Not-So-Optimal Transport Flows for 3D Point Cloud Generation
Authors: Ka-Hei Hui
Chao Liu
Xiaohui Zeng
Chi-Wing Fu
Arash Vahdat
Venue: ICLR 2025 Poster (2025)
Keywords: Generative models
3D point cloud generation
flow matching
optimal transport flows
Primary area: generative models

Abstract: Learning generative models of 3D point clouds is one of the fundamental problems in 3D generative learning. One of the key properties of point clouds is their permutation invariance, i.e., changing the order of points in a point cloud does not change the shape they represent. In this paper, we analyze the recently proposed equivariant OT flows that learn permutation invariant generative models for point-based molecular data and we show that these models scale poorly on large point clouds. Also, we observe learning (equivariant) OT flows is generally challenging since straightening flow trajectories makes the learned flow model complex at the beginning of the trajectory. To remedy these, we propose not-so-optimal transport flow models that obtain an approximate OT by an offline OT precomputation, enabling an efficient construction of OT pairs for training. During training, we can additionally construct a hybrid coupling by combining our approximate OT and independent coupling to make the target flow models easier to learn. In an extensive empirical study, we show that our proposed model outperforms prior diffusion- and flow -based approaches on a wide range of unconditional generation and shape completion on the ShapeNet benchmark.

Published as a conference paper at ICLR 2025
NOT-SO-OPTIMAL TRANSPORT FLOWS FOR 3D POINT CLOUD GENERATION
Ka-Hei Hui'
Chao Liu
Xiaohui Zeng²
Chi-Wing Fu'
Arash Vahdat
'The Chinese University of Hong Kong 2NVIDIA
https://researes wwreaaarrc..iiiiiiii.. nvidia. com/coo/aaaoooooo labs/ http://ennaaaa-aeeergrrr
ABSTRACT
Learning generative models of 3D point clouds is one of the fundamental prob- lems in 3D generative learning. One of the key properties of point clouds is their permutation invariance, i.e., changing the order of points in point cloud does not change the shape they represent. In this paper, equivariant OT flows that learn permutation invariant generative models for point- based molecular data and we show that these models scale poorly on large point clouds. Also, we observe learning (equivariant) OT flows is generally challeng- ing since straightening flow trajectories makes the learned flow model complex propose not-so-optimal
analyze the recently proposed
at the beginning of the trajectory. To remedy these, transport flow models that obtain an approximate OT by an offline OT precompu- tation, enabling efficient construction of OT pairs for training. During training, we can additionally construct hybrid coupling by combining our approximate OT and independent coupling to make the target flow models easier to learn. In an extensive empirical study, we show that our proposed model outperforms prior diffusion- and basedased approaches and shape completion on the ShapeNet benchmark.
wide range of unconditional generation
INTRODUCTION
Generating 3D point clouds one of the fundamental problems in 3D modeling with applications in shape generation, 3D reconstruction, 3D design, and perception for robotics and autonomous systems. Recently, diffusion models (Sohl-Dickstein et al., 2015; Ho et al., 2020) and flow match- ing (Lipman et al., 2022) have become the de facto frameworks for learning generative models for 3D point clouds. These frameworks often overlook 3D point cloud permutation invariance, implying the rearrangement of points does not change the shape that they represent.
In closely related areas, equivariant optimal transport (OT) flows (Klein et al., 2024; Song et al., 2024) have been recently developed for 3D molecules that can be considered as sets of 3D atom coordinates. These frameworks learn permutation invariant generative models, iee. all permutations of the have the same likelihood under the learned generative distribution. They are trained using optimal transport between data and noise samples, yielding several key advantages including low sampling trajectory curvatures. low-variance training objectives, and fast sample generation (Poola- dian et al., 2023). Albeit these theoretical advantages, our examination of these techniques for 3D point cloud generation reveals that they scale poorly for point cloud generation. This is mainly due to the fact that point clouds in practice consist of thousands of points whereas molecules are as- sumed to have tens of atoms in previous studies. Solving the sample-level OT mapping between batch of training point clouds and noise samples is computationally expensive. Conversely, ignoring permutation invariance when solving batch-level OT (Pooladian al., 2023; Tong et al., 2023) fails to produce high-quality OT due to the excessive possible permutations of point clouds.
In this paper, we propose a simple and scalable generative model for 3D point cloud generation using flow matching, coined not-so-optimal transport flow matching, shown in Fig 1. We first propose an efficient way to obtain an approximate OT between point cloud and noise samples. Instead of searching for optimal permutation between point cloud and noise samples online during training, which is computationally expensive, we show that we can precompute an OT between a dense point superset and dense noise superset offline. Since subsampling superset preserves the

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Traiiin Batch
Sussanple
Acd ros0
Subsamp
(a) Independent Coupling (b) Minibatch OT
(c) Equivariant OT
(d) Ours
Figure 1: Different coupling types between Gaussian noise (left) and point clouds (right), where coupled noise and surface points share the same color: (a) Independent Coupling randomly maps noises to point clouds. (b) Minibatch OT computes OT map in batches of noises and point clouds. (c) Equivariant OT follows the similar minibatch OT but aligns points via permutation. (d) Our approach precomputes dense OT data and noise supersets, then subsamples it to couple point clouds with slightly perturbed noise. Note that only (c) and (d) can produce high-quality OT.
underlying shape, we can simply subsample the point superset and obtain corresponding noise from the precomputed OT to construct batch of noise-data pairs for training the flow models.
We demonstrate that approximate OT reduces the pairwise distance between data and noise significantly and benefits from the advantages of OT flows, e.g., straightness of trajectories and fast sampling. However, a careful examination shows that learning (equivariant) OT flows is generally challenging since straightening flow trajectories makes the learned flows com- plex at the beginning of the trajectory. Intuitively, in the OT coupling, the flow model should be able to switch between different target point clouds (i.c., different modes in the data distribution) with small variations in their input, making the flow model have high Lipchitz (see Fig 2).
Po(
(())
pwfat)
Figure 2: In the OT flow model,
the vector field vx(Xo) admits large change in its output with small perturbation of Xg at t=0.
To remedy this, propose simple approach to construct a less "optimal" hybrid coupling by blending our approxi- OT and independent coupling used in the flow matching model. In particular, we suggest perturbing the noise samples obtained from our approximate OT with small Gaussian noise. While this remedy makes our map- ping less optimal from the OT perspective, we show that it empirically shows two main advantages: First, the target flow model less complex and the generated points clouds have high sample quality. Second, when reducing the number of inference steps, the generation quality still degrades slower than other competing techniques, indicating smoother trajectories.
In summary, this paper makes the following contributions: (i) We show that existing OT approxi- mations either scale poorly or produce low-quality OT for real-world point cloud generation. (ii) We show that albeit the nice theoretical advantages, equivariant OT flows have to learn complex function with high Lipchitz at the beginning of the generation process. (iii) To tackle these issucs, propose not-so-optimal transport flow matching approach that involves offline superset OT precomputation and online random subsampling to obtain an approximate OT, and adds a small per- turbation the obtained noise during training. (iv) We empirically compare method against diffusion models, flows, and OT flows on unconditional point cloud generation and shape comple- tion on the ShapeNet benchmark. We show that our proposed model outperforms these frameworks for different sampling budgets over various competing baselines on the unconditional generative task. In addition, we show that we can obtain reasonable generation quality on the shape completion task in less than five steps, which is challenging for other approaches.
RELATED WORKS
Score-based Generative Models. Recently, diffusion models
et al., 2015; Ho et al.,
2020: Song & Ermon, 2019) have gained popularity in generating data of various formats, especially images (Rombach et al., 2022; Ramesh et al., 2022) and videos (Blatmann et al., 2023; Brooks et al., 2024). These generative models employ iterative process to transform known distribution,
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typically Gaussian, into desired data distribution. Song al. (2020) demonstrate that these models can be generalized to a continuous time setting, solving an SDE with an ODE that shares the same marginal. Lipman al. (2022) train vector field to trace linear interpolation between training pairs from data and noise distribution (further details in Section 3.1). We also focus on the flow matching models and aim to employ them effectively for 3D point cloud generation.
