ation, sample only the fundamental domain d and recover the full part by applying the learned symmetry group as defined Eq. (9). This approach reduces the number of points to generate and, more importantly, guarantees symmetry in the resulting shapes.
To learn the distribution of symmetry groups present in the dataset, we first construct a metric space (M, dvv) of reflectional symmetries following Je al. (Je et al. 2024). We then obtain the ground truth for symmetry groups S using mean-shift clustering (Mitra et al. 22066] Fukunaga & Hostetler, 1975 Comaniciu & Meer 2002), nonparametric method based gradient ascent over density
5

---

Published as a conference paper at ICLR 2026
function in M (see Appendix B.1 for more details). Finally, we leverage diffusion model (Ho et al. 2020 to learn p (S).
More specifically, we apply the diffusion process (Eq. 26 in Appendix B.3 to S, and note that each
transition kernel q(S; S ~ (4,S,-1,0,11 is Gaussian. Therefore, intermediate samples S can be expressed in closed form
S,
11
Pr
So
Σ
II jmi+1
(11)
where N (0, I). Setting Ae and 7
Ei= of simplifies this to S So + n.c. Thus,
the distribution of S becomes a convolution:
p (S.) (po (o.)) (S.) po (S) Pa.n (S. S po (S) Ps (S.)
(12)
where po the distribution of So, (PS, is the probability density function for N (S, mtl) and
denotes convolution. With this, we train a diffusion model (Ho et al. 2020) sc (S,t) parametrized
by approximate the time-dependent score function (Song et 2020b) (i.e. the gradient of the log-density of noisy data) via an empirical approximation (Je et al.
2024):
[EREM PR.v. (S) R
Jc (S, ~ V log (())
5)
(13)
TF ΣREM PR. (S)
Since symmetry can vary across shapes, practice, we condition the diffusion model on the shape latent modeling the symmetry group distribution as Pc (S z).
After training, symmetries can be sampled from *c (S, t) through annealed stochastic gradient Langevin dynamics (Song & Ermon 2019;: We initialize Sf N (0. I) and sequentially sample from nerturbe distributions $ç (8,t) for t = J, - 1...., using L Langevin steps:
5
-
S(L) otherwise
(0)
ift =T
(14)
s s(1+1) <- s( Bisc (8)),1) 23,,,, (8) i= 0..,....! L,
(15)
(1) . (0, 1), and B is the step size. The final sample s/1) is the generated symmetry.
where e
Notably, substituting Eq. 13 into Eq. 15 and setting Br = 22 yields the following update rule:
s(1t1) - s. 81
ETEMPTT (5()) T
5!(0)
+ 2B Peuo? ΣTEM (s!5) T
PT
222ne
ErEM PT
(5(0)
EreM PT
(S(1)
(16)
This update rule resembles mean-shift clustering (Eq. infinite neighborhood B(S.) M and uses Gaussian kernel with bandwidth 7x:
22 in Appendix B. except it assumes an
1
1
Ki(S)
(17)
V
Another difference is the injected noise 22 in Fq. 16 which has been shown to improve ro- bustness, sample quality, and mitigate mode collapse (e.g. Je al. 2024).
Figure 3 shows examples of identified symmetry group generators. Most parts exhibit symmetry of a single reflection. In addition, we note that classical symmetries, such as reflectional and rota- tional, are special cases of our formulation, where the symmetry group S is generated by single transformation. Our approach generalizes this by allowing symmetries composed of multiple trans- formations, such sequential reflections (e.g., chair legs in (c)) rotation followed by reflection (e.g., car wheels). While our model does not explicitly capture circular symmetry, can be approx- imated using discrete rotations with minimum angle of (e.g., chair seat in (f)).

---

Published as a conference paper at ICLR 2026
point point clouds
Full
Fundamental domains
Symmetry
(a)
(c)
(d)
(e)
groups
airplane
chutr
body: Refl
sunroof: Ref whoels: Rot(1), Ref hood: Ref body: Ref
(e): backrest: Ref legs: Ref, Ref seat: Ref, Rou
wings: Ref engines: Ref
(f): legs: Roug
seat: Rott
Figure 3: Point cloud airplanes, cars, and chairs with identified symmetry groups and corresponding fundamental domains for each color-coded part. Ref denotes reflection; Rot(a: denotes rotation by
angle a. Our symmetry formulation allows greater flexibility by supporting symmetries composed of multiple transformations, such as two reflections (chair legs (c)) or rotation followed by reflection (car wheels). Circular symmetry is approximated via small-angle rotations (chair seat (f))-
3.3 PART GENERATION
Given the identified symmetry groups S, and corresponding fundamental domains d, for each part Pj. we model the part distribution via the distribution of fundamental domains. Since each full part
Pj can be reconstructed by applying Sj to d, (Eq. (9)), we have
Pe (D, SS 2) - pe (dl Sz,
(18)
To learn Pg (d, | S). we employ transformer-basee diffusion model (Mo et al. 2023 Peebles & Xie. 2023 Vaswani al. 2017), conditioned the shape latent and symmetry group Sj- The model operates directly on voxelized point clouds, using 3D positional and patch embeddings to capture spatial context. Unlike U-Net architectures (Ho al. 2020), which face limitations in scalability and spatial coherence, our transformer backbone incorporates 3D window attention, reducing computational cost while preserving local structure. This design yields higher efficiency, scalability, and fidelity in point cloud generation (Mo et al. 2023).
3.4 PART ASSEMBLY
To assemble the full point cloud, we learn assemblers Tj that correctly transform each part Py- We first train a part encoder Usb (w, Py: 2) to obtain compact representations W then train diffusion model Pyb (T, Wj.Py.-J.. z) model the assembler distribution Pvp (T, W Sy. z). To simplify learning, we assume that w, captures all necessary information about p, and Sj. making T3 conditionally independent of them given Wj. Consequently, the diffusion model is conditioned only on w, and the global shape latent z:
P& (T) W) P4p
Wy.
(19)
This assumption decouples geometric modeling from transformation learning, reducing complexity and allowing the model to focus on learning flexible, generalizable spatial configurations.
Our assembler diffusion model (Tang et al. 2024) employs U-Net backbone (Ronneberger et al.) 2015) augmented with skip connections and cross-attention layers (Vaswani et al. | 2017). The cross- attention modules integrate contextual information from both part latents and the global shape latent, enabling the model to generate transformations that are coherent at both local and global levels. This architecture effectively handles diverse part configurations and generalizes well to novel shapes. The hierarchical structure of U-Net further supports multi-scale spatial reasoning, which is essential for accurate 3D part placement.

