Unpaired Point Cloud Completion via Unbalanced Optimal Transport
Authors: Taekyung Lee
Jaemoo Choi
Jaewoong Choi
Myungjoo Kang
Venue: ICML 2025 poster (2025)
Keywords: Point Cloud Completion
Unpaired Point Cloud Completion
Unbalanced optimal transport
Optimal transport
Primary area: deep_learning->algorithms
TL;DR: We propose an unpaired point cloud completion model based on the unbalanced optimal transport map.

Abstract: Unpaired point cloud completion is crucial for real-world applications, where ground-truth data for complete point clouds are often unavailable. By learning a completion map from unpaired incomplete and complete point cloud data, this task avoids the reliance on paired datasets. In this paper, we propose the \textit{Unbalanced Optimal Transport Map for Unpaired Point Cloud Completion (\textbf{UOT-UPC})} model, which formulates the unpaired completion task as the (Unbalanced) Optimal Transport (OT) problem. Our method employs a Neural OT model learning the UOT map using neural networks. Our model is the first attempt to leverage UOT for unpaired point cloud completion, achieving competitive or superior performance on both single-category and multi-category benchmarks. In particular, our approach is especially robust under the class imbalance problem, which is frequently encountered in real-world unpaired point cloud completion scenarios.

Unpaired Point Cloud Completion via Unbalanced Optimal Transport
Taekyung Lee' Jaemoo Choi?
Jaewoong Choi
Myungjoo Kang" 14
Abstract
2020; Hong et al., 2023). Existing approaches can be cat- egorized into paired (supervised) and unpaired (unsuper- vised) methods. The paired methods rely on training data that explicitly aligns incomplete point clouds with their corresponding complete versions (Yu et al., 2021; Wang et al., 2022; Tchapmi et al., 2019; Xia et al., 2021; Zhou et al., 2021). However, acquiring such paired datasets can be both expensive and challenging. To address these limita- tions, the unpaired point cloud completion has emerged. These approaches aim to train a completion model from the independently sampled incomplete and complete point clouds without explicit one-to-one correspondence. This is achieved by leveraging shared semantic information, such as object class (Ma et al., 2023; Chen et al., 2020; Wen et al., 2021), or through domain adaptation using paired synthetic data (Liu et al., 2024). However, existing unpaired approaches primarily rely on heuristic techniques without rigorous theoretical formulation of the unpaired point cloud completion problem.
Unpaired point cloud completion is crucial for real-world applications, where ground-truth data for complete point clouds are often unavailable. By learning a completion map from unpaired incomplete and complete point cloud data, this task avoids the reliance on paired datasets. In this paper, we propose the Unbalanced Optimal Transport Map for Unpaired Point Cloud Comple- tion (UOT-UPC) model, which formulates the unpaired completion task as the (Unbalanced) Optimal Transport (OT) problem. Our method employs a Neural OT model learning the UOT map using neural networks. Our model is the first attempt to leverage UOT for unpaired point cloud completion, achieving competitive su- perior performance on both single-category and multi-category benchmarks. In particular, our ap- proach is especially robust under the class im- balance problem, which is frequently encoun- tered real-world unpaired point cloud comple- tion scenarios. The code is available at https: / //github. wtt.:/cc///////ee. LtTp///EEEE////TTUUOO UUOT-UC.
In this paper, we formulate this unpaired point cloud com- pletion through the Optimal Transport Map (OT Map) problem. The OT Map is defined the cost-minimizing transport map that bridges two probability distributions. Re- cently, several works proposed methods for learning the OT Map using neural networks (Neural OT) (Rout et al., 2022: Fan et al., 2022: Choi et al., 2024a). These models have been applied to various machine learning tasks, such as generative modeling (Choi et al., 2023) and image-to-image translation (Fan et al., 2022: Gazdieva et al., 2024). A key compo- nent of Neural OT is the cost function, which determines how each input æ is transported to T(x). However, existing approaches mainly focus on variants of the quadratic cost function l2 (Rout et al., 2022; Fan et al., 2022; Choi et al., 2024a; 2023). To address this limitation, we analyze vari- ous candidate cost functions for unpaired completion task and show that this theoretical analysis closely aligns with experimental results.
1. Introduction
The direedimensional (3D) point cloud is a fundamental representation in 3D geometry processing (Guo et al., 2020). However, acquiring complete point cloud data of real-world objects remains a significant challenge due to the limita- tions of the scanning process (Yuan et al., 2018). Conse- quently, diverse approaches have been proposed for point cloud completion, which aims to reconstruct a complete (full) point cloud from incomplete (partial) data (Yu al., 2021; Wang et al., 2022: Tchapmi et al., 2019; Chen al.,
"Equal contribution 'IPAT (Ineedisciplinnary Program in Arti- ficial Intelligence, Seoul National University) Georgia Institute of Technology "Sungkyunkwan University 'Department of Mathe- matical Sciences and RIMS, Seoul National University. Correspon- dence to: Jaewoong Choi <jacwoongchoie @ skku.edu>, Myungjoo Kang <mkang @ kr?>
Based on this OT Map formulation, we propose novel unpaired point cloud completion model based on the Un- balanced Optimal Transport (UOT) framework. We refer to our model as the Unbalanced Optimal Transport Map for Unpaired Point Cloud Completion (UOT-UPC). Our experiments demonstrate that UOT-UPC achieves state-of- the-art performance on unpaired point cloud completion
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada, PMLR 267, 2025. Copyright 2025 by the author(s).

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Unpaired Point Cloud Completion via Unbalanced Optimal Transport
benchmarks across both single-category and multi-category settings. Furthermore, UOT-UPC exhibits particularly ro- bust performance under class imbalance, where incomplete and complete distributions consist of multiple categories in different proportions. The UOT framework provides our model with inherent robustness against class imbalance, fur- ther enhancing its effectiveness in real-world scenarios. Our contributions are summarized as follows:
Kantorovich introduced relaxed formulation of the OT problem (Kantorovich, 1948). Formally, this Kantorovich formulation is expressed in terms of coupling rather than transport map T, as follows:
Coe(u,)) = inf U c(x, (d).,))T(Z.))) (2) =Ella,.)
where is cost function and [(,, v)) is coupling of I and v. In contrast to the Monge problem, the minimizer ' of Eq 2 always exists under some mild assumptions on (X, μr), ()) v) and the cost function c (Villani et al., 2009). Note that under our assumptions that μ and v are absolutely continuous with respect to the Lebesgue measure, the op- timal transport map T exists and the optimal coupling is given by x² (Id x TT)# (Villani et al., 2009).
UOT-UPC is the first unpaired point cloud completion model based on the Unbalanced Optimal Transport framework, formulating the task as finding the optimal transport map.
We provide comprehensive analysis of suitable cost functions and prove a strong alignment between theo- retical insights and empirical results.
UOT-UPC attains otate-of-the-art performance on both single-category and multi culti-c benchmarks.
Rout et al. (2022); Fan et al. (2023) proposed a method for learning the optimal transport map T* using the semi-dual formulation of OT. This neural network-based approach for learning the OT Map is referred to as Neural Optimal Transport (Neural OT). In specific, these models parametrize the potential function e and the transport map T as follows:
UOT-UPC shows robust performance under severe class imbalance problem between incomplete and com- plete point clouds.
Notations and Assumptions Let X. y be two compact complete metric spaces, j1 and be probability distributions on X' and y' respectively. μ and are assumed to be ab- solutely continuous with respect to the Lebesgue measure. Throughout this paper, we denote the source distribution as 1a and the target distribution as v. Since the focus of this paper is on point cloud completion, and v repre- sent the distributions of the incomplete and complete point clouds, respectively. For a measurable map T,, T represents the pushforward distribution of μ.. I[,,, 2/) de- note the set of joint probability distributions X' y whose marginals are μ and v, respectively. Additionally, f" indicates the convex conjugate of a function f. i.e., (") = suPwER ( (z } for f: IR- + oo).
[/x inf (2, - us (Ta(x))) du(x)
Lug,To = sup
To
+ 1 es va(w)au(s)
(3)
Unbalanced Optimal Transport The classical OT prob- lem assumes an exact transport between two distributions Ii and v, i.e. mo = (4, Ti V. However, this exact match- ing constraint results in sensitivity to outliers (Balaji et al., 2020; Séjourné et al., 2022) and vulnerability to class im- balance in the OT problem (Eyring et al., 2024). To mitigate this issue, a new variation of the OT problem is introduced, called Unbalanced Optimal Transport (UOT) (Chizat et al., 2018; Liero et al., 2018b). Formally, the UOT problem is expressed as follows:
2. Preliminaries
Optimal Transport The Optimal Transport (OT) problem investigates the task of transporting the source distribution E P()) to the target distribution E PP))). This prob- lem was initially formulated by Monge (1781) using a de- terministic transport map T : X' + y such that Tap v:
In We c(z, ()d,)) v) (H, inf
sEM+ (xM+((x))
+ De, + Dus (mall)))) (4)
C(H,V)= = inf U/ c(x, ((o:T()))dp())) Tyu=
(1)
where M+ x )) denotes the set of positive Radon measures on X' x Y. Dv, and De, represents two f- divergences generated by convex functions V., and are defined as (my(7)) = V (x) dnf(x). These f- divergences penalize the discrepancies between the marginal distributions To. क, and μ,, v, respectively. Hence, in the UOT problem, the two marginal distributions are softly matched to μ. U, i.e, To as I and मi a v. Intuitively,
Intuitively, Monge's OT problem explores the optimal trans- port map T that connects two distributions while minimiz- ing given cost function c(x,, T(x)). Although Monge's 's OT problem offers an intuitive framework, it has theoretical limitations: this formulation is non-convex and the optimal transport map T may not exist depending on the conditions on f and a (Villani et al., 2009). To address these issues,
drjt))
2