Relation of Flow Matching with Optimal Transport (OT), Flow matching is closely related to the Optimal Transport (OT) problem, which aims find a map with minimal transport cost between two distributions. As for flow matching, the trajectory flows can be taken to define the map that solves the OT problem. However, when the flow models are trained with random pairs, the trajectories are usually curved, so using pairs from the OT map can lead to straighter trajectories that improve the efficiency, as Pooladian et al. (2023); Tong al. (2023) show. Since the OT map is usually unavail- able, various methods are introduced to obtain better training pairs as we discuss in Section 3.1, 1. Another approach to obtain better pairs is to iteratively straighten the trajectory from a flow match- ing model. Rectified Flow (Liu et al., 2022) obtains training pairs by simulating ODE trajectories of pre-trained models. However, this involves an expensive ODE simulation and introduces errors in the straightening process. In this work, mainly study the limitations of existing OT solutions the first approach and derive a new solution that is specific for point cloud generation.
3D Point Cloud Generation. Point cloud generation has been studied extensively using various generative models including VAES (Kim et al., 2021), Energy-hased models (Xie et al., 2021), GANs (Achlioptas
al., 2018; Shu et al., 2019; Li et al., 2021), and flow-based models (Yang et al., 2019; Kim et al., 2020). Recently, diffusion models (Zeng et al., 2022: Zhou et al., 2021; Luo & Hu, 2021; Peng et al., 2024: Zhao et al., 2025) have achieved great success in producing high- quality generation results. However, these methods often ignore point clouds' permutation-invarian nature in training by treating them as high-dimensional flat vectors. PSF (Wu et al., 2023) uses Rectified flow (Liu et al., 2022) to generate point clouds, addressing permutation issues implicitly through flow straightening. This approach is computationally expensive due to multiple sample in- ferences needed for constructing training pairs. In this work, we focus on constructing an efficient, high-quality OT approximation for permutation-in invarrint point cloud generation based on the flow matching models.
METHOD
In Section 3.1, we begin by covering preliminaries of training continuous normalizing flow and recent OT flows. Section 3.2 explores the challenges of applying existing OT approximation meth- ods to 3D point clouds. To tackle these challenges, we introduce our approximate OT approach in Section 3.3 that precomputes OT maps in an offline fashion, and in Section 3.4, we explore simple hybrid and less optimal coupling approach that makes target flows easier to learn.
3.1 PRELIMINARIES
Continuous Normalizing Flow (CNF) (Chen et al., 2018) morph a base Gaussian distribution qo into a data distribution qI using a time-variant vector field va, [0, 1] Rd + Rd parameterized by a neural network 0, The mapping is obtained from an ordinary differential equation (ODE):
d dt x; = (x.)
Conceptually, the ODE transports an initial sample X go. where xo E Rd with pt denoting the distribution of samples at step and px() == (x)). Usually, the vector field vø. is trained to maximize the likelihood p1 assigned to training data samples x1 from distribution Q1. This procedure is computationally expensive due to extensive ODE simulation for each parameter update.
Flow Matching (Lipman et al., 2022) avoid the computationally expensive simulation process for training CNFs. In particular, we define a conditional vector field (x(-)x)) and path (+(+ that transform qu into Dirac delta x1 t = 1. Lipman et al. (2022) show that Vo, be learned via a simple conditional flow matching (CFM) objective:
CCF Ey (x1))) (tto:l))..ooollle.e.e.. (xx:))x)))))
un (xx|xx)11
(2)
A common choice for the conditional vector field is (x(xx)) := X, - -Xo- which can be easily
simulated by linearly interpolating the data and Gaussian samples via x, = = 1) Xn (x]-
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OT Distance Batch Size
OT Computation Time Batch Size
110
OT Compatation Time (c)
Minibatch OT Equivariant
Distence
Minibatch OT Equivoriant OT Independent Coupling Our OT Agproxlmation
107
10
Our Hybrid Coupling y 0.2)
66-
20
30
10
Batch
Batch
Figure 3: Comparison of OT Approximation Methods. Left: Average OT distance across batch sizes. Minibatch OT (blue) fails to reduce distances much compared to independent coupling (red dash). Equivariant OT (orange) significantly reduces distance values. Our OT approximation is on par with Equivariant OT. Right: Computational time for OT across batch sizes. Minibatch OT (blue) maintains reasonable computational time (~1 second) with batch size B 256. Equivariant OT (orange) grows quadratically starting from 2.2 seconds with B = 1.
Optimal Transport (OT) Map. In the CFM objective in Eq. 2, the training pair (Xo, 1 is sampled from an independent coupling: gXx,, x1) = qo (x(x))91 (x1). However, Tong et al. (2023); Poola- dian et al. (2023) show that marginal constraint: J q(x,, X1 di)xx; = (x())) and Sq( [Xo, xi )dx, = q1 (x1). They show an opti- mal transport (OT) map π that minimizes f||xo x <1l1sx(xo, x <1)xxddx1 is a good choice for data coupling, leading to straighter trajectory. Yet, obtaining the optimal transport map is often difficult for complex distributions. Next, we review two main directions for approximating the OT map:
sample the training pair from any coupling that satisfies the
Tong et al. (2023); Pooladian et al. (2023) approximate the actual OT by com- puting it at the batch level. Specifically, they sample a batch of Gaussian noises (x]. ... xx) ~ qo and data samples {x..... xx²
(i) Minibatch OT:
Q1, where B is the batch size. They solve discrete optimal transport problem, assigning noises to data samples while minimizing cost function C(x,, x1). The cost function typically the Euclidean distance, i.e., C(xg, the assignment problem is often solved using the Hungarian algorithm (Kuhn, 1955). After comput- ing the assignment, we can use the assigned pairs to train the vector field network via Eq. 2. As the batch size B approaches infinity, this procedure converges to sampling from the true OT map.
||x0 x 11 and
(ii) Equivariant OT Flow Matching: Song et al. (2024); Klein et al. (2024) also approximate the OT at the batch level, but they focus on generating elements invariant to certain group G, such as permutations, rotations, and translations. Specifically, they propose replacing the afore- mentioned cost function C(xo, x1) with one that accounts for these group elements: C(xo, x1) mingea ||xo - p(g)x1|f I|3, where p(g) is the matrix representation of the group element g. This approach significantly reduces the OT distance
with small batch size, demonstrating success in generating molecular data. Intuitively, using the cost function defined above allows us to align data and noise together (in our case via permutation) when computing the minibatch OT.
So far, we consider generic unconditional generative learning. It is worth noting that mini- batch OT does not easily extend conditional generation problems, i.e., learning p(xly) for generic input conditioning y. when there is only one training sample x for each input conditioning y. This is because the OT assumes access to batch of training samples for each y.
3.2 EXISTING OT APPROXIMATION FOR POINT CLOUD GENERATION
We focus on generating 3D shapes represented as point clouds. A point cloud x E N3 x3 is a set of points sampled from the surface of shape S, where N is the number of points. Unlike 2D images, point clouds have unique properties that pose challenges for existing OT methods:
(i) Permutation Invariance. A point cloud, while arranged in matrix form, is inherently set. Shuffling points in x, should represent the same shape. Mathematically, given a permutation matrix P(g), the sampling probability remains unchanged, iee., Q1 ((g))x)) q (x1).
(ii) Dense Point Set. Point clouds are finite samples on surfaces. However, similar to low-resolution images, sparse point sets may miss fine geometric structures and details. Thus, most works use dense point sets (say N 2 2048) to accurately capture 3D shapes.