---

Published as a conference paper at ICLR 2026
Table Quantitative comparison of point cloud generation. PA denotes part awareness; SA denotes symmetry awareness. Our Quartet is the only model that supports both, achieving significant im- provements over most baselines and setting a new state of the
Airplanc
Car
Chair
Model
SA
I-NNA (1)
SDI (1)
I-NNA (1)
SDI (L)
I-NNA ())
SDI (J)
CD
EMD
CD
EMD
CD
EMD
CD
EMD
CD
EMD
CD
EMD
Training set
64.4
64.1
0.954
4.90
51.3
54.8
7.49
1.18
51.7
50.0
5.56
1.68
PointFlow
Yang ct al.
2019
X
75.7
70.7
3410
782
62.8
60.6
679
347
58.1
56.3
7290
530
ShapeGF
Cal ct 2020
X
81.2
80.9
332
98.6
58.0
61.3
645
40.9
61.8
57.2
1100
101
Net
Klokov
2020
75.2
65.6
4256
245
58.5
827
452
62.4
54.5
5234
245
SetVAE
Kim
76.5
67.7
2830
824
58.8
60.6
1240
327
59,9
5320
673
DPC (Luo & HD, 2021
X 76.4
86.9
187
44.2
60.1
74.8
217
30.3
68.9
80.0
335
50.6
PVD 7ibo of al. A02
X 73.8
64.8
150
42.0
56.3
53.3
213
31.2
54.6
53.8
275
58.4
LION Vatiaal alr A022
X
67.4
61.2
97.2
40.6
53.7
52.3
168
30.8
53.4
51.1
201
55.2
SPAGHETTI (Hert
2022
78.2
77.0
1530
529
72.3
71.0
581
284
70.7
69.0
5930
582
DiT-3D
(Mo 20755
64.7
105
42.4
52.7
50.2
206
327
52.5
53.1
235
49.0
SALAD KDo 20233
73.9
71.1
198
45.1
59.2
57.2
236
29.4
57.8
58.4
308
52.6
FrePolad Zhou 2024
X
65.
62.1
94.1
38.1
52.4
53.2
173
29.6
51.9
50.3
252
50.9
Quartet (ours)
V
63.3
59.7
25.7
1.87
50.1
518
25.7
2.28
51.6
53.7
28.9
2.86
For the part encoder, adopt the SVAE architecture (see Sec. 3.1 with equivariance fine-tuning (EFT): During training, random rigid transformations, including translations, rotations, and reflec- tions, are applied to each part, and the encoder is encouraged to produce the latent transformed accordingly. This promotes geometry-aware but pose-invariant part embeddings, enhancing their suitability for the part assembly task. As result, the diffusion model receives more stable and semantically meaningful latents, leading to more accurate and coherent assembler predictions.
EXPERIMENT
4.1 DATASET
We evaluate our method on the ShapeNetPart dataset Yi et al., 2016), subset of ShapeNet (Chang af., (2033) with semantic part annotations. We focus on three representative categories: airplanes (body, tail, wings, engines), cars (sunrooff, wheels, hood, body), and chairs (backrest and armrests, legs, seat). Segmentation examples are shown in Fig. 3 We use the official train/validation/tss split provided with the dataset.
Following common practice, we use point clouds with 2048 points. Each part is resized to match the mean number of points per part within its category. During preprocessing, parts with more points are randomly subsampled, while those with fewer are upsampled via random duplication. This normalization ensures consistent input dimensions across training batches, enhancing the stability and efficiency of both the encoder and generative models.
4.2 EVALUATION METRICS
Following prior work (e.g. Zhou et al. 2024). assess the quality and diversity of generated point clouds using 1-nearest neighbor (I-NNA) (Lopez-Paz & Oquab] 2016). a retrieval- based metric computed with Chamfer Distance (CD) and Earth Mover's Distance (EMD). 1-NNA measures how often shape's 's nearest neighbor belongs to the same distribution (Yang et al. 2019). A balanced score near 50% indicates close alignment between generated and real shape distributions.
To assess whether a generated 3D shape is symmetric, we introduce the symmetry discrepancy index (SDI), which quantifies how well 3D shape p conforms to given symmetry group S: For normalized shape SDI is defined as the distance d - either CD or EMD between p and the shape reconstructed from its fundamental domain d under
(20) Lower SDI values indicate stronger symmetry. For methods without explicit symmetry modeling, we compute SDI using the simplest yet most common symmetry: reflection across the vertical bisector plane. For readability, report SDI-CD scaled by 10 and SDI-EMD scaled by 103.
(SDI(P) d Sd).
8

---

Published as a conference paper at ICLR 2026
Table Per-part SDI-CD (L) for point cloud generation. With explicit symmetry enforcement, the Quartet achieves significantly lower SDI scores, indicating stronger symmetry in generated parts.
Airplanc
Car
Chair
Model
body
tail
wings
engines
roof
wheels
hood
body
back
legs
seat
Training set
0.877
0.700
0.883
0.717
3.06
3.15
5.27
4.11
2.65
3.04
SPAGHETTI Hertz et al. 2022
318
268
125
258
339
127
417
2857
2948
1358
SALAD (Koo al. A025
26.2
30.5
18.5
68.5
256
163
93.5
104
Quartet (ours)
7.86
4.08
7.76
5.84
9.72
4.10
5.76
9.90
10.3
6.19
10.4
PVD
LION
DiT-3D
SALAD
FrePolad
Quartet (ours)
Figure 4: Point cloud generation. Samples from the Quartet are visually appealing, diverse, and exhibit strong structural consistency. The last three columns illustrate targeted manipulation.
4.3 POINT CLOUD GENERATION
We benchmark the Quartet against several competitive 3D generative models. Implementation and training details are provided in Appendix C and additional experimental results can be found in Ap- pendix D Quantitative results based on I-NNA and SDI reported in Tab Notably, the Quartet is the only method that explicitly models both part structure and symmetry. Per-part SDI scores for part-aware models are presented Tab. The Quartet consistently outperforms state-of-the- methods, achieving high fidelity and diversity while closely matching the real shape distribution. Remarkably, due to explicit symmetry enforcement, the Quartet achieves significantly lower SDI scores, indicating strong alignment with ideal symmetric structures. Qualitative results in Fig. g show that the Quartet generates visually coherent and diverse point clouds obeying symmetry across all three object categories, demonstrating its effectiveness.
Targeted manipulation The Quartet models the distribution of each part separately, enabling tar- geted lert-level manipulation while guaranteeing disentanglement. In Fig. [ and in the last three columns of Fig. 4 we present point cloud samples in which the yellow-highlighted parts vary while remaining structure is held fixed.
4.4 ABLATION STUDY
Our Quartet consists of four diffusion models responsible for shape latents, symmetries, parts, and assemblers. As an ablation study, in Tab. we evaluate several simplified variants with some diffusions or key techniques removed:
NoSVAE: replaces SVAEs (Sec. 3.1 with standard VAEs for full point cloud and part encoding.
NoEFT: removes the equivariance fine-tuning (Sec. (.44 applied to the part SVAE.
Trio Var. I: replaces shape latent diffusion with a simple Gaussian prior;
Trio Var. 2: removes the symmetry diffusion with part diffusion generating the full part directly;
Duet Var. 1: removes both shape latent and symmetry diffusions;
Duet Var. 2: removes symmetry diffusion and merges part and assembler diffusions into single
full-shape diffusion:
Solo: uses single diffusion generating full point clouds.

---

Published as a conference paper at ICLR 2026
Table Ablation study. denotes latent diffusion, S symmetry diffusion, and part and assembler diffusions. All four members in the Quartet are essential; removing any degrades performance.
Airplane
Car
Chair
Variation
L
S
I-NNA (4)
SDI (L)
1-NNA (1)
SDI (4) EMD
I-NNA (4)
SDI (L)
CD
EMD
CD
EMD
CD
EMD
CD
CD
EMD
CD
EMD
Solo
X
X
69.2
64.2
154
44.2
55.5
53.6
241
351
53.2
53.8
295
52.8
Duet Var.
X
/
76.3
82.1
927
614
70.2
72.9
522
460
68.3
67.2
3261
615
Duet Var. 2
/
X
X
66.8
63.3
103
42.4
52.3
52.7
192
28.1
53.2
51.6
184
49.2
Trio Var.
X
V
85.1
85.8
3516
1450
83.9
83.6
836
326
92.5
85.2
6286
562
Trio Var.
V
63.6
95
39.2
32.3
52.3
53.9
174
52.5
NoSVAE J / 64.3
61.8
25.2
2.51
51.6
52.0
25.9
2.55
52.6
53.9
29.3
3.15
NoEFT
< V
63.9
62.4
32.7
9.27
51.8
52.0
27.2
5.98
52.5
54.0
30.2
4.15
Quartet
V
63.3
59.7
25.7
1.87
50.1
51.8
25.7
2.28
51.6
53.7
28.9
2.86
Our results show that all four diffusion models are indispensable members of the Quartet, each con- tributing to the generation of high-quality, diverse point clouds with strong symmetry. Two key observations emerge: First, comparing Solo and Duet Var. 1, we see that generating parts without central shape latent significantly degrades performance. This echoes previous findings that simple Gaussian prior is insufficient for capturing the structural complexity of 3D shapes (Vahdat et al., 2021 Tomezak & Welling, 2018: Rosca al. 2018 Zhou et al., 2024 Vahdat al.j 2022). The performance drops further in Trio Var. I, where symmetry diffusion is added, likely compound- ing the mismatch between prior and part representations. Second, comparing Trio Var. 2 and the full Quartet, we observe that while symmetry enforcement substantially improves SDI, it does not consistently improve I-NNA even with latent diffusion present. This may be due to the added constraint increased learning complexity introduced by enforcing symmetry.
4.5 RUNTIME ANALYSIS
Training the Quartet on each object category takes approximately 50 hours on a single GPU. GPU.Figure presents the generation performance vs. training time for models trained on the airplane category. Some baselines require longer training due to joint optimization Yang et al. 2019). operation on full point clouds (Zhou et al. 2021a]] Luo Huj, 2021). the use complex, latent spaces (Vahdat et al.) 2022). 2. In contrast, despite using four diffusions, the Quartet trains faster than most baselines. This is because the shape latent, symmetry, and assembler diffusions operate on low-dimensional representations (124-, 12M and 3MM-dimnsiional respectively, where M is the number parts). The majority of training time is spent on part diffusion, which is accelcrated using efficient transformer-base diffusion (Mo et al. 2023) applied to fundamental domains that are typically only half or a quarter the size of the full parts.
dimensioon
5 CONCLUSION AND DISCUSSION
We presented the Quartet of Diffusions, a structure-awar framework for 3D point cloud generation that explicitly models part composition and symmetry. At its current stage, work focuses unconditional generation, but it highlights the importance of integrating symmetry- and part-based reasoning into structure-aware models. The model's ability disentangle and manipulate individual parts makes well-suited for interactive shape editing and user-guided design applications.
10