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Unpaired Point Cloud Completion via Unbalanced Optimal Transport
Incomplete
GT
cd
InfoCD
TV
lamp
bed
Figure I: Visualization the incomplete point cloud v, the ground truth completion (I)), and three complete point clouds off (x) that minimize the cost c(z, yf(x)) for two cost functions: cd? and InfoCD, in the multi-category setting.
the UOT problem can be seen as the OT problem between To Ri j and IT R v, rather than between the exact distri- butions y and (Choi et al., 2023). This flexibility offers robustness to outliers (Balaji et al., 2020) and adaptability to class imbalance problem between μ and y (Eyring al., 2024) to the UOT problem (See Sec 4.2 for details). We refer to the optimal transport map T* from To to тi as the unbalanced optimal transport map (UOT Map). Note that, under our assumption that the source and target distributions are absolutely continuous, the existence of this UOT Map is guaranteed (Liero et al., 201 8a, Thm. 3.3).
based model that maps the latent features of the incomplete point cloud to those of the complete point cloud. Wu et al. (2020) proposed a conditional GAN model that generates multiple plausible complete point clouds, conditioned on the incomplete point cloud. ShapeInv (Zhang et al., 2021) employed an optimization-base GAN-inversion approach (Xia et al., 2022). ShapeInv finds the optimal input noise to reconstruct the complete point cloud from the given in- complete point cloud. This is conducted by minimizing the distance between the input incomplete point cloud, which is for completion, and the partial point cloud, which is ob- tained by degrading the generator's 's output. Cycle4 (Wen et al., 2021) proposed two cyclic transformations between the latent spaces of incomplete and complete point clouds, utilizing missing region coding. USSPA (Ma et al., 2023) proposed a symmetric shape-preserving method based on GAN.
Choi et al. (2023) introduced a Neural OT model for the UOT problem into generative modeling, called UOTM (See Sec 4.2 for details). In this paper, we adapt the UOT Map to the task of unpaired point cloud completion. Unlike gen- erative modeling, in unpaired point cloud completion, each incomplete source sample x' should be transported its corresponding complete target sample y. Therefore, it is important to set appropriate cost function c(z, 2) in Eqq 4, because this cost governs how each x is transported to y. In Sec 4.1, we compare diverse cost functions to ensure that the UOT Map works as a valid completion model.
Unlike prior heuristic-driven methods, work is the first to provide theoretical formulation of unpaired point cloud completion as the OT Map problem. Based on this formu- lation, we propose a theoretically grounded UOT approach for unpaired point cloud completion.
3. Related Works
4. Method
Point cloud completion has been investigated through vari- ous approaches. The paired (supervised) methods leverage explicit correspondences between incomplete and comple- tion point clouds to train their models, such as ASFM-Net (Xia et al., 2021) and PVD (Zhou et al., 2021). In contrast, the unpaired (unsupervised) approaches suggest methods that do not rely on paired data. In this regard, unpaired point cloud completion is problem.
In this paper, our key idea is to train our model to learn the UOT Map from the incomplete point cloud distribu- tion μ to the complete point cloud distribution v. In Sec 4.1, we formulate unpaired point cloud completion as an (U)OT Map problem and present an extensive comparison to identify the most suitable cost function. In Sec 4.2, we introduce our max-min learning objective. In Sec 4.3, we provide implementation details, such training algorithm.
general and challenging
Unpaired (Chen et al., 2020) is one of the first approaches for unpaired point completion. This model introduced a GAN-
3