Existing approach to estimating OT maps face these challenges on point clouds:
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Curvature Measure
Independent Coupling
and
Da No4
Our OT Approximation
Hyand Coupling (5 0.2)
Curvature
Timesteps
(6) Independent Coupling
(b) Our OT Approximation
(c) Our Hybrid Coupling (* 0.2)
Figure 4: Analysis of trajectory straightness using different couplings obtain training pairs. Left: We plot the square norm of the difference between successive vector fields, i.e., ||00,0+ (Iz+1) 1) eg [I. as a measure of trajectory curvature. Right (a-c): Trajectory samples obtained by models trained with (a) independent coupling, (b) our OT approximation, and (c) our hy coupling with 33 0.2. Note that we subsample the point cloud to 30 points for a better trajectory visualization.
Ineffectiveness of Minibatch OT. Minibatch OT, effective in low-dimensional and image domains, fails for point clouds due to property (i). There are N! equivalent representations of the same point cloud, implying N! equivalent training pairs (Xo, x1). An OT-speeled pair should minimize the cost: C(xo, X1) = mingeG C(xp, p(g)x)). However, in Minibatch OT's with no permutation, the assignments grow quadratically with batch size, while the number of permutations grows exponen- tially. As shown Figure (left), Minibatch OT achieves only about 6% reduction the cost even with batch size 256, indicating limited effectiveness of this approach in point cloud generation.
Inefficient OT Maps in Equivariant OT. Equivariant OT produces high-quality maps, but is com- putationally expensive for point cloud generation. Figure 3 (left) shows 48.7% reduction even with a batch size 1, showing the importance of aligning points and noise via permutation. However, un- like molecular data, which has limited size (N=55 in (Klein et al., 2024)), representing 3D shapes needs a larger N, following property (ii). Solving the optimal transport cost takes an O N3) computational complexity because of the quadratic number of noise and point cloud pairs in a batch of B examples, and O ( for the the Hungarian algorithm (Kuhn, 1955). Figure (right) shows how this grows rapidly even for a typical point cloud size (N 2048). It takes around 2.2 seconds for the OT computation even for B = 1, leading to a significant bottleneck in the training process that is more than 40x slower than independent coupling.
3.3 OUR OT APPROXIMATION
simple approach to generate training point clouds is to re-sample the points from the object surface in each training iteration. However, most point cloud generation methods avoid this tedious online sampling by pre-sampling a dense point superset X1 E R MMxx x3 with M >> N. During training, random subsets of X1 are selected training targets. This procedure converges to the true sam- pling distribution, following a straightforward extension of the law of large numbers (see Appendix proposition 1 for details). In a similar spirit, we compute an offline OT map between a dense point superset X1 E RM and a dense randomly-sample Gaussian noise superset Xo E RMx3, and during training, subsample data-noise pairs from the supersets based on the offline OT map.
Superset OT Precomputation. Given supersets Xo and X1, we compute bijective map 11 between them, i.e., 11 Xo + X1. When M is small, following (Song et al., 2024; Klein al., 2024), we compute the bijective map using the Hungarian algorithm (Kuhn, 1955). However, this algorithm scales poorly for large point clouds, ie.., M 10K. For such large point clouds, we use Wasserstein gradient flow to transform Xo into X2 by minimizing their Wasserstein distance iteratively. Using an efficient GPU implementation (Feydy et al., 2019), the OT precomputation takes only ~00 seconds for 100K points, showing its high scalability. See Appendix C for the details in procedure.
Online Random Subsampling. Given precomputed coupling IT(,, X1); we randomly sample data-noise pair (Xo.x) ~ I1(Xo.,11 X1) and we train the flow matching model according to Equa- tion 2. As we show in Figure (left), this significantly reduces the transport cost, while introducing negligible training overhead. In Appendix 1.1.2, show that the sampled training pair converges to correct marginals if M is sufficiently large.
In practice, using these pairs for training results in straighter sampling trajectories, measured by the curvature of the sampling trajectory, as shown in Figure 4 (left). The model trained with our OT approximation (orange curve) exhibits much lower maximum curvature compared to the one with
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independent coupling (blue curve). We also visualize the sample trajectories of these two models in Figures 4 (right) (a-b), confirming straighter trajectories for our model.
3.4 HYBRID COUPLING
Though training flows with OT couplings comes with appealing theoretical justifications (e.g., straight sampling trajectories),
identify key training challenge with them that is often over- looked in the OT flows literature. Our experiments (e.g. Section 4.1) indicate that flows trained with equivariant OT maps are often outperformed by those with independent coupling in terms of sample quality, especially when the number of sampling steps is large. We hypothesize this is due to the increasing complexity of target vector fields for OT couplings that makes their approximation harder with neural networks.
Intuitively, as we make target sampling trajectories straighter using more accurate OT couplings, the complexity of generation shifts toward smaller time steps. As Figure shows, the limit of straight trajectories, the learned vector field ve.o(xo) should be able to switch between different target point clouds with small variation in Xo. forcing Ve,0 to be complex at t=0. This problem is fur- ther exacerbated in the equivariant OT flows with large N where permuting Gaussian noise cloud in the input makes it virtually the same for all target point clouds. To verify this, we measure the trained vector field's complexity for 3D point cloud generation us- ing the Jacobian Frobenius norm in different timesteps in Figure 5. As hypothesized above, switching from independent coupling (blue curve) to our OT approximation (orange curve) shifts the high Ja- cobian norm at f. for independent coupling to t 0 for OT coupling. This motivates us to develop a method to make it easier for neural networks to approximate the target vector field, while still maintaining a relatively straight path.
Jacabian Frobenluss norn
Independent Coupling
Our OT Approximation Our Hybrid Coupling i8 - 0.2)
Timesteps to 1)
Figure 5: We show Jaco- bian Frobenius Norm for dif- ferent trained Va, over differ- ent time intervals, which mea- sures the model complexity as in (Dockhorn et al., 2022).
Hybrid Coupling. Given the different behavior of independent and OT couplings in Figure 5, we aim to reduce the complexity of the vector field at early timesteps by combining our OT approxima- tion with independent coupling. To do so, we propose injecting additional random Gaussian noise into Xg, making our OT couplings even less "optimal". The new training x² is defined as:
x /1 - 9 + VBe, 6 e~ O,,
where 3 E [0, 1] is the blending coefficient. Intuitively 3 allows us to switch smoothly between independent and OT couplings. Specifically, for O, the coupled data and noise pairs converge to our OT couplings, whereas when 3 + 1, they follow the independent coupling.
We empirically observe that 8 0.2 in most experiments strikes good balance between learning complexity (as shown by the green curve in Figure 5), low curvature for the sampling trajectories (the green curve in Figure (left) and Figure (right) (c)), and sample generation quality (shown later in Section 4.3). In the next section, we refer to this hybrid coupling as our main method.
EXPERIMENT
In this section, we present our experimental results for unconditional and conditional 3D point cloud generation, i.e., shape completion, after reviewing dataset and evaluation details.
Dataset. Following Yang et al. (2019); Klokov et al. (2020); Cai et al. (2020); Zhou et al. (2021), we employ the ShapeNet dataset (Chang et al., 2015) for training and evaluating our approach. Specifically, we train separate generative models for the Chair, Airplane, and Car categories with the provided train-test splits. Additionally, we provide more visualizations in Appendix on training a single model on all three categories to validate the generalization of our approach. To form our training point clouds, we randomly sample superset of M 100K points each input 3D shape. Then, during the online random subsampling, we randomly subsample the superset to N = 2,048 points, following the procedure in Section 3.3.

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1-NNA-CD for Chair Category
1-NNA-CD for Airplane Category
I-NNA-C for Car Category
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1-NNN-EMD for Airplane Calegory
1-NINVEMD for Car Category
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Inference Steps 11 to 100)
Figure 6: Quantitative comparisons of generation quality for different training paradigms using 1-NNA-CD (top) and I-NNA-EMI (bottom) for Chair (left), Airplane (middle). and Car (right). We present evaluation metrics across various inference steps, L..., from steps to 100 steps, for five methods: (i) ours, (ii) diffusion model with v-prediction (Salimans & Ho, 2022), and three flow matching models with different coupling methods: (iii) independent coupling (Lipman et al., 2022), (iv) Minibatch OT (Tong et al., 2023: Pooladian al., 2023), and (v) Equivariant OT (Song al., 2024: Klein et al., 2024). Note that values closer to 50% indicate better performance.