---

Published as a conference paper at ICLR 2026
REFERENCES
Panos Achlioptas, Olga Diamanti, Ioannis Mitliagkas, and Leonidas Guibas. Learning representa- tions and generative models for 3d point clouds. In International conference on machine learning, pp. 40-49. PMLR, 2018.
Atallah. On symmetry detection. IEEE Transactions on Computers, 100(7):663-666, 1985.
David Berthelot, Thomas Schumm, and Luke Metz. Began: Boundary equilibrium generative ad- versarial networks. arXiv preprint vrXiv:1703.10711 2017.
Maxime Böcher. Plane analytic geometry: with introductory chapters the differential calculus. H. Holt, 1915.
Martin Bokeloh, Michael Wand, and Hans-Peter Seidel. A connection between partial symmetry and inverse procedural modeling. ACM SIGGRAPH 2010 papers, pp. 1-10. ACM New York, NY, USA, 2010.
Trenton Bricken, Adly Templeton, Joshua Batson, Brian Chen, Adam Jermyn, Tom Con- erly, Nick Turner, Cem Anil, Carson Denison, Amanda Askell, Robert Lasenby, Yifan Wu, Shauna Kravec, Nicholas Schiefer, Tim Maxwell, Nicholas Joseph, Zac Hatfield-Dodds, Alex Tamkin, Karina Nguyen, Brayden McLean, Josiah Burke. Tristan Hume, Shan Carter, Tom Henighan, and Christopher Olah. Towards monosemanticity: Decomposing language models with dictionary learning. Transformer Circuits Thread, 2023. https:/ttransformer html.
Tim Brooks, Aleksander Holynski, and Alexei Efros. Instructpix2pix: Learning to follow image editing instructions. In Proceedings of the IEFF/CVF Conference on Computer Vision and Pattern Recognition, pp. 18392-18402, 2023.
Antoni Buades, Bartomeu Coll, and Jean-Michel Morel. Nonlocal image and movic denoising. International journal of computer vision, 76: 123-139, 2008.
Ruojin Cai, Guandao Yang. Hadar Averbuch-Elor, Zekun Hao. Serge Belongie, Noah Snavely, and Bharath Hariharan. Learning gradient fields for shape generation. In Computer Eision-ECCy 2020: 16th European Conference, Glasgow, UK, August 23-28, 2020, Proceedings, Part III 16, pp. 364-381. Springer, 2020.
Hanqun Cao, Cheng Tan, Zhangyang Gao, Yilun Xu, Guangyong Chen, Pheng-Ann Heng, and Stan Li. A survey on generative diffusion models. IEEE Transactions on Knowledge and Data Engineering, 2024.
Angel X Chang, Thomas Funkhouser, Leonidas Guibas, Pat Hanrahan, Qixing Huang, Zimo Li, Silvio Savarese, Manolis Savva, Shuran Song. Hao Su, al. Shapenet: An information-rice 3d model repository. arXiv preprint arXiv:15/2.03001 2015.
Siddhartha Chaudhuri, Daniel Ritchie, Jiajun Wu, Kai Xu, and Hao Zhang. Learning generative models of 3d structures. Computer Graphics Forum, 39(2):643-66 666, 2020.
Minghao Chen, Roman Shapovalov, Iro Laina, Tom Monnier, Jianyuan Wang, David Novotny, and Andrea Vedaldi. Partgen: Part-level 3d generation and reconstruction with multi-view diffusion models. arXiv preprint arXiv:2412. 18608, 2024.
Nanxin Chen, Yu Zhang, Heiga Zen, Ron J Weiss, Mohammad Norouzi, and William Chan. Wave- grad: Estimating gradients for waveform generation. arXiv preprint arXiv:2009.007/ 2020.
Xi Chen, Diederik P Kingma, Tim Salimans, Yan Duan, Prafulla Dhariwal, John Schulman, Ilya Sutskever, and Pieter Abbeel. Variational lossy autoencoder. arXiv preprint arXiv:1611.0274 2016.
Daniel Cohen- Or and Hao Zhang. From inspired modeling to creative modeling. The Visual Com- pufer, 32:7-14, 2016.
11

---

Published as a conference paper at ICLR 2026
Dorin Comaniciu and Peter Meer. Mean shift: A robust approach toward feature space analysis. IEEE Transactions on pattern analysis and machine intelligence, 24(5):603 619, 2002.
Florinel-Alin Croitoru, Vlad Hondru, Radu Tudor lonescu, and Mubarak Shah. Diffusion models in vision: A survey. IEEE Transactions on Pattern Analysis and Machine Intelligence, 45(9): 10850-10869, 2023.
Imre Csiszár.
1-divergence geometry of probability distributions and minimization problems. The annals of probability, pp. 146 158, 1975.
Hoagy Cunningham, Aidan Ewart, Logan Riggs, Robert Huben, and Lee Sharkey. Sparse autoen- coders find highly interpretable features in language models. arXiv preprint arXiv:2309.0600) 2023.
Prafulla Dhariwal and Alexander Nichol. Diffusion models beat gans on image synthesis. Advances in neural information processing systems, 34:8780-8794 2021.
Anastasia Dubrovina, Fei Xia, Panos Achlioptas, Mira Shalah, Raphaël Groscot, and Leonidas J Guibas. Composite shape modeling via latent space factorization. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 8140-8149, 2019.
Richard O Duda and Peter Hart. Use of the hough transformation detect lines and curves in pictures. Communications of the ACM, 15(1):11-15, 1972.
Vassiliy A Epanechnikov. Narmparacc estimation of a multivariate probability density. Theory of Probability Its Applications, (4(1):153-158 1969.
Noa Fish, Melinos Averkiou, Oliver Van Kaick, Olga Sorkine-Hornung, Daniel Cohen-Or, and Niloy Mitra. Meta-reprresentation of shape families. ACM Transactions Graphics (TOG), 33(4):1-11,001 2014.
Keinosuke Fukumaga and Larry Hostetler. The estimation of the gradient of a density function, with applications in pattern recognition. IEEE Transactions on information theory, 21(1):32-40, 1975.
Thomas Funkhouser, Michael Kazhdan, Philip Shilane, Patrick Min, William Kiefer, Ayellet Tal, Szymon Rusinkiewicz, and David Dobkin. Modeling by example. ACM transactions graphics (TOG), 23(3):652-66333 2004.
Ran Gal, Olga Sorkine, Niloy J Mitra, and Daniel Cohen- Or. iwires: An analyze-and-edi approach to shape manipulation. In ACM SIGGRAPH 2009 papers, pp. 1-10. ACM New York, NY, USA, 2009.
Jean Gallier and Jean Gallier. The cartan-dicudonné theorem. Geometric Methods and Applications: For Computer Science and Engineering, pp. 201-23112200 2011.
Lin Gao, Jie Yang, Tong Wu, Yu-Jie Yuan, Hongbo Fu, Yu-Kun Lai, and Hao Zhang. Sdm-net: Deep generative network for structured deformable mesh. ACM Transactions Graphics (TOG), 38 (6):1-15, 2019.
Lin Gao, Ling-Xia Zhang, Hsien- Yu Meng, Yi-Hui Ren, Yu-Kun Lai, and Leif Kobbelt. Prs-net: Planar reflective symmetry detection net for 3d models. IEEE transactions visualization and computer graphics, 2766)3007--3088 2020.
Kyle Genova, Forrester Cole, Daniel Vlasic, Aaron Sama,, William T Freeman, and Thomas Funkhouser. Learning shape templates with structured implicit functions. In Proceedings of the IEEE/CVF international conference on computer vision, pp. 7154-7164, 2019.
Kyle Genova, Forrester Cole, Avneesh Sud, Aaron Sarna, and Thomas Funkhouser. Local deep implicit functions for 3d shape. In Proceedings of the IEFF/CVF conference on computer vision and pattern recognition, pp. 4857-4866, 2020.
Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recog- nition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp- 770-778, 2016.
12