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Unpaired Point Cloud Completion via Unbalanced Optimal Transport
Table I: Cost function evaluation by comparing the cost-minimizer gi(x) and the ground-truth completion yo for each incomplete point cloud x. We evaluate the suitability of each cost function for UOT-UPC by measuring the L1 Chamfer distance (ed 102(4) between of (2) and yd(())
(a) Multi cateory
Cost Function
AVG
chair
table
trash bin
TV
cabinet
bookshelf
sofa
lamp
bed
tub
USSPA
8.64
7.40
8,88
9.13
8.70
11.48
7.61
6.52
10.01
8.72
8,30
lg
23.97
12.52
31.21
29.17
26.65
22.29
22.96
20.51
24.64
27.03
21.80
edf2
9.78
8.07
7.69
14,00
5.91
18.86
7.88
7.34
6.23
8.76
7.07
ed Fand
8.87
9.48
8.62
9.38
7.80
10.55
7.73
5.63
14.59
10.32
7.28
InfoCD
8.46
7.43
6.41
11.69
5.69
17.35
6.52
6.25
2.70
6.91
4.92
(b)
Single-category
Cost Function
AVG
chair
table
trash hin
TV
cabinet
bookshelf
sofa
lamp
bed
tub
USSPA
7.18
7.44
7.15
6.98
6.08
10.02
7.00
6.12
8,35
7.90
4.79
14.88
11.21
12.52
22.37
8.29
20.46
17.87
8.69
11.57
19.55
7.07
od
6,65
7.17
7.35
8.35
5.46
10.59
5.77
6.39
3.70
6.46
5.28
2d Imdd
6.12
7.29
7.41
7.23
5.18
9.03
6.45
4.64
2.82
6.75
4.44
InfoCD
5.58
6.84
5.90
6.91
5.29
7.86
4.37
5.75
2.72
5.78
4.51
Table 2: Class imbalance in the USSPA benchmark dataset (Ma et al., 2023). The Incomplete and Complete rows indicate the proportion of each class in the respective datasets. The Ratio represents the proportion ratio (incomplete/compl A Ratio * indicates the presence of class imbalance.
class
chair
table
trash hin
TV
cabinet
bookshelf
sofa
lamp
bod
tuh
Incomplete
43%
21.3%
8.0%
6.4%
6.0%
6.1%
3.9%
1.1%
2.9%
1.2%
Complete
22.2%
22.2%
1.9%
6.1%
8.7%
2.5%
17.6%
12.9%
1.3%
4.7%
Ratio
1.94
0.96
4.21
1.05
0.69
2.44
0.22
0.09
2.23
0.26
4.1. Motivation
cloud samples. i.c., T(x) ~ 1/ for x ~> j that (2) optimally minimizes the given cost function c(x, T(x)). Therefore, condition (i) is inherently satisfied. The key remaining ques- tion is:
Task Formulation as OT Map We begin by defining our target task: Unpaired point cloud completion. Consider two sets of point cloud data: the incomplete set X {x; x; E X 1.... N} and the complete set Y {yı y/ E V.j 1.... M). Note that X and Y are unpaired, i.e., X and Y are independently sampled from μ and v. In real- world scenarios, obtaining incomplete-complete pairs of point clouds is prohibitively expensive, making this unpaired approach essential. For instance, one can train a model on incomplete real-world objects with complete synthetic data, then perform completion on real-world inputs (Ma et al., 2023). Formally, our objective is to train a completion model T from the unpaired datasets:
Q. Can we induce the OT Map to satisfy condition (ii) by selecting an appropriate cost function c(---)?
If we can identify such cost function c(:,)) that induces an explicit bias in 2 to satisfy condition (ii), then 7 can serve as the point cloud completion model.
Cost Function Comparison By definition, the OT Map T is the cost minimizer among all target distribution gener- ators (Eq. 1). Hence, in order to satisfy condition (ii). the chosen cost function c(.,-) should assign lower cost to c(x. Txx)) when T(x:) is the correct completion of a and higher cost to c(xr, y) when y is not the correct correspond- ing completion. In this regard, we test various cost function candidates, including 4, L2-Chamfer distance (cd"2) (Fan et al., 2017), one-directional L2- -h-mmfffer distance (od (wd), and InfoCD (Lin et al., 2024). For an incomplete (partial) point cloud x; {rmn E R$ and complete point cloud Vj (y/m E R3,, each cost function is defined as follows:
T: T: I I(T).
(5)
where e and T(x) denote the input incomplete point cloud and its corresponding completion. This completion model T must satisfy the following two conditions:
(i) T should generate a complete point cloud sample. (ii) T should transport each incomplete point cloud to its appropriate complete counterpart y-
12(21, Uj) = En, || V2nl12.
In this regard, the optimal transport map (Eq. 1) is suitable for the point completion model. By definition, the optimal transport map T (1) a generator of the complete point
. 2 (1t.1)) ()t,)))
En minm || - 9ym I1+ min, | - Vjm 112-

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Unpaired Point Cloud Completion via Unbalanced Optimal Transport
Lod w5) ) = En minm |0 Vim
Unfortunately, the standard OT problem (Eq. 1) is suscep- tible to this class imbalance problem (Fyring et al., 2024). The standard OT Map transports each source sample μ to a target sample y ~ y without rescaling. Consequently, in this class imbalance case, 20% of the 'Table' incomplete point cloud data would be transported to 20% of the 'Chair' complete point cloud. This behavior is undesirable for a completion model. In practice, this class imbalance prob- lem is prevalent in the unpaired point cloud completion benchmark (Table 2). The proportion of some categories, e.g., 'lamp' and 'trash bin' classes, significantly differs by more than threefold between the incomplete and complete datasets. To address this, we propose using the UOT Map as our point cloud completion model. The robustness of UOT to class imbalance will be explained in See 4.2 and empirically validated through experiments in See 5.2.
InfoCD(I; Wj) - (nntocD 1/)) + EnnooD v.).
where LinooDD (It.
exp(- 3 min. d( (bz))
-- 13 Em log
Σm exp min, d( ((H.m.M))))
For each partial point cloud x and a given cost function G, we select k-nearest complete samples (z) for 1 I S. k based on c(x;,) .) on the target (completion) dataset. Then, we compare them with the ground-truth completion (x). Our goal is to evaluate each cost function by testing whether the k-nearest neighbor v/{ (1)) is indeed similar to the ground- truth completion yd (z) ). If so, this suitable cost function can be exploited to train our OT-based completion model via the optimal transport map. The experiment is conducted on paired completion data from ShapeNet (Chang et al., 2015).
Fig. visualize the incomplete point cloud x, the ground- truth completion (z), and the 3-nearest neighbor (z(z)) for the cd and InfoCD cost functions (See Appendix B for additional results for the single-category setting). Fig. 1 show that selecting the cost- cost-minimizing pair based on InfoCD retrieves an appropriate vf(z), which closely resembles ya(x), (z:), in the multi-category setting. Table presents similar results. Table I reports the LI chamfer dis- tance between yot (z) and the nearest neighbor (z) for each cost function. The results indicate that the 12 cost per- forms the worst. This result shows that 22 cost is unsuitable for the point cloud completion task. In contrast, the InfoCD achieves competitive results, performing comparably or bet- ter than USSPA on the majority of datasets. Therefore, in Sec 4.2, we propose an OT Map approach using the In- foCD cost function for the point cloud completion task, based on our investigation of the most suitable cost fune- tion. Furthermore, we conduct an ablation study on the cost function in Sec 5.3 to demonstrate how this cost function comparison closely aligns with the completion performance of UOT-UPC.
4.2. Proposed Method
In this section, we introduce our unpaired point cloud com- pletion model, which is based the UOT Map, called UOT-UPC Our approach is to learn the UOT Map T* from the incomplete point cloud distribution μ to the complete point cloud distribution // using a neural network Tg. To achieve this, we adopt the UOTM framework (Choi et al., 2023), which is based on the semi-dual formulation (Eq. 6) of the UOT problem (Vacher & Vialard, 2023)).
1/x
Cuoa(H, = sup vEC
Vi (-v( du(x)
+ +/ʃ -VV((-oo())dddt)))
(6)
where the c-transform of v is defined as (2)) inf (c(z, v) v(y)). We refer to the optimal maximizer wEy o² of Eq. 6 as the optimal potential function for the UOT problem. Following prior approaches for learning the opti- mal maps (Korotin et al., 2021; Fan et al., 2022; Rout et al., 2022; Choi et al., 2023), we introduce Ta to approximate the UOT Map T as follows:
UOT Map for Class Imbalance Problem In this para- graph, we clarify the motivation for adopting the UOT Map, instead of the classical OT Map. Our goal is unpaired point cloud completion, where the training data X and Y are not provided as paired samples. This inherently introduces the class imbalance problem. For instance, consider point cloud data consisting of the 'Chair' and 'Table' categories, where incomplete and complete point clouds originate from different sources, such as real-world scans and synthetic datasets. In this case, the ratio of these two classes may differ between the incomplete point cloud distribution μ and the complete point cloud distribution v. For example, while the incomplete point cloud data might consist of 50% 'Chair" and 50% 'Table,' the complete point cloud data could be composed of 70% "Chair" and 30% 'Table."
(x) E arginf [C(a, v) - -vy)]
yEy
v (x) (o,TT( - (To())) (7)
Note that the UOT Map T satisfies the above conditions (Eq. 7) with the optimal potential v* (Choi et al., 2023). By parametrizing the optimal potential 5² with a neural network us and substituting rf using the right-hand side of Eq. 7, we arrive at the following learning objective Leg,To:
Lng.Te
inf vi (- inf [c (a. To(x)) - -vs (Io(z))) du(x) Uo /
To
1/8 us V3(( du()) (8)
+
5