Inference Steps
100 Inference Steps
Ours
Diffusion Model
Independent Coupling
Minibatch OT
Equivariant OT
Diffusion Model
Independent Coupling
Minibatch OT
Equivariant OT
Figure/ 7:
Qualitative comparisons of generation quality for Chair (top) and Airplane (bottom) categories. We present inference results with 10 steps (left) and 100 steps (right).
In addition, we prepare a partial input shape for the shape completion task for each training shape. We use the GenRe dataset (Zhang et al., 2018) for depth renderings of ShapeNet shapes. Partial point clouds are obtained by unprojecting and sampling up to 600 points from these depth images.
Evaluation Metrics. We use LION (Zeng al., 2022)'s evaluation protocol, focusing 1-NN classifier accuracy (1-NNA) with Chamfer Distance (CD) and Earth Mover's Distance (EMD) met- rics. I-NNA measures how well the nearest neighbor classifier differentiates generated shapes from test data. Optimal generation quality is achieved when the classifier accuracy is ~00%. As discussed by Yang et al. (2019), 1-NNA more robust and correlates better with generation quality.
For shape completion, since we have ground-trute truth (GT) shape for each given condition, we fol- low (Zhou et al., 2021) to measure the similarity of our generated shape against the GT shape. In particular, we randomly sample 2,048 points on the GT shape surface and use CD and EMD distance metrics, where lower values indicate a higher similarity and thus a better performance.
4.1 UNCONDITIONAL GENERATION
For unconditional generation, we first evaluate our framework against alternative training paradigms, including diffusion models and flow matching models with different couplings:
Baselines. We consider both quantitative and qualitative comparisons. In this setting. We maintain the same architecture and hyperparameters, changing only the training procedure or objective for fair comparison. We then compare with four training alternatives: (i) the diffusion model objective using v-prediction (Salimans & Ho, 2022), and (ii)-iiv three flow matching models with different coupling methods: independent coupling (Lipman al., 2022), Minibatch OT (Tong et al., 2023;
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1-NM~CD for Chair Calegory
1-NNA--C for Airplane Category
1-NNA-CD Car Category
PVD
PVD
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inference Steps (1 100)
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inference Steps (1 100)
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1-NN--EM for Car Category
PVD
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PVD
1ANNA END END
LON
PSF
PSF
PSF
DIT-3D
inference Steps (1 100)
Inference Steps (1 to 100)
Inference Steps (1 100)
Figure 8: Quantitative comparisons with other point cloud generation methods using 1-NNA-CD
(top) and 1-NNA-EMD metrics (bottom) for Chair (left), Airplane (middle), and Car (right). We present evaluation metries across various inference steps, i.e., from 1 step to 100 steps, for five methods: (i) ours, (ii) PVD (Zhou et al., 2021), (iii) LION (Zeng et al., 2022), (iv) PSF (Wu al., 2023) without rectified flow, and (v) DiT-3D (Mo et al., 2023).
Inference Steps
100 Inference Steps
DIT-30
PVD
LION
Figure 9: Qualitative comparisons of generation quality for Chair (top) and Airplane (bottom) categories. We present inference results with 10 steps (left) and 100 steps (right).
Pooladian et al., 2023), and (iv) Equivariant OT (Song et al.. 2024; Klein et al., 2024). For Minibatch OT, we obtain the OT using a batch size of on each GPU but compute the training losses across all four GPUs. For Equivariant OT, due to its high computational demand, we can only consider permutations with a batch size of I, and train the model with the same training time (4 days).
Quantitative Comparisons. Figure 6 plots 1-NNA-CD & EMD for varying computation budget (inference steps) over the Chair, Airplane, and Car categories. Overall, our approach (blue curve) achieves similar better performance across all metries and categories, particularly when given sufficient number of steps, e.g., 100. Our approach performs best with fewer inference steps (10-20) due to a straighter trajectory, demonstrating the advantages of our approximate OT. Minibatch OT's generation performance matches original flow matching, possibly due to small OT distance reduc- tions between training pairs. Equivariant OT (orange curve) shows inferior performance compared to others, likely due to slow training within training budget.
Qualitative Comparisons. Figure 7 shows visual comparisons of different training methods for the Chair and Airplane categories, which feature more distinguishable characteristics. To facilitate easier comparisons, we use our generation results to retrieve the closest results from other methods. We display the generation results with 10 inference steps on the left. Unlike other methods that often add noise to distinct shape parts such as chair's 's armrests or airplane's 's wings, our approach better preserves these structures. With sufficient steps (right), nearly all methods (except Equivariant OT) can produce high-quality results with thin structures and fine details, e.g., the back of the chair.
Next, compare with other point cloud generation methods that require multi-steps generation:
Baselines. We also compare our framework with the following four baselines under the same in- ference budget: (i) PVD (Zhou al., 2021) and (ii) DiT-3D (Mo et al., 2023) models that directly generate point clouds, whereas (iii) LION (Zeng et al.. 2022) employs a diffu- sion model to produce set of latent points that are later decoded into point cloud. Since these
diffusiin-basee
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Cladr Catsgery
8
Inference Stace 1501
Portial Input
Steps
Cher Categery
END
100 Sheps
Inference Steas it 10001
Model
Coupling
Figure 10: Comparisons with other training paradigms on the shape completion task. Left: Quan- titative comparisons with other alternatives using different inference steps. Right: Qualitative com- parisons with other methods show the completion generated by 2, 5, and 100 steps, respectively.
models are trained for more inference steps (1,000), we employ DDIM sampler at inference, which is considered to be effective with fewer inference steps. (iv) Lastly, PSF (Wu et al., 2023), similar to our approach, uses a flow-based generative model, but additionally applies a rectified flow pro- cedure (Liu et al., 2022) to progressively straighten the inference trajectory, taking significantly additional training time. Since our method can be further accelerated with rectified flows, we only compare performance with PSF before the expensive rectification step. Note that we only report DiT-3D's result on the Chair category, as the pre-trained models for other categories are unavailable.
Quantitative Comparisons. Figure 8 shows results I-NNA-CD & EMD for different inference steps. Our method shows a comparable performance with existing flow-based and diffusion- generative models when given 100 inference steps. Additionally, cantly better generation quality when the inference is reduced to around 20 steps, without relying on additional expensive straightening procedures, e.g., rectified flow.
method can achieve signifi-
Qualitative Comparisons. Figure 9 shows visual comparisons with other point cloud generation methods for the Chair and Airplane categories, which feature more distinguishable characteristics. Overall, all existing methods can generate high-quality 3D shapes when given large number of inference steps. However, when the number inference steps is reduced, the generation quality of other methods declines significantly, as illustrated by the jet planes example on the left.
4.2 SHAPE COMPLETION
Baselines. We evaluate against the same baselines from Figure 6 for the shape completion task.
Quantitative Comparisons. Figure 10 (left) shows the evaluation metrics (CD & EMD) against various inference steps. We observe that our model can achieve reasonable generation quality with only five inference steps, while other approaches typically require about 50 steps to obtain similar quality. For Equivariant OT, it can produce comparable performance to our approach two infer- ence steps but fails to improve further with additional steps. Minibatch OT shows poor metrics, even with inference steps, as its batch-level OT is computed under different partial shapes. This violates assumptions that OT should be computed between training instances with the same condition.
Qualitative Comparisons. We present a visualization of shape completion results for various meth- ods in Figure 10 (right). Our approach produces reasonable-looking shape with only five inference steps, while other methods still generate noisy shapes. Furthermore, Minibatch OT generates a shape that does not correspond the input due violating the training assumption. Equivariant OT fails to produce shapes of similar visual quality even with increased inference steps.
4.3 MODEL ANALYSIS
In the followings, perform analysis major modules in approach, including the blending coefficient 3 of the hybrid coupling and the effect of the superset size M for the OT computation.