---

Published as a conference paper at ICLR 2026
Amir Hertz, Or Perel, Raja Giryes, Olga Sorkine-Homunn, implicit shapes through part aware generation. ACM Transactions on Graphics (TOG), 41(4):1- 20, 2022.
Hornung, and Daniel Cohen-Or. Spaghetti: Editing
Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. Advances in neural information processing systems, 33:6840-6851, 2020.
Jonathan Ho, Tim Salimans, Alexey Gritsenko, William Chan, Mohammad Norouzi, and David Fleet. Video diffusion models. Advances in Neural Information Processing Systems, 35:8633- 8646, 2022.
Donald D Hoffman and Whitman A Richards. Parts of recognition. Cognition, 18(1-3):65 1984.
Haibin Huang, Evangelos Kalogerakis, and Benjamin Marlin. Analysis and synthesis of 3d shape families via deep-lleared generative models of surfaces. Computer Graphics Forum, 34(5):25-38, 2015.
Qixing Huang, Fan Wang, and Leonidas Guibas. Functional map networks for analyzing and ex- ploring large shape collections. ACM Transactions Graphics (TOG), 13,
Ka-Hei Hui, Ruihui Li, Jingyu Hu, and Chi-Wing Fu. Neural template: Topology-aware reconstruc- tion and disentangled generation of 3d meshes. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 18572-18582, 2022.
Aapo Hyvärinen, Jarmo Hurri, Patrik Hoyer, Aapo Hyvärinen, Jarmo Hurri, and Patrik O Hoyer. Estimation of non-normalized statistical models. Natural Image Statistics: A Probabilistic Ap- proach to Early Computational Vision, pp. 119-426,24 2009.
John Illingworth and Josef Kittler. A survey of the hough transform. Computer vision, graphics and image processing, 44(1):87-116, 1988.
Jihyeon Je, Jiayi Liu, Guandao Yang, Boyang Deng, Shengqu Cai, Gordon Wetzstein, Or Litany, and Leonidas Guibas. Robust symmetry detection riemannian langevin dynamies. In SIGGRAPH Asia 2024 Conference Papers, pp. 1-11, 2024.
Myconghun Jeong, Hyeongju Kim, Sung Jun Cheon, Byoung Jin Choi, and Nam Soo Kim. Diff-tts: A denoising diffusion model for text-to-speech. arXiv preprint arXiv:2/04.01 044000000999..... 2021.
Penglei Ji and Xinguo Liu. A fast and efficient 3d reflection symmetry detector based on neural networks. Multimedia Tools and Applications, 78(24) 2019.
Evangelos Kalogerakis, Siddhartha Chaudhuri, Daphne Koller, and Vladlen Koltun. A probabilistic model for component-base shape synthesis. Acm Transactions on Graphics (TOG), 3144):1-1, 2012.
William Karush. Minima of functions of several variables with inequalities side constraints. M. Sc. Dissertation. Dept. of Mathematics, Univ. of Chicago, 1939.
Michael Kazhdan, Thomas Funkhouser, and Szymon Rusinkiewicz. Symmetry descriptors and 3d shape matching. Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Ge- ometry processing, pp. 115-123, 2004.
Hyeongju Kim, Hyeonseung Lee, Woo Hyun Kang, Joun Yeop Lee, and Nam Soo Kim. Softflow: Probabilistic framework for normalizing flow on manifolds. Advances in Neural Information Processing Systems, 33:16388-163977 2020.
Jinwoo Kim, Jaehoon Yoo, Juho Lee, and Seunghoon Hong. Setvae: Learning hierarchical compo- sition for generative modeling of set-structured data. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 15059-15068, 2021.
Vladimir G Kim, Wilmot Li. Niloy J Mitra, Siddhartha Chaudhuri. Stephen DiVerdi, and Thomas Funkhouser. Learning bast- templates from large collections of 3d shapes. ACM Transac- fions Graphics (TOG), 32 2013.
13

---

Published as a conference paper at ICLR 2026
Diederik P Kingma and Max Welling. arXiv:13/2.611,, 2013.
Auto-encoding variational
bayes.
arXiv preprint
Roman Klokov, Edmond Boyer, and Jakob Verbeek. Discrete point flow networks for efficient point cloud generation. In European Conference on Computer Vision, pp. 694-710. Springer, 2020.
Juil Koo, Seungwoo Yoo, Minh Hieu Nguyen, and Minhyuk Sung. Salad: Part-level latent diffu-
sion for 3d shape generation and manipulation. In Proceedings of the TEEF/CVF International Conference on Computer Vision, pp. 14441-14451, 2023.
H Kuhn and A Tucker. Nonlinear programming. In Proceedings of 2nd Berkeley symposium (pp. 481 481-492), 1951.
Yann LeCun, Sumit Chopra, Raia Hadsell, M Ranzato, and Fujie Huang. A tutorial on energy-based learning. Predicting structured data, 1(0), 2006.
Guo Li, Ligang Liu, Hanlin Zheng, and Niloy J Mitra. Analysis, reconstruction and manipulation using arterial snakes. ACM Trans. Graph., 29(6):152, 2010.
Jun Li. Kai Xu, Siddhartha Chaudhuri, Ersin Yumer, Hao Zhang, and Leonidas Guibas. Grass: Generative recursive autoencoders for shape structures. ACM Transactions on Graphics (TOG), 36(4):1-14, 2017.
Jun Li, Chengjie Niu, and Kai Xu. Learning part generation and assembly for structure-aware shape synthesis. In Proceedings of the AAAI conference on artificial intelligence, volume 34, pp- 11362-11369, 2020.
Shidi Li, Miaomiao Liu, and Christian Walder. Editvae: Unsupervised parts-aware controllable 3d point cloud shape generation. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 36, pp. 1386-1383 1394, 2022a.
Shidi Li, Christian Walder, and Miaomiao Liu. Spa-vae: 3d point cloud generation. arXiv preprint arXiv:2203.07825, 2022b.
for unsupervised
Songlin Li, Despoina Paschalidou, and Leonidas Guibas. Pasta: Controllable part-aware shape generation with autoregressive transformers. arXiv preprint arXiv:2407.3667, 2024.
Songxiang Liu, Dan Su, and Dong Yu. Diffgan-tts: High-fidelity and efficient text-to-speech with denoising diffusion gans. arXiv preprint arXiv:2201.11972 2022.
Zhijian Liu, Haotian Tang, Yujun Lin, and Song Han. Point-voxel enn for efficient 3d deep learning. Advances in Neural Information Processing Systems, 32, 2019.
David Lopez-Paz and Maxime Oquab. Revisiting classifier two-sample tests. arXiv preprint
1rX1::1610.06545, 2016.
Shitong Luo and Wei Hu. Diffusion probabilistic models for 3d point cloud generation. In Proceed- ings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 2837-2845, 2021.
Zhengxiong Luo. Dayou Chen, Yingya Zhang, Yan Huang, Liang Wang, Yujun Shen, Deli Zhao, Jingren Zhou, and Tieniu Tan. Videofusion: Decomposed diffusion models for high-quality video generation. arXiv preprint 23Xiv:330..08320 2023.
Aleksandar Makelov, George Lange, and Neel Nanda. Towards principled evaluations of sparse autoencoders for interpretability and control. arXiv preprint arXiv:2405.0883 08340000066,, 2024.
Luke Marks, Alasdair Paren, David Krueger, and Fazl Barez. Enhancing neural network inter- pretability with feature feature-aligned sparse autoencoders. arXiv preprint arXiv:2411.01220. 2024.
Aurelien Martinet, Cyril Soler, Nicolas Holzschuch. and François X Sillion. Accurate detection of symmetries in 3d shapes. ACM Transactions Graphics (TOG), 25(2):439 464, 2006.
14