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Unpaired Point Cloud Completion via Unbalanced Optimal Transport
Algorithm 1 Training algorithm of UOT-UPC
point clouds. We conducted ablation studies on the mix- ture trick and the cost function in Sec 5.3.
Require:
The mixture of the incomplete and complete point cloud distribution μ. The complete point cloud distribution V. V. (z) = Softplus(r)). Generator net- work Ta and the discriminator network Ug. di is density loss. Total iteration number K.
5. Experiments
In this section, we evaluate our model from various per- spectives. For implementation details of experiments, please refer to Appendix A.
1: for k 0, 1,2,..... K do
2: Sample batch X p. Y ~.V.
3: LT = [XT Ezex c(2, To(x)) us (To
In Sec 5.1, we evaluate our model on the unpaired point cloud completion benchmarks, considering both single- category and multi-category settings.
dl (To(x))
4:
Update 0 by minimizing the loss LT.
5: C. -- + Vi (x + u (To (x)))
In Sec 5.2, we demonstrate our model's robustness to the class imbalance problem.
+ Ever (z))v((v))
6:
Update e by minimizing the loss Ly
In Sec 5.3, we conduct ablations studies to investigate the effect of different cost functions and the source mixture trick.
7:
end for
5.1. Unpaired Point Completion Benchmark
Note that the learning objective Log.T, reduces to the stan- dard OT Map when V;; is the identity function (Eq. 3), which is equivalent to setting V as the convex indicator function at {1}. Hence, the UOT Map is a generalization of the OT Map. Furthermore, the UOT Map can be interpreted as the OT Map between the rescaled distributions <o(x)) = and T7 (1) = V2'(- (()="(()))t))) (f))v(y), where o" denotes the optimal potential (Choi et al., 2023). These rescaling factors (.) offer the flexibility of the UOT map to handle the class imbalance problem, which is a key challenge in unpaired point cloud completion (Eyring et al., 2024).
We assess our UOT-UPC model on the unpaired point cloud completion benchmarks under two settings: (1) Real Data Completion (USSPA dataset (Ma et al., 2023)) and (2) Syn- thetic Data Completion (PCN dataset (Yuan et al., 2018)). For quantitative evaluation, we measure the LI Chamfer distance (Fan et al., 2017) (ed'l) and F-scores (Tatarchenko et al., 2019) (FO and F.%.). These scores assess the quality of the completion results by comparing them against the ground-truth completions in the test dataset.
(1""""
Real Data Completion The primary advantage of un- paired point cloud completion is its ability to train models using incomplete and completion point cloud data from different sources. In this respect, USSPA (Ma et al., 2023) introduced a benchmark, where incomplete point clouds are obtained from real scans, while complete point clouds come from synthetic data. This dataset consists of ten object categories, including chairs, trash bins, lamps, etc. We eval- uate our model in two settings: the single-category setting, where performance is assessed separately for each object category, and the multi-category setting, where the model is trained and tested across all categories combined. The multi-category setting is particularly challenging, as the model should learn to complete partial point clouds from diverse categories. Our UOT-UPC is compared to diverse paired (Poin'lr, Disp3D, and TopNet) and unpaired models (Shapelnv, Unpaired, Cylce4, and USSPA). For the paired approaches, each model is trained using the paired label provided in the USSPA dataset.
4.3. Implementation Details
As described in Algorithm 1, Log.22 can be computed by the Monte Carlo approximation with mini-batch samples from the incomplete point cloud and the complete point cloud y. Intuitively, our learning objective is similar to the adversarial training in GANs (Goodfellow et al., 2020). Our potential vs and completion model To play similar roles as the discriminator and generator in GANs, respectively. This is because the minimization with respect to To in Eq 8 is equivalent to the maximization of Leg.T.
We parametrize the generator and discriminator using the similar backbone network as USSPA (Ma et al., 2023) (See Appendix A for the implementation details). InfoCD(..)) (Lin et al., 2024) is adopted as the cost function -(.--) in the learning objective Leg.Te- Moreover, in practice, we set the source distribution jı as a mixture of the incomplete point
cloud distribution js and complete point cloud distribution v. with a mixing probability of 50%, i.e., ja = 0.5y + 0.5v. Then, we train the UOT Map between f and v. This mixture trick helps our generator to produce high-fidelity complete
Fig. 2 illustrates completion samples in the single category setting (See Appendix C. for qualitative results in the multi- category setting). Table 3 and 4 presents the LI Chamfer distance (cd'l) and F-scores (F0.1% 1% and F1%) ). The AVG column in Table indicates the average cd scores across all ten categories. Across almost all settings and metries, our
Since assume V; he convex and non-negative, its convex conjugate V is an increasing function.
6

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Unpaired Point Cloud Completion via Unbalanced Optimal Transport
Incomplete
GT
USSPA
UOT-UPC
booksheif
table
bed
Figure 2: Comparison of generated samples from UOT-UPC and USSPA on USSPA dataset in the single-category setting. Table Point cloud completion comparison on USSPA dataset in the single-category setting, assessed by LI Chamfer Distance (ed") x 10 (L)). The boldface denotes the best performance among unpaired methods. All scores taken from (Ma et al., 2023).
Method
AVG
chair
table
trash bin
TV
cabinet
bookshell'
sofa
lamp
bed
tub
PoinTr (Yu et al. 2021)
14.37
13.65
12:52
15.26
12.69
17.32
13.99
12.36
17.05
15.13
13.77
Paired
DispI1 (Wang et al., 2022)
7.78
6.24
8,20
7.12
7.12
10.36
6.94
5.60
14.03
6.90
5.32
TopNet (Tchapmi et al., 2019)
7.07
6.39
5.79
7.40
6.26
8.37
7.02
5.94
8.50
7.81
7.25
Shapelnv (Zhang et al., 2021)
21.39
17.97
17.28
33.51
15.69
26.26
25.51
14:28
16.69
32,33
14.43
Unpaired (Chen et al., 2020)
10.47
8.41
7.52
12.08
6.72
17.45
9.95
6.92
19.36
10.04
6,22
Unpuired
Cycle4 (Wen et al., 2021)
11.53
9.11
11:35
11.93
8.40
15.47
12.51
10.63
12.25
15.73
7.92
USSPA (Ma et al., 2023)
8.56
8.22
7.68
10.36
7.66
10.77
7.84
6.14
11.93
8.20
6.75
UOT-UPC (Ours)
7.60
7.51
6.33
8.83
6.07
11,54
7.32
6.61
7.30
9.00
5.45
Table 4: Point cloud completion comparison on USSPA
Table 5: Point cloud completion comparison on the PCN
dataset in the single-category setting and the multi-category setting, assessed by L1 Chamfer Distance (ed" x 102 (L))
dataset in the single-category setting, assessed by LI Cham-
fer Distance (cd') x 102 (L)).
and F-scores (F0.1% x 102 F1%
x 102 (t)).
soore
Method
AVG
chair
table
cabinet
sofa
lamp
Method
Singlo-cntegory (AVG)
Multi-category
PoinTr
5,49
5.61
5.68
6.08
5.67
4.44
p0.1%
F
oa!
F9.1%
Paired
Disp3D
2.51
2.42
2.30
2.38
2.44
3.00
TopNet
5.92
6.34
5.45
6.06
5.80
5.95
PoinTr
14.37
18.35
80.41
Paired
Disp3D
7.78
30.29
78.26
Shapelnv
19.05
23.18
15.66
17.14
22.85
16.40
TopNet
7.07
12.33
80.37
Unpaired
14.87
12.87
8.14
14.30
18.23
20.82
Unpaired
Cycle4
17.60
14.25
15.73
21.06
21.54
15.40
Shapelny
15.58
66.53
19.35
16.98
69.66 06.66
USSPA
12.63
13.52
9.66
8.89
15.51
15.57
Unpaired
12:20
64.35
10.12
10.86
66.65
Unpaired
Cycle4
9.98
60.14
12.00
8.61
56.57
UOT-UPC
7.87
9.96
8.74
6.41
7.83
6.42
USSPA
17.49
73.41
8.96
16.88
72.31
UPC
18.43
75.59
8.96
19.25
71.52
single-category setting. Our model significantly outperforms other unpaired approaches, such as USSPA (Ma al., 2023). Note that the unpaired approaches tackle more challeng- ing problem, they do not rely on the pair information between incomplete and complete point clouds. As result, paired approaches generally achieve higher scores compared to unpaired methods. Nevertheless, our UOT-UPC model achieves competitive performance.
model outperforms other unpaired models. In particular, in the single-category setting, our model attains the best results across all metrics the AVG column. In the multi-category setting, our model achieves the best performance in cd'1 and 10.1% while showing comparable results in F1% (See Appendix C.2 for qualitative results on the real-world KITTT dataset (Geiger et al., 2012)).
5.2. Robustness to Class Imbalance of UOT approach
Synthetic Data Completion We also evaluate our UOT- UPC on the PCN dataset (Yuan et al., 2018). 1).. Similar to the USSPA dataset, we compare our model with the same paired and unpaired approaches. Table reports the results for the
In this section, we explore the robustness of our model in class-imbalanced settings. Specifically, we examine how the performance of existing point cloud completion models
7