Blending Coefficient B. We examine the impact of different couplings on flow matching model training. Specifically, we train models with coupling blending coefficient / from (OT approxi- mation) to 1.0 (independent coupling). For each value, train model from scratch on the Chair Category and evaluate using the I-NNA-CD EMD metrics with 100 steps. To avoid approxima- tion errors in larger supersets, adopt superset size of 10, 000 with exact OT in this experiment.

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Table 1: I-NNA-CD & EMD over 100 steps for models trained with different 5.
Interpolation Coefficient (3)
Metries
0.001
0.01
0.1
0.2
0.5
1.0
I-NNA-CD
0.7289
0.6918
0.6223
0.5968
0,5853
0,5861
0,5989
I-NNA-EMD
0.6647
0.6563
0.6156
0.5884
0.5741
0.5780
0.5869
Table 2: I-NNA-CD & EMD over 100 steps for models trained using different superset sizes (M).
OT Superset Size (M)
Metrics
2048 0.6352
5000
10000
20000
50000
100000
I-NNA-CD
0.5853
0.5853
0,5853
0.5921
0.5725
I-NNA-EMD
0.6254
0.5853
0.5741
0.5627
0.5695
0.5627
Table presents the results. We can observe that directly employing our OT approximation (B 0.0) can lead to inferior performance, which can be gradually improved by injecting small amount of noise into the coupling. Interestingly, the best performance achieved when compared with other cases at around B 0.2. Injecting more noise until arriving independent coupling does not yield further improvement. These results demonstrate that neither our OT approximation nor independent coupling is optimal in terms of generation quality, and the hybrid coupling is necessary.
OT Supersets Size M. Next, we investigate the importance of constructing sufficiently large su- persets for OT computation. Here, we try supersets of varying sizes, starting from 2,048 (number of points to be generated) and progressively increasing the number towards 100,000. As outlined in Section 3.3, we employ the exact OT method for superset sizes of 10,000 or fewer points, and the OT approximation method for larger sets. For each superset size, we also train flow matching model on the Chair Category and evaluate also its generation quality over 100 inference steps.
Table 2 reports the evaluation results. Overall, we notice that small superset size usually leads to slightly worse performance, potentially due to overfitting the same target generation points. Increas- ing the number of points in the superset helps improve the performance. Notably, despite using an approximate OT (introduced in Section 3.3), we still observe some improvement in the generation quality when using large superset (M 100, 000), indicating the benefit of using a large superset outweighs the errors introduced by the approximation.
CONCLUSION
In conclusion, we proposed novel framework, coined as not-so-optimal transport flow matching for 3D point cloud generation. We demonstrated that existing OT approximation methods scale poorly for large point cloud generation and showed that target OT flow models tend to be more complex, and thus, harder to be approximated by neural networks. To address these challenges, our approach introduces an offline superset OT precomputation followed by an efficient online subsam- pling- To make the target flow models less complex, we proposed hybrid coupling that blends our OT approximation and independent coupling, making our OT intentionally less optimal. We demon- strated that our proposed framework can achieve generation quality par with existing works given sufficient inference steps, while achieving superior quality with smaller sampling steps. Addition- ally, we show our approach is effective for conditional generation tasks, such shape completion, achieving good generation even with five steps. Despite these advancements, there are still several limitations and potential extensions of this work. First, we do not consider other forms of invariance, such as rotational invariance that is present in large shape datasets such as Objaverse (Deitke et al., 2023). Second, we only consider point clouds representing coordinates. It would be interesting to explore the generation of point clouds with additional information, such as points with colors or even Gaussian splitting. Third, we currently focus on generation with a fixed resolution, and would be interesting to extend our method for iipoluttioninvarian cases, such as those in condi- tional tasks (Huang et al., 2024). Forth, we assume our dataset does not contain outliers in the point cloud following existing works, and developing a robust learning procedure with noisy point cloud dataset would be another interesting direction. Fifth, we show empirically that hybrid coupling can reduce the trained models' complexity (as shown in Figure 5), theoretical connection between / and the complexities is yet to be studied. Last but not least, we hope that this work can inspire further development around OT maps that easier to leam and that proposal, especially for the bybrid coupling, can translate to success at generating other forms of large point clouds such as large molecules or proteins.
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6 REPRODUCIBILITY STATEMENT
We demonstrate our efficiency in enabling the reproduction this work's results. particular, we describe the dataset (ShapeNet (Chang al., 2015)) and the data splits employed in training and evaluation of this work in Section 4. We also cover the evaluation protocol used in this work for assessing performance. Additionally, we include implementation details, such training specifies, network architecture, and shape normalization for evaluation in Appendix B. Lastly, we outline the procedure of our superset computation and describe the hyperparameter choices in Appendix
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A APPENDIX
A.1 PROOF FOR SATISFACTION OF MARGINAL. CONSTRAINTS.
A.1.1 LAW LARGE NUMBERS
Proposition Given (X1,... Xa), which are independently and identically distributed (IID) real d-diemsnion random variables, following probability distribution p(X), i.e., X; ~ p(X), X E RR.. We have an additional random variable Y that is random uniform sample of these variables, i.c. P(Y X;) = 1. The cumulative distribution function (CDF) F(t) of random variable Y will converge to the F()), ie., CDF of X.
Proof: We first define an empirical cumulative distribution function (x) over the random vari- ables (X1, ;x)):
(l(() - - Σ 1x.g
(4)
2=1
where 1x,5t is indicator for X < 1d for all dimensions {1,--. .d).
The Glivenko- theorem states that this empirical distribution function Fx will converge to the cumulative distribution F(X) if n is sufficiently large:
sup (a(() F()) 0
(5)
If we have an additional random variable Y that its value is a random subsample of the variables (X1, ... X.)::
P(Y = X2) = Vi =
1121, 1,2,
(6)
The CDF of this variable F(t) is:
-(t)= - s ()
Σ P(Y Xi) 1x,5t
Σ 1x = 1x:s==Fa00 n
1
Therefore, the CDF of Y also converges to the original underlying CDF F(t) if n is sufficiently large.
PL, and another random samples (Y,, "Y,,",,,,, Yy) ~ and we are also given an arbitrary bijective map between random variables. iee, II (1.....,n) {1.... ,n.. If we construct random variable Z (X,Y) follows the following couplings:
Proposition 2. Assume have random samples (X1,
P(X= = Y,) x,)= H
Tf= else j + II(i;;
The CDF of the marginal P(X) will converge the CDF of Pv, while the CDF of the marginal P(Y) will converge the CDF of p2-
Proof: Since II is bijective, we can compute the marginal P(X = X2) directly:
P(X = xx)= = Σ P(X = =,,==Y,))
1=1
P(X X1, X,, = Σ
P(X X = Y,)
(8)
j+T((()
+0
Similarly, we can show the marginal of P(Y) also 1. By leveraging Proposition 1, we show that
P(X) will converge the CDF of p1, and the CDF of P(Y) will converge to the CDF of p2-
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A.1.2 PROOF OF OUR OT APPROXIMATION
We first give a definition of coupling q( 21) in our case before showing its marginal fullfils the marginal requirements. In particular, we denote zo E R x3 and 21 E R x3 as two random vari- ables following the distributions, go (2o) and qi (x1), respectively. It is noted that Qo := [N, Z), which is the standard Gaussian for each dimension in 20. and (a1) ) is the distribution all pos- sible point clouds, which involves the joint modeling of point cloud distribution given a shape (q1 (S) and the distribution of shape (g(S)), i.e., q1 (z1) وq ( (S))SS
We can notice that each row in zg independently and identically distributed (IID), i.e, qo(20) = II do (z)), where we denote the i-th row of Eg as zó and distribution of zó as (z)),, which 3-dimensional unit Gaussian. We can also assume each point in x1 is IID given shape, i.e. (z) II. ái (x]])), where we denote the i-th row of 01 as of and the distribution of of as n (xf(S).