---

Published as a conference paper at ICLR 2026
Ravish Mehra, Qingnan Zhou, Jeremy Long, Alla Sheffer, Amy Gooch, and Niloy J Mitra. Abstrac- tion of man-made shapes. ACM SIGGRAPH Asia 2009 papers, pp. 1-10. ACM New York, NY, USA, 2009.
Niloy J Mitra, Leonidas Guibas, and Mark Pauly. Partial and approximate symmetry detection for 3d geometry. ACM Transactions on Graphics (TOG), 25(3):560-568, 2006.
Niloy J Mitra, Leonidas J Guibas, and Mark Pauly. Symmetrization. ACM Transactions on Graphics (TOG), 26 2007.
Niloy J Mitra, Mark Pauly, Michael Wand, and Duygu Ceylan. Symmetry in 3d geometry: Extrac- tion and applications. Computer graphics forum, 32(6):1-253 2013a.
Niloy J Mitra, Michael Wand, Hao Zhang, Daniel Cohen-Or, and Martin Bokeloh. Structure-aware shape processing. Computer Graphics Forum (Eurographics State-of-the-ars Report), pp. 175- 197. Wiley and Eurographics, 2013b.
Niloy Mitra, Michael Wand. Hao Zhang, Daniel Cohen-Or, Vladimir Kim, and Qi-Xing Huang. Structure-aware shape processing. In ACM SIGGRAPH 2014 Courses, pp. 1-21. Wiley and Eu- rographics, 2014.
Kaichun Mo, Paul Guerrero, Li Yi, Hao Su, Peter Wonka, Niloy Mitra, and Leonidas J Guibas. Struc- turenet: Hierarchical graph networks for shape generation, arXiv preprint arXiv:1908.000 2019.
Shentong Mo, Enze Xie, Ruihang Chu, Lanqing Hong. Matthias Niessner, and Zhenguo Li. Dit-3d: Exploring plain diffusion transformers for 3d shape generation. Advances in neural information processing systems, 36:67960 (6667960-67971, 1199112002 2023.
George Kiyohiro Nakayama, Mikaela Angelina Uy, Jiahui Huang, Shi-Min Hu, Ke Li, and Leonidas Guibas. Difffacto: Controllable part-based 3d point cloud generation with cross diffusion. In Proceedings of the IEEE/CVF International Conference Computer Vision, pp. 14257-14267, 2023.
Charlie Nash and Christopher KI Williams. The shape variational autoencoder: A deep generative model of part-segmented 3d objects. Computer Graphics Forum, 36(5):1-12, 2017.
Andrew Ng al. Sparse autoencoder. CS294A Lecture notes, 72(2011):1-19, 2011.
Charles O'Neill, Christine Ye, Kartheik Iyer, and John F Wu. Disentangling dense embeddings with sparse autoencoders. arXiv preprint arXiv:2408. 00657, 2024.
Daniele Panozzo, Yaron Lipman, Enrico Puppo, and Denis Zorin. Fields on symmetric surfaces. ACM Transactions on Graphics (TOG), 31(4):1-12, 2012.
Mark Pauly, Niloy Mitra, Joachim Giesen, Markus H Gross, and Leonidas J Guibas. Example- based 3d sean completion. In Symposium on geometry processing, pp. 23 23-32, 2005.
Mark Pauly, Niloy J Mitra, Johannes Wallner, Helmut Pottmann, and Leonidas J Guibas. Discover- ing structural regularity in 3d geometry. ACM SIGGRAPH 2008 papers, pp. 1-11. ACM New York, NY, USA, 2008.
William Peebles and Saining Xie. Scalable diffusion models with transformers. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 4195-4205, 2023.
Cody Phillips, Matthieu Lecce, and Kostas Daniilidis. Seeing glassware: from edge detection to pose estimation and shape recovery. In Robotics: Science and Systems, volume 3. pp. 3. Michigan, USA, 2016.
Joshua Podolak, Philip Shilane, Aleksey Golovinskiy, Szymon Rusinkiewicz, and Thomas Funkhouser. planar-reflective symmetry transform for 3d shapes. In ACM SIGGRAPH 2006 Papers, pp. 459, ACM New York, NY, USA, 2006.
15

---

Published as a conference paper at ICLR 2026
Joshua Podolak, Aleksey Golovinskiy, and Szymon Rusinkiewicz. eenmetty-enhanced remeshing of surfaces. In Proceedings of the fifth Eurographics symposium on Geometry processing, pp- 235-242, 2007.
Aditya Ramesh, Prafulla Dhariwal, Alex Nichol, Casey Chu, and Mark Chen. Hierarchical text- conditional image generation with clip latents. arXiv preprint arXiv:2204.061125, 1(2):3 2022.
Daniel Reisfeld, Haim Wolfson, and Yehezkel Yeshurun. Context-free attentional operators: The generalized symmetry transform. International Journal of Computer Vision, 14:119-130, 1995.
Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Bjöm Ommer. High- resolution image synthesis with latent diffuusin models. In Proceedings of the IEEE/CVF Con- ference on Computer Vision and Pattern Recognition, pp. 10684-10695, 2022.
Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomed- ical image segmentation. In Medical Image Computing and Computer-Assisted Intervention- MICCAI 2015: 18th International Conference, Munich, Germany, October 5-9, 2015, Proceed- ings, Part III 18, pp. 234-241. Springer, 2015.
Mihaela Rosca, Balaji Lakshminarayanan. and Shakir Mohamed. Distribution matching in varia- tional inference. arXiv preprint arXiv: 18Xiv:/802.06847. 2018.
Chitwan Saharia, Jonathan Ho, William Chan, Tim Salimans, David J Fleet, and Mohammad Norouzi. Image super-resolution via iterative refinement. IEEE transactions on pattern anal- ysis and machine intelligence, 45(4):4713-4722 2022.
Nadav Schor, Oren Katzir, Hao Zhang, and Daniel Cohen-Or. Componet: Learning to generate the unseen by part synthesis and composition. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 8759-8768, 2019.
Zeyun Shi, Pierre Alliez, Mathieu Desbrun, Hujun Bao, and Jin Huang. Symmetry and orbit detec- tion via lie-algebra voting. Computer graphics forum, 35(5):217-222) 2016
Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsupervised learning using nonequilibrium thermodynamics. In International conference on machine learn- ing. pp. 2256-2265, PMLR, 2015.
Jiaming Song. Chenlin Meng. and Stefano Ermon. Denoising diffusion implicit models. arXiv preprint arXiv:2010.02500 2020a.
Yang Song and Stefano Ermon. Generative modeling by estimating gradients of the data distribution. Advances in neural information processing systems, 32, 2019.
Yang Song, Jascha Sohl-Dickstein, Diederik Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. arXiv preprint arXiv:2011. 13456,, 2020b.
Kevin Swersky, Mare' Aurelio Ranzato, David Buchman, Nando Freitas, and Benjamin M Mar- lin. On autoencoders and score matching for energy based models. In Proceedings of the 28th international conference on machine learning, pp. 1201-1208, 2011.
Nicolas Talabot, Olivier Clerc, Arda Cinar Demirtas, Doruk Oner, and Pascal Fua. Partsdf: Part- based implicit neural representation for composite 3d shape parametrization and optimization. arXiv preprint arXiv:2502. 12985, 2025.
Jiapeng Tang, Yinyu Nie, Lev Markhasin, Angela Dai, Justus Thies, and Matthias Nießner. Dif- fuscene: Denoising diffusion models for generative indoor scene synthesis. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 20507-205 20518, 2024.
Sebastian Thrun and Ben Wegbreit. Shape from symmetry. In Tenth IEEE International Conference on Computer Vision (ICCV'05) Volume 1, volume 2, pp. 1824-1831. IEEE, 2005.
Jakub Tomezak and Max Welling. Vae with a vampprior. In International Conference on Artificial Intelligence and Statistics, pp. 1214-1223. PMLR, 2018.
16

---

Published as a conference paper at ICLR 2026
Shubham Tulsiani, Or Litany, Charles R Qi. He Wang, and Leonidas J Guibas. Object-centric multi- view aggregation. arXiv preprint arXiv:2007. 2020.
Arash Vahdat, Karsten Kreis, and Jan Kautz. Score- generative modeling in latent space. Advances in Neural Information Processing Systems, 34:1 1287-1877111 1177-1102,, 2021.
Arash Vahdat, Francis Williams, Zan Gojeic, Or Litany, Sanja Fidler, Karsten Kreis, et al. Lion: La- tent point diffusion models for 3d shape generation. Advances in Neural Information Processing Systems, 15:10011-1003, 2022.
Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. Advances in neural informa- tion processing systems, 30, 2017.
Hao Wang, Nadav Schor, Ruizhen Hu, Haibin Huang, Daniel Cohen-Or, Or, and Hui Huang. Global- to-local generative model for 3d shapes. ACM Transactions on Graphics (TOG), 37(6):1-10, 2018.
Lei Wang, Yuchun Huang, Pengjie Tao, Yaolin Hou, and Yuxuan Liu. Learning geometry-image representation for 3d point cloud generation. arXiv preprint urXiv:2011.14288 2020.
Yue Wang, Yongbin Sun, Ziwei Liu, Sanjay E Sarma, Michael M Bronstein, and Justin M Solomon. Dynamic graph enn for learning on point clouds. ACM Transactions on Graphics (TOG), 38(5): 2019.
Qiuhong Anna Wei, Sijie Ding, Jeong Joon Park, Rahul Sajnani, Adrien Poulenard, Srinath Sridhar, and Leonidas Guibas. Lego-net: Learning regular rearrangements objects in rooms. In Pro- ceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 19037- 19047, 2023.
Max Welling and Yee W Tch. Bayesian learning via stochastic gradient langevin dynamics. In Proceedings of the 28th international conference on machine learning, pp. 681-688. Citeseer, 2011.
Jan D) Wolter, Tony C Woo, and Richard A Volz. Optimal algorithms for symmetry detection in two and three dimensions. The Visual Computer, 1:37-48, 1985.
Lemeng Wu, Dilin Wang, Chengyue Gong, Xingchao Liu, Yunyang Xiong, Rakesh Ranjan, Raghu- raman Krishnamoorthi, Vikas Chandra, and Qiang Liu. Fast point cloud generation with straight flows. In Proceedings of IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 9445-9454, 2023.
Rundi Wu, Yixin Zhuang, Kai Xu, Hao Zhang, and Baoquan Chen. Pq-net: A generative part seq2seq network for 3d shapes. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 829-838, 2020a.
Shangzhe Wu, Christian Rupprecht, and Andrea Vedaldi. Unsupervised learning of probably sym- metric deformable 3d objects from images in the wild. In Proceedings of the IEEE/CVF confer- ence on computer vision and pattern recognition, pp. 1-10, 2020b.
Zhijie Wu, Xiang Wang, Di Lin, Dani Lischinski, Daniel Cohen-Or, and Hui Huang. Sagnet: Structure-aware generative network for 3d-shape modeling, ACM Transactions on Graphics (TOG), 38(4):1-14, 2019.
Zhen Xing, Qijun Feng, Haoran Chen, Qi Dai, Han Hu, Hang Xu, Zuxuan Wu, and Yu-Gang Jiang- A survey video diffusion models. ACM Computing Surveys, 57(2):1-42, 2024.
Jiale Xu, Weihao Cheng, Yiming Gao, Xintao Wang. Shenghua Gao, and Ying Shan. Instantmesh: Efficient 3d mesh generation from a single image with sparse-view large reconstruction models. arXiv preprint ar axXhv:2404.07111 2024.
Kai Xu, Hao Zhang, Andrea Tagliasacchi, Ligang Liu, Guo Li. Min Meng, and Yueshan Xiong. Partial intrinsic reflectional symmetry of 3d shapes. In ACM SIGGRAPH Asia 2009 papers, pp- 1-10. ACM New York, NY, USA, 2009.
17