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Unpaired Point Cloud Completion via Unbalanced Optimal Transport
Table 6: Comparison of class imbalance robustness
Table 8: Ablation study on the source mixture trick, i.e., the complete input. Complete Input
(cd") x 102 (4)) on (Datal, Data2) = (TV, Table).
0.3
0.5
0.7
Category
od 4
score
f
scoree
USSPA
7.60
6.97
8,08
7.97
7.94
18.12
73.49
Single
OT-UPC
23.77
23.74
29.79
27.21
7.60
18.43
75.59
Ours
6.86
6.75
6.75
6.94
8.98
18.04
71.75
Multi
8,96
19.25
71.52
5.3. Ablation Study
Table 7: Ablation study on the cost c(.,-) (ed x 10 (4)).
Effect of Appropriate Cost Functional We conduct an ablation study by modifying the cost function e(;;) .) in our model (Eq. 8). Each model is evaluated in the multi-category setting and in the single-category settings for the 'trash bin' and "TV' classes. Table 7 demonstrates that model achieves the best performance using the InfoCD cost func- tion, followed by (cdl ed and 1 (See Table 10 for the cost ablation results on the PCN dataset). Note that this rank- ing closely aligns with our cost function analysis in Table 1. This consistency suggests a strong correlation between our theoretical cost function evaluation (Table 1) and actual model performance. Furthermore, these findings suggest that further exploration of alternative cost functions could potentially enhance our model's performance. We leave this exploration for future work.
Cost function
Multi-category
trush bin
TV
24.16
45.57
23.71
od?
10.12
10.40
6.47
od? Iwdd
13.58
10.16
7.39
InfoCD
8.96
8.83
6.07
changes with different class imbalance ratios. We select two categories of datasets: Datal (category: TV) and Data2 (category: Table). These categories are selected because of their relatively abundant training samples and the distinct differences in their shape. For the incomplete point cloud samples, we use the entire training data for both Data and Data2, maintaining their ratio of 6.4 21.3 in Table 2. For the complete point cloud samples, we manipulate the
imbalance ratio r, i... Datal and Data2 are sampled at a ratio of 6.4 21.3 x r.
Add Complete Sample to Source As described in Sec 4.3, we introduce the source mixture trick into our model, where the source distribution is given as a mixture of incomplete and complete point cloud data with 50% mixing proba- bility. Here, we conduct an ablation study to evaluate the effect of this source mixture trick. The results are presented in Table 8. Across both single-category and multi-category experiments, our model almost consistently exhibits im- provements in both ed and F scores when utilizing the source mixture trick. Intuitively, this trick aims to assist the transport map in generating the target distribution better. Specifically, when given complete point cloud inputs, the optimal transport map should ideally learn an identity map- ping, which is relatively easier compared to completing an input incomplete point cloud. We hypothesize this property encourages the training process, enabling the model to gen- erate higher-fidelity complete point clouds more efficiently.
(Datal: Data2) (6.4 : 21.3) - (6.4: : 21.3 x r).
Then, each model is evaluated across diverse values of r to explore the effects of class imbalance. We compare our model to (i) the OT counterpart of our model (OT-UPC) and (ii) USSPA, the state-of-the-art method for unpaired point cloud completion. Note that, discussed in See. 4.2, our model reduces to the OT counterpart when V = Id. For detailed hyperparameter settings, See Appendix A.
As shown in Table 6, our model outperforms the two alter- native models across various class imbalance settings. (See Table 9 in the Appendix for results on other class combi- nations.) Note that we tested significantly larger total number of training samples, more than three times that of Datal (Table 2). Hence, setting >> 1 would result in discarding too many training data samples. Our model consistently demonstrates stable performance, ranging between 6.75 and 6.94 across various class imbal- ance ratios r, while USSPA shows considerably greater variance. In contrast, the standard OT generally performs poorly. We hypothesize that this phenomenon occurs due to the unstable training dynamics of the standard OT. The sta- ble training dynamics in learning the transport map is also another advantage of the UOT over OT (Choi et al., 2024b). In summary, these results indicate that our UOT-UPC offers strong robustness to class imbalance.
1, because Data2 has
6. Conclusion
In this paper, we introduced UOT-UPC, an unpaired point cloud completion model based on the UOT map. We for- mulated the unpaired point cloud completion task as the (Unbalanced) OT problem and investigated the cost function for this task. Our experiments demonstrated a strong corre- lation between cost function selection and completion per- formance. Our UOT-UPC attains competitive performance compared to both unpaired and paired point cloud comple- tion models. Moreover, our experiments showed that UOT-
8