In our superset OT precomputation for a given shape S, we pre-sample a set of random variables Qı. and have
(a).... ..). xrd") ~ qo. and set of random variables (at.... ... .i.... xx precomputed bijective mapping Il {1,... .M <> {1,-.. .M. With these defined, our coupling =|)) for one row in the training pair (2 21) given S can be formulated as:
= (0. { 1
if = (I) else j + IT(A:::
((ag = 2 - = =11s)
0,
Since the each row in the training pairs are independently subsampled, the coupling of the train- ing pair (1) given shape is defined g(z,, #1|5) II ( In the end, the coupling over all training pairs can be obtained by marginalize over all possible shapes, i.e. If (1))))())))(())))))
Theorem 1. Our coupling without blending converge the following marginal if the superset size M is sufficiently large:
/ x(xx) = =x0(x)), / ((xx, x1 )1)xx = =x1(x).
(9)
Proof:
We first show the left constraint:
LHS = / (1)ox1))d11 = 1 g(00, ((20.21))(((S)dSdm
(10)
fo(S)
((S)) / ((20, x1 (S)dddss
change the order of integration
(11)
-f 0(5)) / [Ia(ss a yas
independent assumption of each row in training pair
(12)
- =f ((s) II x1
integrals of independent products
(13)
= =f (SS)) II ( o S)ds
restricting to discrete values in supersets
(14)
= =f ((S) II (n(
Proposition 2
(15)
= =ʃ (((S))((20) = =g0(o)
independent assumption of each row in Gaussian noises
(16)
(17)
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Similarly, perform the same computation the right constraint:
LHS
((z0,x1)dro = g(00, 2(=0.21(5)(($)d5ddr
(18)
((S) / g(200=11|)))) 21 2115)d2odS -
change the order integration
(19)
=fo(s)
((s)) / [ .. , ydS
independent assumption of each row in training pair
(20)
N - = (())) / o(S)dd(ss
integrals of independent products
(21)
M = f (S)) II
( 1sss
restricting to discrete values in supersets
(22)
= - ((S) II SS)
Proposition 2
(23)
= =f ( = (x1)
independent assumption of each row in point cloud
(24)
(25)
A.1.3 PROOF OF HYBRID COUPLING
In the last, we would like to show even with our hybrid coupling, the marginal still fulfills the requirements. particular, we define a new noises x after perturbation:
1 v -1 Brg BBe, N(e, 0, T),
(26)
where 3 E [0, 1] the blending coefficient. We denoted this as a conditional distribution g(2/) (z)), which has form of Na /1 Bro, (3). Is noted that since N(c, 0, 0, I), each row of a is also IID given oo, i.e, go (2)/(=) = II; (a(((11=)) Due to the independent properties, is sufficient to show that:
/ (((r)=(1$)d = a (x](s), / +11))d) = =00()))-
(27)
For the sake of simplicity, remove all the index i and shape S in the folloings. We first show the left constraint:
I ((12/==))dts
/ / go (26 vo, x )dzndx'
(28)
/ qo (0(26o00))( in dd
(29)
/ (000. 22) / (o(6blo0))) )o)
(30)
/ q(00. x (1)ddo
(31)
g(x1)
(32)
By Proposition 1, show ((x1) still converge the right CDF if M is sufficient large.
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On the other hand, we show that:
/
( al )dxı = / Qo (2/z vo, )droda
(33)
= {
4o (2). ro)dro
(34)
= (20o
(35)
(0,
(36)
where the last equality is obtained by inserting q(zu)) = N( I) and qu(0/|ao)
=
(œ Bro BI).
B IMPLEMENTATION DETAILS
Training Details. We implement our networks using PyTorch (Paszke et al., 2019) and run all ex- periments GPU cluster with four A100 GPUs. We employ the Adam optimizer (Kingma, 2014) to train our model with a learning rate of 2 x 10 and an exponential decay of 0.998 every 1,000 iterations. Following LION (Zeng al., 2022), our model with decay of 0.9999. Specifically, we train our unconditional generative model for approximately 600,000 iterations with batch size of 256, taking about four days complete. It is noted that we use larger batch sizes and more training iterations compared to existing work, includ- ing (Zhou et al., 2021; Wu et al., 2023), effectively compare various training paradigms (Figures & 7). This choice ensures our training procedure has higher stability and converges properly. Addi- tionally, we want highlight that the online subsampling procedure introduces negligible overhead in the training process (merely requiring additional indexing of a cached array).
exponential moving average (EMA) of
Network Architecture. For the network architecture, we adopt the same structure as PVD (Zhou et al., 2021) and employ PVCNN (Liu et al., 2019) as our vector field network for the unconditional generation task. In the shape completion task, we use an additional 256-dimensional latent vector to represent the input partial point cloud, which is then injected into PVCNN. To do we use another PVCNN follow LION (Zeng et al., 2022) to extract the latent vector from the partial point cloud.
Generated Shape Normalization. To ensure fair evaluation among different baselines in the unconditional task, we convert the inference results of various generative methods into a common coordinate domain. For all baseline methods, including PVD) (Zhou et al., 2021), DIT-3D (Mo al., 2023), LION (Zeng et al., 2022), and PSF (Wu et al., 2023), we respect the original normalization adopted in their training procedures. Since all these methods compute global mean coordinate and global scale ratio across all training shapes, we use these two quantities to reverse the normalization on the generated shape, based on the values obtained from the training set. This procedure aligns with existing baselines, such as (Yang et al., 2019; Zeng et al., 2022).
C APPROXIMATED OT
C.I IMPLEMENTATION DETAILS
In the following, we outline the procedure for the approximated Optimal Transport (OT) employed
for supersets with size M > 10, 000. We are motivated to use an approximated OT because the computational time for the exact method, i.e., the Hungarian algorithm (Kuhn, 1955), is prohibitively large for large superset size. Given supersets of size 10,000, already takes 220 seconds. Consider- ing the complexity of O(13), this is unlikely to be scalable for larger set sizes, e.g., M 100, 000, for the thousands of training shapes in each ShapeNet category, even if we move this computation offline. Therefore, resort
approximation procedure that utilizes Wasserstein gradient flow to obtain our point superset X, by progressively transporting a noise Xo.
OT Distances. To enable optimization procedure to move the points, we need to define an objective to be optimized, which is the 2- Wassersteei distance in our case. In particular, the OT distance can be defined as:
(go: 91)== == min qEI((o.11) / ((00, ((la)o ddodda
(37)
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1-NNA-CD for Chair Category
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Das
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Figure 11: Quantitative comparisons of generation quality for our framework against the training pairs obtained by 1-step Wasserstein Gradient Flow. We also show I-NNA-CD (top) and 1-NNA- EMD (bottom) for Chair (left), Airplane (middle), and Car (right). Note that a value closer to 50% indicates better performance.
where [' 91) represents all couplings with marginals 9o and q1- However, explicitly computing the OT is not tractable, so a relaxation of this cost is introduced by adding an entropic barrier term weighted by for solving various problem, including (Léonard, 2012). This relaxation enables an efficient solution by employing the Sinkhorn algorithm, which can be highly parallelized on the GPU. As shown in (Feydy al., 2019), this entropic barrier introduces bias in measuring the distance. Even when the marginals in the above equation are equal, i.e., = the distance might not be zero, leading to poor gradients. To address this issue, Genevay et al. (2018) introduce correction terms to the objective, forming the Sinkhorn divergence. Additionally. Feydy et al. (2019) provide an efficient, memory-saving GPU implementation that scales millions of samples.
Optimization Procedure. With the defined objective, our optimization will compute the gradient with respect to this objective to update the current point samples. Initially, we initialize the coor- dinates of a set of points as the noise superset Zg. At each optimization iteration, we compute the Sinkhorn divergence between the current point set and the target superset X/. By repeatedly mini- mizing the OT distance in the form of Sinkhorn divergence, we obtain deformed point set that fits the given superset well. Finally, we use this deformed point set as our point superset to represent the shape S.
Note that using this computed superset as X₁ introduces approximation errors compared to true samples X However, empirically observe that the procedure usually converges well with only small errors, and in our experiments, the large superset size typically outweighs the introduced error (as shown in Section 4.3).