---

Published as a conference paper at ICLR 2026
Guandao Yang, Xun Huang, Zekun Hao, Ming- Yu Liu, Serge Belongie, and Bharath Hariharan. Pointflow: 3d point cloud generation with continuous normalizing flows. In Proceedings of the IEEE/CVF international conference computer vision, pp. 4541 2019.
Jing Yang, Kyle Fogarty, Fangcheng Zhong, and Cengiz Oztireli. Sym3d: Learning symmetric triplanes for better 3d-awareness of gans. arXiv preprint arXiv:2406.06432, 2024.
Ruihan Yang. Prakhar Srivastava, and Stephan Mandt. Diffusion probabillstic modeling for video generation. Entropy, 25(10)11469 2023.
Li Yi, Vladimir Kim, Duygu Ceylan, I-Chao Shen, Mengyan Yan, Hao Su, Cewu Lu, Qixing Huang, Alla Sheffer, and Leonidas Guibas. A scalable active framework for region annotation in 3d shape collections. ACM Transactions on Graphics (TOG), 35(6):1-12, 2016.
Mourad Zerroug and Ramakant Nevatia. Part-based 3d descriptions of complex objects from single image. IEEE Transactions on Pattern Analysis and Machine Intelligence, 21(9):83-848, 1999.
Guangyao Zhai, Evin Pinar Örnek, Shun-Cheng Wu, Yan Di, Federico Tombari, Nassir Navab, and Benjamin Busam. Commonscenes: Generating commonsense 3d indoor scenes with scene graph diffusion. Advances Neural Information Processing Systems, 36:30026-30038, 2023.
Qian Zheng, Andrei Sharf, Guowei Wan, Yangyan Li, Niloy Mitra, Daniel Cohen-On and Baoquan Chen. Non-local scan consolidation for 3d urban scenes. ACM Trans. Graph., 29(4):94 2010.
Chenliang Zhou, Fangeheng Zhong, and Cengiz Öztireli. Clip-pae: Projection-aaugmentaiio em- bedding to extract relevant features for a disentangled, interpretable and controllable text-guided face manipulation. In ACM SIGGRAPH 2023 Conference Proceedings, pp. 1-9, 2023.
Chenliang Zhou, Fangcheng Zhong, Param Hanji, Zhilin Guo, Kyle Fogarty, Alejandro Sztrajman, Hongyun Gao, and Cengiz Oztireli. Frepolad: Frequency-rectified point latent diffusion for point cloud generation. European Conference on Computer Vision, pp. 434-453. Springer, 2024.
Linqi Zhou, Yilun Du, and Jiajun Wu. 3d shape generation and completion through point-voxel diffusion. In Proceedings of the IEEE/CVF International Conference Computer Vision, pp- 5826-5835, 2021a.
Yichao Zhou, Shichen Liu, and Yi Ma. Nerd: Neural 3d reflection symmetry detector. In Pro- ceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 15940- 15949, 2021b.
Chuhang Zou, Ersin Yumer, Jimei Yang, Duygu Ceylan, and Derek Hoiem. 3d-prmn: Generat- ing shape primitives with recurrent neural networks. Proceedings of the IEEE International Conference Computer Vision, pp. 900-909, 2017.
18

---

Published as a conference paper at ICLR 2026
Quartet of Diffusions: Structure-Aware Point Cloud Generation through Part and Symmetry Guidance Appendix
A RELATED WORK ON SYMMETRY DETECTION
Symmetry is fundamental geometric property commonly observed in both natural and human- made objects. As a result, detecting and leveraging symmetry has long been a central problem in the fields of computer vision and computer graphies (Mitra et al.) 2013a). Early work in this area focused on detecting exact symmetries in 2D or 3D planar point sets (Atallah) 1985 Wolter et al., 1985). However, the reliance on exact symmetry limits the practicality of these methods, as real- world objects often exhibit only approximate symmetry due to noise, occlusion, or design variations.
Traditional voting- based approaches, such as those based on the Hough transform (lllingworth & Kittler, 1988). attempt to accumulate votes for potential symmetries from point correspondences. While effective in idealized settings, these methods are known be sensitive to noise and often produce unreliable results when applied to imperfect incomplete data.
To address these limitations, Mitra et al. (Mitra et al., 2006) proposed more robust technique that replaces voting with mean-shift clustering (Fukunaga & Hostetler 1975| Comaniciu & Meer] (002) in a transformation space (for more details see Appendix [.I).. This method searches for modes in the space of rigid transformations, making it more resilient to noise and better suited for partial symmetries. Their approach laid the groundwork for several subsequent improvements (Pauly et al., 2008 Shi et al. 2016), which further enhanced the robustness and generality of symmetry detection.
In recent years, learning-bast methods have also been introduced (Ji & Liu 2019 Zhou et al.) 2021b Gao al. 2020), leveraging neural networks detect symmetry directly from 3D data. These methods benefit from data-driven representations and have shown improved generalization across object categories and varying conditions.
Building on these trends, Je el al. (Je et al. 2024) proposed a hybrid approach that combines ele- ments of both traditional and tearning-based techniques. Their method redefines the symmetry space and applies Langevin dynamics-a sampling technique from generative modeling-to iteratively re- fine symmetry estimates. This formulation provides both robustness and efficiency and serves strong foundation for the symmetry detection component in our framework.
B BACKGROUND
B. SYMMETRY DETECTION BY MEAN- SHIFT CLUSTERING
A common approach to detecting symmetry in 3D objects involves using the Hough trans- form (lllnggwrth Kittler 1988) to vote on the parameters potential symmetry planes (Mitra) et al. 2013a 2006). In this framework, each pair of points in the shape casts vote for candidate symmetry in a predefined transformation space M. For example, in the case of reflectional symme- try, point pair a and b casts vote for the plane that passes through their midpoint a+b and has normal vector given by ||a-b|| the direction that would reflect one point onto the other.
In an ideal setting, where the object and its symmetries are exact, votes from all symmetric point pairs would concentrate discrete points in M. In such cases, detecting symmetry reduces to identifying the peak with the highest vote count. However, most real-world objects and 3D datasets only exhibit approximate symmetry. As result, the votes form smooth, continuous distribution in M rather than sharp peaks. Detecting symmetry in this context thus requires identifying clusters in M that correspond to the most prominent approximate symmetries.
Mitra et al. MMira et al. 2006 proposes to use the mean-shift clustering (Fukunaga Hostetler,
1975 Comaniciu & Meer 2002), nonparametric clustering method based on gradient ascent
19