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Unpaired Point Cloud Completion via Unbalanced Optimal Transport
UPC presents robustness the class imbalance problem, which is prevalent in the unpaired point cloud completion tasks. One limitation of our work is that while we explored various candidate cost functions, there may exist better cost functions for this task, e.g.. parametrized using neural net- works. We leave this exploration for future work. Also, our model sometimes exhibits unstable training dynamics due to adversarial training, which is commonly observed in GAN models.
balanced optimal transport: Dynamic and kantorovich formulations. Journal of Functional Analysis, 274(11): 3090-3123, 2018.
Choi, J., Choi, J., and Kang, M. Generative modeling through the semi-dual formulation unbalanced opti- mal transport. In Thirty-seventh Conference on Neural Information Processing Systems, 2023.
Choi, J., Chen, Y., and Choi, J. Improving neural optimal transport via displacement interpolation. arXiv preprint arXiv:2410.03783, 2024a.
Acknowledgements
This work was supported by the NRF grant [RS-2024- 00421203] and the IITP grant funded by the Korea gov- ernment (MSIT) 1SS-2021-1211343-G6
Choi, J., Choi, J., and Kang, M. Analyzing and improv- ing aetimal-transpont bbsssp adversarial networks. In The Twelfth International Conference on Learning Represen- tations, 2024b.
Artificial Intel- ligence Graduate School Program (Scoul National Univer- sity)]. Jaewoong supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (RS-2024-003346646) Jaemoo is sup- ported by National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT)(RS-200)
Eyring, L.,, Klein, D., Uscidda, T., Palla, G., Kilbertus, N., Akata, Z., and Theis, F, Unbalancedness in neu- ral monge maps improves unpaired domain translation. In The Tweifth International Conference on Learning Representations, 2024. URL https:://////. M //pprr/vee/.......... net / ftt.forrmmmidddnnnaaaaa
00342044).
Impact Statement
Fan, H., Su, H., and Guibas, L. J. A point set generation net- work for 3d object reconstruction from single image. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 605-613 2017.
The point cloud completion research contributes positively to various fields, including autonomous driving, robotics and virtual/augmented reality. Also, is applicable to urban planning and cultural heritage preservation. Our research does not involve personal data human subjects, and we have carefully addressed potential data bias issues. We also risks related to illegal surveillance
Fan, J., Liu, S., Ma, S., Chen, Y., and Zhou, H.-M Scal- able computation of monge maps with general costs. In ICLR Workshop on Deep Generative Models for Highly Structured Data, 2022.
ensure that there or privacy violations. As a result, we believe that this re- search is conducted ethically and poses no social or ethical concerns.
Fan, J., Liu, S., Ma, S., Zhou, H.-M., and Chen, Y. Neural monge map estimation and its applications. Transac- tions Machine Learning Research, 2023. ISSN 2835- 8856. URL https: /penreviewwwweeeeooeeeeeeeeeeewe wweeteettttooee forum?
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Unpaired Point Cloud Completion via Unbalanced Optimal Transport
A. Implementation Details
Unless otherwise stated, our implementation follows the experimental settings and hyperparameters of USSPA (Ma et al., 2023).
A.1. Network
We adopt the generator and discriminator architectures from the USSPA framework as completion model T3 and potential vgg For the potential Ug. the final sigmoid layer of the discriminator omitted to allow for the parameterization of the potential function, enabling outputs assume any real values. Additionally, we remove the feature discriminator streamline the architecture. In the potential Ve. implement the encoder proposed by (Yuan al., 2018) in their Point Cloud Networks (PCN). Following the encoder, we employ an MLPConv layer specified as MLPConv(CHn, MLPConv(1024, [256, 256, 128, 128, 1)), which indicates that the output is computed as follows:
y = Conv MawvID,--a 128.0s (ReLU ReLUU (eLLU 1024
1024,C) (256 ...)....
(9)
Here, Conv MaavIDD,,cc Can. represents ID convolutional layer with Cin input channels and Cour output channels.
The completion model To receives as input a concatenation of the incomplete point cloud and a complete point cloud. These imputs are processed independently to generate distinct complete point cloud samples. The completion model Ta follows an Encoder-Decoder architecture, augmented by an upsampling refinement module (upsampling module) in sequence. The upsampling module is implemented using a 4-layer MLPConv network, where the final MLPConv layer is responsible for refining and adding detailed structures to the output (Ma al., 2023). Specifically, the inputs to the last MLPConv layer are composed of the skeleton point cloud produced by the Encoder-Decode structure and the features extracted from the third MLPConv layer.
A.2. Implementation detail
Motivation Optimal Cost Function The incomplete and complete point clouds utilized in the optimal cost function outlined in Sec 4.1 are sourced from the dataset proposed by (Ma et al., 2023). This dataset consists of paired incomplete and complete point clouds. For fair comparison, we shuffle the complete point clouds to create an unpaired setting. We then use these shuffled point clouds as artificial complete data to train the USSPA model.
Training Concerning the loss function L. We employ Infocd as the cost function with a coordinate value of r = 0.044. For the hyperparameters of InfoCD, we set Tinfoed to 2 and AIafoDD to 1.0 x 10-7 The functions V and V are defined using the Softplus activation, SP(x) = 2 log(1 + e*)) - log 2. As a regularization term, we incorporate the density loss dl proposed by (Ma al., 2023), and we designate a coordinate value of 10.5 for dl. The objective of Potential vy is to assign high value to target sample y while assigning lower values generated sample ÿ. We utilize the Adam optimizer with 31 = 0.95, 32 0.999 and learning rates of 1.0 10-5 for both the potential Ui and completion model To, respectively. The training is conducted with batch size 4. The maximum epoch of training is 480. We report the final results based on the epoch that yields the best performance.
Ablation study Effect of Appropriate Cost Functional 2 fwdd 2 and 12
We set cost function coordinate value T = 100 for cost function
Ablation study Add Complete Sample to Source For a fair comparison, we set the learning rates of 2.0 x 10-5 for both the potential Us and completion model Té. respectively, when not using the source mixture trick.
"The softplus function is translated and scaled to satisfy SP(0) = and SP' (0) = 1.
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Unpaired Point Cloud Completion via Unbalanced Optimal Transport
Evaluation Metries
L1-Chamfer Distance ed' (Fan et al., 2017)
od ( Wj) =
Σ
min I Minl12 +
Σ
min | Vanll2.
(10)
[
n
y5
m
where each of Mj is point cloud
F score F score
(Tatarchenko et al., 2019)
2 x P(a)) x R(aa)
Fa score
(11)
P(a) R(a)
where P(a) [Z Em; min, 2m Bin [2) measures the accuracy of zz.
and R((:)= (y Eayl min
(I [la)
measures the completeness of 24.
lasl
A.3. OT-UPC
For the potential Mg. we implement MI.PConv(522, [128, 128, 1]) following the PCN encoder (Yuan et al., 2018). We incorporate RI regularization (Roth et al., 2017) and R2 regularization (Mescheder et al., 2018) to the loss function L,, Both regularization terms are assigned coordinate values r1 = r2 = 0.2. The density loss di is excluded from the L gradient clipping value of 1.0 is applied. We use Adam optimizer with 31 0.9, fa = 0.999 and a learning rate IrT, 5.0 x 10 for the completion model Tg. In addition, we use Adam optimizer with 31 0.9, 32 0.999 and learning rate lr.g 1.0 10-7 for the potential vg. All other settings not explicitly mentioned follow those of our model, UOT-UPC.
13

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Unpaired Point Cloud Completion via Unbalanced Optimal Transport
B. Cost Function Evaluation
Incomplete
GT
cd
InfoCD
booksheif
chair
trash bin
Figure 3: Visualization of the incomplete point cloud x, the ground-truth completion (x)), and the three complete point clouds (z) that minimize the cost c(x, (z) for two cost functions: cd and InfoCD, in the single-category setting.
Incomplete
GT
cd
12
bookshelf
chair
trash bin
Figure 4: Visualization of the incomplete point cloud x, the ground-truth completion (z), and the three complete point clouds (z) that minimize the cost c(x, of (x)) for two cost functions: ed22 fund and 12, in the single-category setting.
14

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Unpaired Point Cloud Completion via Unbalanced Optimal Transport
Incomplete
GT
cd fdd
12
TV
lamp
bed
Figure 5: Visualization of the incomplete point cloud x, the ground-truth completion y (I), and the three complete point clouds yf (z) that minimize the cost c(x, yf (z)) for two cost functions: ed fud and 12, the multi-category setting.
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Unpaired Point Cloud Completion via Unbalanced Optimal Transport
C. Additional Results
C.I. Additional Qualitative Results
Incomplete
GT
USSPA
UOT-UPC
chair
table
trash bin
TV
cabinet
Figure 6: Comparison generated samples from our UOT-UPC and USSPA the single-category setting.
16

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Unpaired Point Cloud Completion via Unbalanced Optimal Transport
Incomplete
GT
USSPA
UOT-UPC
bookshelf
sofa
lamp
bed
tul
Figure 7: Comparison of generated samples from UOT-UPC and USSPA in the single-category setting.
17

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Unpaired Point Cloud Completion via Unbalanced Optimal Transport
Incomplete
GT
USSPA
UOT-UPC
chair
table
trash bin
TV
cabinet
Figure 8: Comparison of generated samples from UOT-UPC and USSPA in the multi-category setting.
18

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Unpaired Point Cloud Completion via Unbalanced Optimal Transport
Incomplete
GT
USSPA
UOT-UPC
bookshelf
sofa
lamp
bed
tub
Figure 9: Comparison of generated samples from UOT-UPC and USSPA in the multi-category setting.
19