C.2 TRAINING PAIRS BY 1 1-STEP APPROXIMATION
We also compare our approach with training pairs obtained through the optimization procedure described above. In this experiment, we perform only single optimization iteration point clouds and noises of size 2, 048 and set 3 to 0.0 For each training iteration, we use the optimized results, Xo and X1, our training pair. The quantitative comparison results are shown in Figure 11. Our approach consistently outperforms the 1-step Wasserstein Gradient Flow across all categories and most inference timesteps.
D ADDITIONAL. QUANTITATIVE COMPARISON
D.1 COVERAGE METRIC (COV)
Following Zhou et al. (2021); Zeng et al. (2022), we also evaluate coverage (COV), a metric that measures the diversity of generated 3DD shapes. COV calculates the proportion of testing shapes that
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COV-CD for Chair Category
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Figure 12: Quantitative comparisons of generation diversity for different training paradigms using COV-CD (top) and COV-EMI (bottom) for Chair (left), Airplane (middle), and Car (right). We present evaluation metrics across various inference steps, i.e., from I steps to 100 steps, for five methods: (i) ours, (ii) diffusion model with v-prediction (Salimans & Ho, 2022), and three flow matching models with different coupling methods: (iii) independent coupling (Lipman et al., 2022), (iv) Minibatch OT (Tong et al., 2023: Pooladian al., 2023), and (v) Equivariant OT (Song al., 2024: Klein et al., 2024). Note that a higher value indicates better performance.
COV-CD for Chair Category
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Figure 13: Quantitative comparisons with other point cloud generation methods using the COV- CD (top) and COV-EMD metrics (bottom) for Chair (left), Airplane (middle), and Car (right). We present evaluation metrics across various inference steps, i.e., from I step to 100 steps, for five methods: (i) ours, (ii) PVD (Zhou et al., 2021), (iii) LION (Zeng et al., 2022), (iv) PSF (Wu et al., 2023) without rectified flow, and (v) DiT-3D (Mo et al., 2023).
can be retrieved by generated shapes, with higher values indicating higher diversity. However, Yang et al. (2019); Zhou et al. (2021); Zeng et al. (2022); Wu et al. (2023) have noted that this metric is not robust training set shapes can have worse COV than generated results. Moreover, Yang et al. (2019) suggest that perfect coverage scores are possible even when distances between generated and testing point clouds are arbitrarily large. Given these limitations, COV should be considered only as reference metric, while I-NNA provides a more reliable measure that captures both generation quality and diversity.
Evaluation Results. We present quantitative comparisons with different baselines in Figure 12 and 13. Our approach generates shapes with reasonable diversity even with a limited number of steps (10-20), demonstrating the framework's 's effectiveness. With sufficient inference steps (100), our approach achieves comparable performance to other baselines. However, we observe contradictory conclusions between 1-NNA and COV metries (as shown by simultaneously high 1-NNA and COV
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Table 3: We provide evaluation results (I-NNA-CD and I-NNA--MD) using 1000 inference steps for DiT-3D (Mo et al., 2023), PSF (Wu et al., 2023) without rectified flow, PVD (Zhou et al., 2021), and LION (Zeng et al., 2022). The best-performing method is highlighted in red, while the second-
best is shown in blue.
Chair
Airplane
Car
I-NNA-CD
I-NNA-EMD
I-NNA-CD
I-NNA-EMD
I-NNA-CD I-NNA-EMD
DiT-3D
0.6072
0,5604
PSF
0.5612
0.5642
0.7457
0.6617
0.5682
0.5412
PVD
0.5626
0.5332
0.7382
0.6481
0.5455
0.5383
LION
0.537
0,5234
0.6741
0.6123
0.5341
0,5114
Ours
0.5551
0.5763
0.6864
0.6185
0.5966
0.5355
scores for Equivariant OT in the Airplane category), which aligns with the previously discussed limitations of the COV metric.
D.2 MORE INFERENCE STEPS
In addition to the baseline comparisons presented in the main paper (Figures 8 and 9), we provide additional comparisons with 1000 inference steps, matching the original settings used by the baseline methods. We employ the DDPM sampler (Ho et al., 2020) rather than the DDIM sampler (Song & Ermon, 2019) in this experiment and refer to the original values of PVD and LION reported in (Zeng et al., 2022).
Evaluation Results. We present the evaluation results based on I-NNA-CD and I-NNA-EMD in Table 3. Our method achieves comparable performance with PVD (Zhou et al., 2021), which also directly generates point clouds. When compared to LION (Zeng et al., 2022), which generates latent point cloud representations. our framework performs slightly worse. Note it is known that at a high sampling budget, SDE-based samplers often outperform ODE-basee samples (see (Karras et al., 2022) and (Xu et al., 2023)).
E ADDITIONAL VISUAL RESULTS
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Figure 14: Qualitative comparisons of generation quality for Chair category. We present inference results with 10 steps (left) and 100 steps (right).
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Qualitative comparisons of generation quality for Airplane category. We present infer- ence results with 10 steps (left) and 100 steps (right).
Figure 15:
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Steps
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Figure 16: Qualitative comparisons with other methods show the completion generated by 2, 5, and 100 steps, respectively.
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Qualitative comparisons of generation quality for Airplane (top) and Chair (bottom) categories. We present inference results with 10 steps (left) and 100 steps (right).
Figure 17:
We provide the generation results from model trained with shapes from all three categories (Chair, Airplane and Car) together. The results indicate that our approach can achieve high-quality generation with limited inference steps (20 steps) even when training with mixture of shapes from different categories.
Figure 18:
24


Reviews:
[{"review_id": "MVdaYbDfGm", "reviewer": "Reviewer_gCWN", "summary": "This paper introduces an offline superset OT pre-computation method followed by an efficient online subsampling to reduce the complexity of target flow models which is hard to be approximated by the neural networks. The proposed framework could achieve good shape generation with a few steps.", "questions": "1. Could the author provide a broader range of inference steps in Figure 8? Additionally, is there a comparison available for the models when they have converged?\n2. Why is rotational invariance not considered or discussed in the paper?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "This paper could generate fine 3D point results within limited steps.", "weaknesses": "1. Energy-based models, such as [1][2], are naturally permutation-invariant with respect to the order of point cloud data. However, these models lack sufficient discussion and comparative analysis, which would provide a clearer understanding of their strengths and limitations with the proposed method.\n2. The author asserts that diffusion models lack permutation-invariance in point cloud generation. However, recent studies, including [3], which use point-voxel representations; [4], which incorporate translation- and rotation-invariant features; and [5], which leverage latent diffusion models, are not included in the baselines for comparison.\n3. The author claims that the proposed method achieves high-quality generation with a limited number of inference steps. However, other fast sampling methods, such as [6], are not considered, which would offer a broader perspective on the efficiency of sampling approaches.\n4. While the author suggests that the proposed method scales well, there is no study on its performance across varying resolutions of 3D shapes. Furthermore, high-resolution 3D point generation methods, such as [7] and [8], are not included, which limits the scope of comparison for resolution-dependent generation quality.\n\n[1] Xie, Jianwen, et al. \"Generative pointnet: Deep energy-based learning on unordered point sets for 3d generation, reconstruction and classification.\" Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2021.\n\n[2] Xie, Jianwen, et al. \"Generative VoxelNet: Learning energy-based models for 3D shape synthesis and analysis.\" IEEE Transactions on Pattern Analysis and Machine Intelligence 44.5 (2020): 2468-2484.\n\n[3] Zhou, Linqi, Yilun Du, and Jiajun Wu. \"3d shape generation and completion through point-voxel diffusion.\" Proceedings of the IEEE/CVF international conference on computer vision. 2021.\n\n[4] Peng, Yong, et al. \"SE (3)-Diffusion: An Equivariant Diffusion Model for 3D Point Cloud Generation.\" International Conference on Genetic and Evolutionary Computing. Singapore: Springer Nature Singapore, 2023.\n\n[5] Zhao, Runfeng, Junzhong Ji, and Minglong Lei. \"Decomposed Latent Diffusion Model for 3D Point Cloud Generation.\" Chinese Conference on Pattern Recognition and Computer Vision (PRCV). Singapore: Springer Nature Singapore, 2024.\n\n[6] Wu, Lemeng, et al. \"Fast point cloud generation with straight flows.\" Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. 2023.\n\n[7] Huang, Zixuan, et al. \"PointInfinity: Resolution-Invariant Point Diffusion Models.\" Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2024.\n\n[8] Wen, Xin, et al. \"Point cloud completion by skip-attention network with hierarchical folding.\" Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. 2020.", "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": "mNX1ZypJL2", "reviewer": "Reviewer_F3TG", "summary": "The paper proposes a paradigm for training a flow-based generative model for permutation-invariant data such as 3D point clouds using a simple and efficient approximation of optimal transport (OT).