---

Published as a conference paper at ICLR 2026
density function p(S) in M defined as
p(S) = Σ K S R. REM
(S-R) )
(21)
where K a kernel function (e.g., Gaussian or Epanechnikov kernel (Epanechnikoy 1969) with
bandwidth h. The significant modes of p, and hence the significant symmetries, can be determined using gradient ascent: The algorithm first initializes from any of the candidate transformation So M and performs the following iteration until convergence:
S f EREB(S.) K (5)=)) R
(22)
EREB(S.) (S--R)
where B(S)) is neighborhood of St.
While it has shown promising results in detecting symmetries (e.g., Chang et al. 2015), it may fail under noisy shapes noisy transformation space (Je et al. 2024). Inspired by Je el al. (Je al. 2024), we leverage diffusion process (Ho et al., 2020) to establish the connection between the iteration in Eq. (22) and the stochastic gradient Langevin dynamics (Welling & Teh (01I). where stochastic noise is injected to improve the sample quality and robustness.
B.2 VARIATIONAL AUTOENCODER
We use variational autoencoders ( (VAEs) Kingma & Welling 2013) as our latent distribution model as it provides access a low-diiensionnl latent space and has been successfully applied generate point clouds (Li et al. 2022a Wang et al. 2020 Li et al. 20226 Zhou et al. 2024). VAEs are
probabilistic generative models that can model a probability distribution of given dataset X.
Starting with known prior distribution p(z) of shape latents z R", the parametric decoder of VAE models the conditional distribution Pn (x z) parametrized by n.. However, training the decoder to maximize the likelihood of data is not possible as
p(x) - f Pn (x z) p(z) dz
(23)
is intractable. Instead, a parametric encoder Pn (x z) is used to approximates the posterior distri- bution. Both networks are jointly trained to maximize lower bound on the likelihood called the evidence lower bound (ELBO):
CELBO (m) x) == (x)x)) log (x z)] Dki. (9,, (z xxxx) ,zz))
(24)
where DkL, the Kullback-Leibler divergence between the two distributions (Csiszár 1975).
B.3 DENOISING DIFFUSION PROBABILISTIC MODEL
Our generative framework leverages four denoising diffusion probabilistic models (DDPMs or dif- fusion models) DSchl-Dickstein et al. 2015) Ho et al. 2020) to model distinct data distributions. Given a data sample ~ p(z), diffusion models progressively corrupt = Zo into a noisy sample 2, through Markovian forward diffusion process. At each time step L 1, 2, . T, Gaussian noise is added according to a predefined variance schedule {athe:
q (Z1:r 200 = II g(zz 11
(25)
g zt-1) N (pi2)-15a11),
(26)
where N (n, σr) denotes multivariate Gaussian distribution with mean ş and variance In practice, we set μ4 == VI- ot. If T is sufficiently large (e.g., 1000 steps), p(z,) will approach the standard Gaussian distribution A' (0, I).
Diffusion models learn a reverse process Po (z,-1 2t), parameterized by 0, which defines a Marko- vian denoising chain that inverts the forward diffusion. This process gradually transforms sample
20

---

Published as a conference paper at ICLR 2026
of standard Gaussian noise 2, back into data sample Zo:
Po(z = p(z,) II Po (zc-1 zx):
(27)
(0 2,) ~ N (no (0ozz)0)) (),s?)),
(28)
where (ag2: (Zt,t) represents the predicted mean for the Gaussian distribution at time step t and { another variance schedule.
DDPMs are trained by maximizing the variational lower bound of log-likelihood of the data zo under ((zo):
log ((1:: 20)
po (o(o):))
2(())) [o 2 Byizor+)
(29)
Expanding Eq. (2 with Eq. 27 and noticing that p(z,) and g(21.r (o) are constant with respect to 0, we obtain our objective tunction to maximize:
(((()))
log po 22x)
(30)
1=1
Since we can factor the joint posterior
2a) II (z
Zz, Zn)
(31)
and both g(zc-1 2,, 2o) and po (2:-1 ze) are Gaussian, maximizing Eq. (30 is equivalent as minimizing the following matching objective for parametric model Eg (z,, t):
Ldiffissio ( (Ep((e)) (=( [lle (z(2, 1)]
(32)
where (LL, -) is the uniform distribution on (1,2...., T}. Intuitively, minimizing this loss corre- sponds to learning to predict the noise required to denoise the diffused sample 2,. Notably, the training objective of diffusion models closely aligns with estimating the gradient of the log data den- sity i.e., the score function
used in score-based energy models (Song et al. 2020a Swersky
jet al. 2011 Hyvärinen et al. 2009 LeCun al. 2006).
During inference, the network allows sampling through an iterative procedure since the learned distribution can be factorized as
po ((=)) = p (2+) po (zc-1 2e) = (z+) II po (2.-1 2z)
(33)
for p(z,) N 10.
C IMPLEMENTATION AND TRAINING DETAILS
Tables and provide the architecture and training details for the various VAE and diffusion models used in the Quartet. The hyperparameter X in Eq. (8), which controls the activation sparsity in the SVAE, set to 0.005. The hyperparameter for the equivariance fine-tuning in the part VAE, discussed in Sec. 3.4 is set to 0.01. For all models, we employ learning rate scheduler with reduce-on-plee -olttaaa policy, which decreases the learning rate by factor if the loss does not improve for 10 consecutive epochs. All training converged successfully by the end of training.
D MORE EXPERIMENTAL RESULTS
Table Figs. and present more results for the point cloud generation experiment in Sec. 4
21

---

Published as a conference paper at ICLR 2026
PVD
LION
DIT:3D
SALAD
FrePoled
Quattet (ours)
Figure 5: Full point cloud generation results. Samples from the Quartet are visually appealing, diverse, and exhibit strong structural and symmetry consistency.
Figure 6: More targeted manipulation results. We vary the yellow yellow-highhe highlighted parts while holding the remaining structure fixed.
22

---

Published as a conference paper at ICLR 2026
Table Model architecture details for the Quartet. denotes the number of parts for each object category (M 4 for airplanes and cars; M for chairs).
Model
Backbone
Data dimensionality
Latent dimensionality
Point cloud SVAE
PVCNN (Liu al. 2019
2048 3
128
Shape latent diffusion
U-Net (Ronncherger et al. 2015
128 12 M
Symmetry diffusion
ResNet He el al. 2016
Part diffusion
Transformer Vaswani ct
2017
2048 3
Part VAE
PVCNN LLu el al. 2019
57 X to 1024 x 9 M
128
Assembler diftusion
U-Net (Ronneberger 2015
Table 5:
Training details for the Quartet.
Model
Batch size
Number of epoch
Learning rate
Point cloud SVAE
64
1000
10-3 3
Shape latent diffusion
32
2000
10-1
Symmetry diffusion
32
2000
10
Part diffusion
16
2000
10-3
Part VAE
64
1000
10-5
Assembler diffusion
32
2000
10-4
Table Quantitative comparison of point cloud generation. PA denotes part awareness; SA denotes symmetry awareness. Our Quartet is the only model that supports both, achieving significant im- provements over most baselines and setting a new state of the art.
Airplane
Car
Chair
Model
PA SA 1-NNA (L)
SDI
1-NNA (1)
SD)
1-NNA (L)
SDI (J)
CD EMD CD
EMD
CD EMD
CD
EMD
CD
EMD
CD
EMD
Training
64.4
64.1 0.954
4.90
51.3
54.8
7.49
1.18
51.7
50.0
5.56
1.68
r-GAN (Achlioptas al.
2018
98.4
96.8
4519
1410
83.7
99.7
1053
362
94.5
99.0
7249
619
I-GAN/CD (Achiiopaas al.
2018
X 87.3
94.0
3629
1353
68.6
83.8
918
352
66.5
88.8
7972
582
I-GAN/EMD Achiropas el
2018
X 89.5
76.9
4129
914
71.9
64.7
982
313
71.2
66.2
7184
521
PointFlow Yang et al 2019
X 75.7
70.7
3410
782
62.8
60.6
679
347
58.1
56,3
7290
530
SoftFlow Kim el al. 2020
X
76.1
65.8
3284
529
59.2
60.1
1549
428
64.8
60.1
5628
420
ShapeGI (Cai al. 2020
81.2
80.9
332
98.6
58.0
61.3
645
40.9
57.2
1100
101
DPF-Net
KKokov
2020
65.6
4256
62.0
58.5
827
62.4
245
SetVAE [umm
76.5
67.7
2830
824
58.8
60.6
1240
5320
673
DPC (Luo & Ht 2021
76.4
86.9
187
44.2
60.1
74.8
217
30.3
68.9
80.0
335
50.6
PVD Zhou al. 2
X 73.8
64.8
150
42.0
56.3
53.3
213
31.2
54.6
53.8
275
58.4
LION \
Vahdat et 2022
X 67.4
61.2
97.2
40.6
53.7
52.3
168
30.8
$3.4
$1.1
201
55.2
SPAGHETTT Henzz et al
2022
X 78.2
77.0
1530
529
72.3
71.0
581
284
70.7
69.0
5930
582
DiT-aD Mo ala 2023
X
64.7
60.3
105
42.4
52.7
50.2
206
327
52.5
53.1
235
49.0
SALAD Koo el 2023
X 73.9
71.1
198
45.1
59.2
57.2
236
29.4
57.8
584
52.6
FrePolad Zhou al. 2024
62.1
94.1
52.4
173
50.9
Quartet (ours)
63.3
59.7
25.7
1.87
50.1
51.8
25.7
2.28
51.6
53.7
28.9
2.86
23