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Unpaired Point Cloud Completion via Unbalanced Optimal Transport
C.2. Qualitative comparison between UOT-UPC and existing methods on the KITTI dataset.
Input
UOT-UPC
Input
UOT-UPC
Figure 10: Point cloud completion results of the UOT-UPC model on the KITTI dataset (Geiger et al., 2012). The model trained on the ShapeNet dataset under the car category and tested on partial point clouds from the KITTI dataset without fine-tuning. From the qualitative comparison with previous approaches (Fig 1I), our UOT-UPC model achieves higher-fiee figher-fidelity point cloud completion, demonstrating better global structure and more evenly distributed points.
KiTH
Input
ShapeInv.
OptDE
ACL-SPC
Input
Shapelnv.
OptDE
ACL-SPC
Figure 11: Point cloud completion results of previous models the KITTI dataset (Geiger al., 2012). The generated samples are taken from ACL-SPC (Hong al., 2023), which is self-supervised model. The others are unsupervised approaches: ShapeInv (Zhang et al., 2021) and OptDE (Gong et al., 2022).
20

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Unpaired Point Cloud Completion via Unbalanced Optimal Transport
C.3. Comparison of class imbalance robustness for diverse class combinations.
Table 9: Comparison of class imbalance robustness (ed x 10 (U)) between UOT-UPC (ours), USSPA, and OT-UPC on diverse class combinations (Datal, Data2). Our UOT-UPC consistently outperforms other models across a wide range of class imbalance ratios in both additional class settings.
(a) (Datal, Data2) = (Lamp, Trash bin) with sample count (1.1 8..0 r).
0.3
0.5
0.7
USSPA
10.16
9.49
10.21
10.21
OT
25.68
21.95
28.41
25.36
Ours
9.42
9.48
9.57
9.44
(b) (Datal, Data2)
(Lamp, Bed) with sample count (1.1 2,9 "
r
0.3
0.5
0.7
USSPA
9.64
9.78
9.27
9.79
OT 18.99
21.23
19.27
22.12
Ours
8.95
8.91
8.98
8.73
C.4. Additional experimental results the PCN dataset
Table 10: Ablation study on the cost function e):,)) on the PCN dataset (ed 102 (4)). The results are consistent with
Table 7. InfoCD achieved the best performance, while the L2 distance yielded the worst results.
Cost function
cabinet
sofa
lamp
l
23.86
19.92
19.22
cd?
8.85
8.40
7.07
cd2 /wd
13.53
11.32
12.78
InfoCD
6.41
7.83
6.42
21

---

Unpaired Point Cloud Completion via Unbalanced Optimal Transport
C.5. Ablation study cost-intensity
We evaluate the robustness of our model with respect the cost-intensity hyperparameter т, defined c(z, y) InfoCD(x;, y). Specifically, we tested our model the multi- cateiry setting and the single-category settings of the "bookshelf" and 'lamp' classes, while changing conditions by setting the maximum T to mmax = 0.25 and the minimum to Twin = 0.02, resulting in ratio of Tiax/ Tmin > 10. As depicted in Fig. 12, our model shows moderate performance across various 7 values. In particular, the sweet spot of r lies roughly between 0.044 and 0.1. The performance deteriorates by approximately 10% when is either too large (Fmax) or too small (Tmin).
{0.02, 0.025, 0.044, 0.1, 0.25). Note that impose challenging
9.5
10P
9.0
Multi category booksheif
8.5
lamp
7.5
0.02
0.025
0.044
0.1
0.25
Figure 12: Ablation study on the cost intensity (ed 10 (!)).
22