\nComputing the optimal transport flows online scales poorly to a large number of points due to the prohibitive cost, while existing works based on approximations perform poorly.\nOn the contrary, the proposed method precomputes Gaussian-to-points OT of point clouds offline, and subsample it online to form the training pairs. \nApart from the OT approximation scheme, the paper also uncovers the issue regarding high Lipchitz at $t=0$, and proposes adding small Gaussian noise during training as a remedy.\nThe proposed method is benchmarked on ShapeNet for point cloud completion and unconditional generation, outperforming existing diffusion and flow-based approaches.", "questions": "* I wonder if it is possible to use an even worse (but fast) OT approximation algorithm (such as Feydy et al., 2019 with fewer iterations) to replace the hybrid coupling and enable efficient online sampling? Would it achieve the same purpose as the proposed Gaussian noise perturbation?\n* How critical is the size of the precomputed superset in terms of the model performance?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "* The proposed method achieves top performance among approximate OT flow and diffusion baselines, especially in the low-iteration regime.\n* The paper is well written, and the analysis on the behavior of the proposed approximation is very comprehensive.\n* It is surprising yet convincing to see that a more optimal OT leads to poorer performance due to high Lipchitz.", "weaknesses": "* The proposed method still requires the computation of a dense OT offline. The computational cost can still be very high for large point clouds. I wonder what is the number of points used for precomputing the OT superset, and how long does it take to process one 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": "WI3lB3DJII", "reviewer": "Reviewer_GZj5", "summary": "The paper designs an optimal transport flow matching method for 3D point cloud generation, addressing the critical challenge of permutation invariance in point clouds. The method incorporates offline OT mapping between data and noise to reduce training time. Additionally, it employs a hybrid coupling strategy that blends independent coupling with optimal transport to improve the alignment of point clouds. The authors provide a theoretical explanation and empirical evidence demonstrating why traditional OT methods struggle with point clouds, showing that their proposed approach effectively overcomes these limitations.", "questions": "- Could you please clarify how Figure 1 effectively illustrates the different coupling types between Gaussian noise and point clouds?\n- How does your approach handle noisy input data, particularly in scenarios where the data may contain outliers?\n- What strategies do you plan to implement to address scalability issues in practical applications?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "- Novelty: The adaptation of optimal transport methods in the context of point cloud generation is a significant and novel contribution. This approach addresses the permutation invariance of point clouds in flow-matching-based point cloud generation.", "weaknesses": "- Complex Computation and Slow Training Speed: Despite the use of offline OT matching, the training process remains computationally intensive due to the random subsampling of data-noise pairs and the iterative training of the vector field​. This results in significant training time, with approximately four days required on a cluster with four A100 GPUs, highlighting the method's complex computation and slow speed issues. \n\n- Scalability Issues: The use of Wasserstein gradient flow and the Hungarian algorithm for optimal transport computation in large-scale point clouds is computationally expensive. Additionally, the method necessitates separate training for each category, which not only diminishes efficiency and scalability but also requires extensive training time for each individual category. This lengthy training process further exacerbates the overall computational burden. Compared to contemporary 3D generation approaches that can efficiently handle multi-category generation, this method does not scale well.", "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": "0Si80UmCTK", "reviewer": "Reviewer_mmfz", "summary": "This paper explores optimal transport (OT) flow for point cloud generation, finding that existing OT approximations are not directly applicable to this task. The authors suggest that this limitation arises because equivariant OT flows must learn a complex, high-Lipschitz function early in the generation process. To address this, they introduce a \"not-so-optimal\" transport flow that combines offline superset OT precomputation with online subsampling and propose a hybrid coupling strategy.", "questions": "1. How is the performance of the proposed method if the time steps reach 1000 as in other baselines? Will it also achieve better results against the baselines?\n2. In Figure 4, “Note that we subsample the point cloud to 30 points for a better trajectory visualization”, how many points are used for training in these visualization experiments, 30 points or more?\n3. In Line 374, what does ‘hyperparameters’ refer to?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper is well-written and easy to follow.\n2. Extensive experiments are conducted to support the claims in the paper.\n3. Experiment results demonstrate the effectiveness of the proposed methods, especially with fewer inference steps.", "weaknesses": "1. 1-NNA CD and EMD are mainly used to measure the quaility. However, another aspect of generation, the diversity, has been ignored in the experiment. Coverage (COV) with CD and EMD should be reported to measure the generation diversity. You can refer to DiT-3D for details of COV.\n2. The experiments are focused solely on single-category generation. It would be more valuable to test the method on multi-category training, such as using the full ShapeNet-13, ShapeNet-55 or Objaverse dataset.", "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": "kLKQvyW0Rn", "reviewer": "Reviewer_o98M", "summary": "This paper presents a novel flow matching methodology for 3D point cloud generation. To overcome the limitations of existing Optimal Transport (OT) based methods (scalability issues, complex flow learning), it introduces 'not-so-optimal' transport flow matching. The proposed method enables efficient learning by combining offline superset OT computation with online subsampling, and reduces flow model complexity through a hybrid approach with independent coupling. The paper makes several significant contributions to the field of 3D point cloud generation. First, it provides a thorough analysis of existing OT-based methods, meticulously identifying their limitations in terms of scalability and computational overhead. Building on this analysis, the authors introduce an innovative approach combining superset OT precomputation with efficient online subsampling, addressing the identified scalability issues. They further enhance their methodology by proposing a hybrid coupling approach that cleverly combines OT and independent coupling, offering a more balanced solution. The effectiveness of these contributions is demonstrated through state-of-the-art performance on the ShapeNet benchmark, showing superior results in both unconditional generation and shape completion tasks.", "questions": "1. Is there a theoretical foundation for determining the optimal superset size in superset OT precomputation?\n\n2. How should the value of $\\beta$ in hybrid coupling vary depending on dataset characteristics and task requirements?\n\n3. Can the proposed method be effectively applied to other types of point cloud data?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper demonstrates balance between theoretical foundations and empirical validation. The authors provide mathematical analysis of their approach while supporting their claims with comprehensive experimental results across multiple benchmarks and metrics.\n\n2. The authors systematically analyze existing methods' limitations, particularly in terms of scalability and computational complexity. They not only identify these challenges clearly but also propose concrete, well-thought-out solutions that directly address each limitation.\n\n3. The authors show remarkable approach in managing the inherent trade-off between computational resources and model performance. Their proposed hybrid approach effectively balances the benefits of optimal transport with the computational efficiency of independent coupling, resulting in a practical solution.", "weaknesses": "1. While the paper addresses permutation invariance in detail, it lacks comprehensive treatment of other important invariances in 3D point cloud generation, particularly rotational invariance.\n\n2. The paper provides insufficient theoretical guidance for determining optimal superset size M and hybrid coupling's $\\beta$ parameter. While empirical results are presented for various superset sizes, blending coefficients. Without robust theoretical foundations for these choices, it becomes challenging to establish meaningful connections with existing theoretical frameworks and related research domains.\n\n3. The experimental validation focuses primarily on 3D ShapeNet datasets and benchmarks, with limited exploration of more challenging real-world applications. The lack of validation on complex domains like large molecular structures or protein configurations leaves questions about the method's broader applicability and scalability in these important areas.", "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)