Reviews:
[{"review_id": "IrODDCzgHt", "reviewer": "Reviewer_rXcP", "summary": "The paper proposed Quartet of Diffusion, a part-level, symmetry conditioned point cloud diffusion model that is trained on Shapenet dataset with segmentation labels. After sampling a global shape latent code, the paper then samples both the part level code and a symmetry group that the shape obeys. Afterward, a set of scales, translations, and rotations are sampled that are used to assemble the shapes together. The method shows superior generation metrics compared to other point cloud generation methods. It also supports part level generation which keeping the rest of the parts fixed", "questions": "1. Are the assembled shapes always valid? Are there intersections or detachment between parts? Some metrics that quantifies this would be helpful (See SeaLION or DiffFacto for example metrics)\n2. Some shapes don’t have all the parts. How does the part diffusion model shapes with missing parts?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper proposed a novel symmetry conditioned point-cloud generation method. The symmetry diffusion formulation proposed in the paper seems novel and could potentially be a contribution to the shape generation field \n2. The generation quality surpasses many strong baseline including LION and ShapeGF. This suggests the overall generation quality.", "weaknesses": "1. I found the application provided by the authors to be insufficient to justify the modeling of parts and symmetry. Part-level generation is not a novel task and could be done without modeling the symmetry (See DiffFacto). It would be great to see further application enabled by the structure-aware modeling. \n2. The training requires part-segmentation labels as well as closed vocabulary part classes. This limits the generalization ability of the network to larger dataset such as Objaverse. \n3. The generated outcome is a point-cloud, which to me is not clear what it would be useful for. Can subsequent surface reconstruction algorithm be used to recover a higher fidelity shape?", "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": "vgzPHw2SEK", "reviewer": "Reviewer_wWCJ", "summary": "This work proposes a 3D point cloud generative model that considers part generation with symmetry. The model uses a four-stage diffusion process: it first generates a global shape latent, then symmetry groups (represented by reflection planes), followed by part geometry, and finally an assembler that composes the shapes. Experimental results show the framework achieves better performance than various point cloud generation baselines while using a lower training budget. The ablation study validates the effectiveness of the proposed method.", "questions": "Overall, I am convinced by the novelty and strong performance of the proposed framework. However, I have concerns about the fairness of the evaluation protocol and the absence of important baselines. Therefore, I am currently leaning toward rejection but am open to raising my rating if the following questions are addressed:\n\n1. Data Split Clarification: Could you please clarify the train/test split used in your experiments? Since you use ShapeNetPart (a subset of the full ShapeNet dataset) while the baseline evaluation protocol follows PointFlow (Yang et al., 2019) which uses the full category dataset, it is unclear whether your split maintains consistency with the original protocol. Specifically, could you confirm that there is no data leakage between your training and test sets, and provide details on how many samples were used for training vs. testing?\n2. Comparison with Part-Aware Baselines: The current baselines (including SALAD and SPAGHETTI) do not use part annotations during training, which makes the comparison potentially unfair given that your method explicitly leverages part labels. Could you include comparisons with some part-aware generative methods such as StructureNet (SIGGRAPH Asia 2019), SDM-NET (SIGGRAPH Asia 2019), PQ-NET (CVPR 2020), DSG-Net (TOG 2021), DiffFacto (ICCV 2023), and PASTA (Arxiv 2024)? This would provide a more comprehensive evaluation of your method's advantages when part supervision is available.", "limitations": "", "rating": 4, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "- The explicit consideration of symmetry in the 3D point cloud generation process is a novel exploration, and the evaluation, especially the SDI metric in Tables 1 and 3, clearly demonstrates the effectiveness of the proposed framework.\n- Additionally, the proposed framework achieves state-of-the-art performance compared to existing point cloud generation methods on standard metrics (1-NNA in Table 1), showcasing the quality and diversity of generated shapes.\n- A thorough ablation study in Table 3 validates the necessity of each introduced module.", "weaknesses": "Despite the strong performance shown, I still have some concerns regarding the evaluation setting.\n\n1. Data Split Concerns. In Table 1, the evaluation protocol follows PointFlow (Yang et al., 2019), which uses the full category dataset for train/test splitting. Since this work uses ShapeNetPart, a subset of that data, it is unclear whether the train/test split remains consistent. If not, the test set may have leaked into the training set. Clarification is needed to ensure fair evaluation.\n2. Comparison Baselines. The proposed framework is compared against many point cloud generation methods. However, none of them (including SALAD and SPAGHETTI) use part annotations for training their generative models. The proposed comparisons do not seem comprehensive. Instead, I would suggest comparing with methods that use part labels when training generative models. Some examples are provided below:\n- StructureNet: Hierarchical Graph Networks for 3D Shape Generation. SIGGRAPH Asia 2019.\n- SDM-NET: Deep Generative Network for Structured Deformable Mesh. SIGGRAPH Asia 2019.\n- PQ-NET: A Generative Part Seq2Seq Network for 3D Shapes. CVPR 2020.\n- DSG-Net: Learning Disentangled Structure and Geometry for 3D Shape Generation. TOG 2021.\n- DiffFacto: Controllable Part-Based 3D Point Cloud Generation with Cross Diffusion. ICCV 2023.\n- PASTA: Controllable Part-Aware Shape Generation with Autoregressive Transformers. Arxiv 2024.\n\nI suggest making minor adjustments to the framework's claims: \n\n- The paper claims the framework supports fine-grained manipulation, but provides no visual examples or quantitative evaluations. This claim should be softened to avoid confusion.\n- Various existing works consider symmetry in 3D shape generation (particularly for feature extraction), especially StructureNet and related approaches. More discussion on this point is recommended.", "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": "Dy6Kh663gS", "reviewer": "Reviewer_6Uas", "summary": "The paper proposes “Quartet of Diffusions,” a structure-aware point-cloud generative framework that factorizes generation into four diffusion models: a global shape latent, a symmetry group, distinct parts, and an assembly model. Each model is learned separately and connected via conditioning, which enforces part symmetry and yields realistic assemblies. The method is reported to be computationally efficient in training (despite four models) and outperforms prior work on standard metrics (e.g., Chamfer / 1-NNA) and on a newly introduced symmetry-awareness metric (SDI).", "questions": "-\tBy explicitly defining diffusion models for the four aspects (shape latent, symmetry, parts, assembly), the authors induce a bias into the model. Is it possible to investigate the impact of each of the four aspects individually?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 4, "strengths": "-\tClear conceptual decomposition: Factorizing generation into interpretable aspects (shape latent, symmetry, parts, assembly) is an elegant way to inject structure and inductive bias that matches human intuition about object design.\n-\tSolid empirical gains: The method shows strong improvements on classical generation metrics (1-NNA / Chamfer) and on the proposed symmetry metric (SDI), indicating both fidelity and structural plausibility.\n-\tSymmetry guarantees: Enforcing symmetry in the part generation produces more realistic, consistent results for symmetric objects.\n-\tReasonable compute: The paper claims modest GPU/training cost despite four separate diffusion models, which makes the approach more practical than it may first appear.\n-\tRigorous presentation: The paper is mathematically well-presented where needed.", "weaknesses": "-\tWhile the authors introduced an evaluation metric for symmetry-awareness, the work does not explicitly evaluate the part-awareness such as in SeaLion (Zhu et al., 2025).\n-\tThe method is only evaluated on three object classes of ShapeNet. \n-\tRelated work could be improved and be made more readable. The section seems only like an assembly of many citations without having a flow content-wise.\n-\tThe figures can be improved as often the point clouds overlap with text boxes, making the figures less readable and enjoyable (Fig. 1, 2, 3)", "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": "U01HZjpQTL", "reviewer": "Reviewer_E7K3", "summary": "The paper proposes a part-level point cloud generation framework called the \"Quartet of Diffusions,\" which integrates part and symmetry awareness into the generative process. The method uses four diffusion models to generate structure-aware 3D shapes by modeling shape latents, symmetries, parts, and assemblers. The Quartet framework is shown to outperform state-of-the-art methods in terms of diversity, quality, and symmetry.", "questions": "See the weaknesses", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- Part-level 3D generation is a critical research topic recently.\n- Using separate diffusion models to represent parts and transformations between them is novel and interesting, ensuring global shape symmetry.\n- The disentangled representation of parts, symmetries, and global structure offers fine-grained control over the generation process, allowing for targeted manipulation of individual parts.\n- The method pipeline is complex but clearly written.", "weaknesses": "- Currently, part generation methods based on implicit representations (e.g., PartCrafter, PartPacker, OmniPart) have achieved very impressive results, almost reaching production-ready levels. However, this paper still insists on using point clouds as the 3D representation, and the experiments are only validated on the ShapeNet dataset. As this approach can only generate a limited number of points and coarse geometric details, this somewhat diminishes the contribution and novelty of the paper. Therefore, I recommend that the authors discuss in detail why they chose this approach and clarify the advantages of using point clouds as a 3D representation for structured 3D generation.\n- The Quartet framework involves four separate diffusion models, which could introduce significant computational overhead during both training and inference, making it potentially less practical for real-time applications.\n- Integration of four diffusion models will lead to accumulated errors, which are not analyzed.", "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)