Reviews:
[{"review_id": "FYhZVwHoQn", "reviewer": "Reviewer_xCQJ", "summary": "This paper introduces Unbalanced Optimal Transport Map for Unpaired Point Cloud Completion (UOT-UPC), which is a novel point cloud completion approach that uses unpaired point clouds during training. Unlike previous approaches that have formulated the point cloud completion task as an optimal transport problem, the authors propose to instead formulate it as an unbalanced optimal transport problem to loosen the exact matching constraint in OT, which helps with training on unbalanced datasets. They train a Neural OT to learn a UOT map that transports incomplete point cloud to complete point cloud using InfoCD as the cost function. The authors evaluate their approach on different, real, synthetic, and hybrid datasets and show impressive performance compared to other models.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for carefully reading our manuscript and providing valuable feedback. Moreover, we appreciate the reviewer for considering \"the most important contribution is that this paper is the first to bring UOT to unsupervised point cloud completion\". We hope our responses to be helpful in addressing the reviewer's concerns. \n\n$ $ \n\n---\n> **Q. [Other Strengths And Weaknesses]** The paper's clarity could be improved by explicitly defining important benchmarks before using them in the text. For example, Section 3.1 first compares UOT-UPC with USSPA without first introducing USSPA.' \n\n\n**A.** We appreciate the reviewer for providing valuable suggestions. Following the reviewer's advice, **we moved the Related Works section (Sec 4) before the Method (Sec 3)**. This reordering allows us to introduce USSPA before presenting experimental results in Section 3.1.\n\n$ $ \n\n---\n> **Q. [Other Comments Or Suggestions]** The name of the approach is misspelled in the bolded section of the third paragraph in the introduction. Also, the beginning of Section 3.1 defined the set $X$ as: $$X = \\{x_i \\mid x_i \\in X, i = 1, \\dots, N\\}$$ for incomplete point clouds and the set $Y$ as: $$Y = \\{y_j \\mid y_j \\in Y, j = 1, \\dots, M\\}$$  for complete point clouds. However, the cost function comparison uses $n$ for complete and $m$ for incomplete point clouds. I believe it would be a bit clearer to change the variables used for the Task formulation section.\n\n**A.** We appreciate the reviewer for the careful comment. (1) We corrected the typo in the bolded section of the third paragraph in the introduction. (2) Following the reviewer's advice, we revised the variable notation in the cost function definitions to ensure consistency. Specifically, we now consistently use the $n$ for the incomplete point clouds $x$ and $m$ for the complete point clouds $y$ throughout the paper.", "claims_and_evidence": "The claims are supported by clear and convincing evidence.", "methods_and_evaluation_criteria": "The methods and evaluation criteria make sense of this problem.", "theoretical_claims": "I have not checked the correctness of any proofs for theoretical claims.", "experimental_designs_or_analyses": "The experimental designs are sound. The datasets used consist of real and synthetic scans of objects, as well as real world scans which cover a variety of different types of data.", "relation_to_broader_scientific_literature": "There has been a growing research focus on training models with unlabeled or minimally labeled data, particularly in 2D feature representation and unsupervised learning. This paper relates to this as it uses unpaired point cloud data to train a model for point cloud completion, which is unsupervised in nature. Another problem that is investigated is unbalanced datasets, which the paper shows UOT-UPC performs well with. I think that the most important contribution is that this paper is the first to bring UOT to unsupervised point cloud completion, where the problem has been typically formulated as a regular OT problem.", "essential_references_not_discussed": "There are not necessarily related works that are essential to understanding the key contributions which are missing.", "other_strengths_and_weaknesses": "This paper introduces an original and well-motivated approach to unpaired point cloud completion by using the unbalanced optimal transport formulation. The authors provide a comprehensive evaluation of their approach through their experiments and ablations, which show impressive performance and robustness on unbalanced datasets. The paper itself is also well written overall, with an algorithm figure that effectively describes their approach. However, the paper's clarity could be improved by explicitly defining important benchmarks before using them in the text. For example, Section 3.1 first compares UOT-UPC with USSPA without first introducing USSPA.", "other_comments_or_suggestions": "The name of the approach is misspelled in the bolded section of the third paragraph in the introduction. Also, the beginning of Section 3.1 defined the set $X$ as $X = \\{x_i | x_i \\in X, i = 1, \\cdots , N\\}$ for incomplete point clouds and $Y = \\{y_j | y_j \\in Y, j = 1, \\cdots , M\\}$ for complete point clouds. However, the Cost Function Comparison uses $n$ for complete and $m$ for incomplete. I believe it would be a bit clearer to change the variables used for the Task formulation section.", "questions_for_authors": "No questions so far", "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": "Emi2Vi44t3", "reviewer": "Reviewer_hu8B", "summary": "This paper studies the problem of reconstructing 3d objects from partial observations, an important problem in real-world graphics applications. While some approaches to this problem consider settings where a large dataset of partial and full observations of the same objects are available, this work focuses on the likely more practical settings where one merely has a dataset of partial observations and another dataset of full observations but the observations don't correspond to the same objects. Their method is based on solving an optimal transport problem with neural networks, with the important modification of allowing for un-balancedness, which improves the robustness of their method to imbalances in the class distribution of the data.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for carefully reading our manuscript and providing valuable feedback. We are especially grateful for the reviewer’s recognition of our contribution, noting that “this paper pushes forward the state of the art of point cloud completion by considering the unpaired problem and designing a strong method based on optimal transport, which is also robust to class imbalances.” Moreover, we appreciate the reviewer's positive assessment of our cost function analysis and empirical evaluations under class imbalance. \n\n$ $\n\n---\n> **Q. [Other Comments Or Suggestions]** You define the \"unbalanced optimal transport map\" on line 129, but does this necessarily exist?\n\n**A.** Thank you for the great question. In this work, we assume that the incomplete and complete point cloud distributions are absolutely continuous with respect to the Lebesgue measure (Lines 76-77). This is a mild assumption, as it is satisfied whenever the distributions admit probability density functions (pdfs). Under this assumption, **the existence of the unbalanced optimal transport map is guaranteed**, as stated in Thm 3.3 in [1].\n\nReference\n- [1] Liero, Matthias, Alexander Mielke, and Giuseppe Savaré. \"Optimal entropy-transport problems and a new Hellinger–Kantorovich distance between positive measures.\" Inventiones mathematicae 211.3 (2018): 969-1117.", "claims_and_evidence": "Their main claims are that their method is roughly state-of-the-art for unpaired data, and at least competitive for paired data, and also that it is robust to imbalances in the class distribution of the data. These claims are well-supported in my view.\n\nI also appreciated their careful exploration of the relative merits of cost functions as well as the performance of other methods under major class imbalances. Overall, I feel that the paper is very careful and empirical solid.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "It is a practical paper so there aren't significant theoretical claims.", "experimental_designs_or_analyses": "The experimental designs seem legitimate but I have not made a careful study of them.", "relation_to_broader_scientific_literature": "This paper pushes forward the state of the art of point cloud completion by considering the un-paired problem and designing a strong method based on optimal transport which is also robust to class imbalances.", "essential_references_not_discussed": "I am not aware of any un-discusses essential references.", "other_strengths_and_weaknesses": "None additional.", "other_comments_or_suggestions": "- You define the \"unbalanced optimal transport map\" on line 129, but does this necessarily exist?", "questions_for_authors": "None.", "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": "sSIC5ZbGv6", "reviewer": "Reviewer_LXHR", "summary": "This paper proposes UOT-UPC, a novel approach to unpaired point cloud completion using the Unbalanced Optimal Transport framework. The model formulates the completion task as an optimal transport problem and trains a neural network-based Neural OT map to learn the transport mapping from incomplete to complete point clouds. UOT-UPC is designed to handle class imbalance, a common issue in real-world unpaired completion tasks.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for carefully reading our manuscript and providing valuable feedback. Moreover, we appreciate the reviewer for considering our work addresses \"limitations in existing heuristic-driven methods\" by \"bridging OT theory and unpaired point cloud completion\". We hope our responses to be helpful in addressing the reviewer's concerns.\n\n$ $ \n\n---\n> **Q.[Questions For Authors]** Why is InfoCD the most suitable for completion tasks? Please provide a theoretical explanation. MSNet and SpareNet previously used EMD to calculate the distance between point sets. Would EMD based on optimal transport be better than InfoCD?\n\n**A.** Thank you for providing the insightful comments. Following the reviewer's suggestion, **we conducted an additional ablation study using the EMD as the cost function for our UOT-UPC model**. The results are as follows (Table 7): \n\n- Comparison of EMD cost function with other cost functions on the USSPA benchmark, assessed by L1 Chamfer Distance $cd^{l 1} \\times 10^2$ ($\\downarrow$).\n \n|Cost function| Multi-category | trash bin | TV |\n|:---|:---|:---|:---|\n|$l_{2}$| 24.16 | 45.57 | 23.71|\n|$cd^{l2}$| 10.12 | 10.40 | 6.47|\n|$cd^{l2}_{fwd}$| 13.58 | 10.16 | 7.39|\n|EMD| 9.66 | 10.46 | 6.41|\n|Ours(InfoCD)| **8.96** | **8.83** | **6.07** |\n\nOur results show that **InfoCD consistently outperforms EMD** across all evaluated categories. Additionally, **training with EMD incurs significantly higher computational costs** due to its iterative computation procedure, as shown below:\n\n- Train time comparison on 'lamp' category from the USSPA [1].\n\n|Train (480 Epoch) | Time (sec) |\n|:---|:---|\n|EMD | 3781.91|\n|Ours(InfoCD) | 1320.80 | \n\nInfoCD is designed to prevent multiple points from being matched to a single point, **encouraging a more evenly distributed alignment** between point sets, compared to EMD (Fig 4 in [1]). This advantage arises from the **contrastive nature of the InfoCD** cost function [1], which introduces an additional repulsion effect between negative pairs.\nThis property aligns well with the goals of point cloud completion, where generating globally coherent and evenly spaced completions is critical. Empirically, this is supported by the higher-fidelity completions of the UOT-UPC model using InfoCD in Fig  2 and 10, where the **completions exhibit more globally evenly separated completions**.\n\n$ $\n\nReference\n- [1] Lin, Fangzhou, et al. \"InfoCD: a contrastive chamfer distance loss for point cloud completion.\" NeurIPS 2023.\n\n$ $\n\n---\n> **Q. [Theoretical Claims]** ... The paper acknowledges that the UOT-UPC training process may exhibit instability, resembling the mode collapse phenomenon observed in GAN training. However, no theoretical analysis is provided to explain why UOT-UPC training might be unstable.\n\n\n**A.** We appreciate the reviewer for providing constructive comments. As the reviewer said, we believe the observed training instability is due to the inherent difficulty of finding a Nash equilibrium in the min-max optimization, similar to challenges encountered in GAN training. In this regard, **several works investigated GAN instability both theoretically [1] and empirically [2]**. We agree that extending this analysis to the UOT-UPC setting would be a valuable future research direction.\n\nReference\n- [1] Mescheder, Lars, Andreas Geiger, and Sebastian Nowozin. \"Which training methods for GANs do actually converge?.\" ICML 2018\n- [2] Salimans, Tim, et al. \"Improved techniques for training gans.\" NeurIPS 2016.\n\n$ $ \n\n---\n> **Q.[Other Comments Or Suggestions]** The results of the proposed approach should be added to Figure 11.\n\n**A.** We agree with the reviewer that including UOT-UPC results in Fig 11, i.e., providing a completion result on the exact same incomplete point cloud, would allow a clearer qualitative comparison. However, we were unable to find the random seed necessary for reproducibility in the ACL-SPC model. As an alternative, **we chose to present multiple completion results on the KITTI dataset in Fig 10**. Across multiple incomplete point clouds, our model consistently produces higher-fidelity completions, exhibiting better global structure and more evenly distributed points.", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "The proposed methods are well-aligned with the problem of unpaired point cloud completion. The paper formulates unpaired point cloud completion as an Optimal Transport (OT) problem, which is theoretically sound for aligning distributions between incomplete and complete point clouds. The introduction of Unbalanced Optimal Transport (UOT) is particularly meaningful, as real-world point cloud datasets often exhibit class imbalance, making traditional OT less effective.", "theoretical_claims": "The paper presents theoretical claims related to the OT and UOT formulation for unpaired point cloud completion. The proofs supporting these theoretical claims are mathematically sound. The paper conducts a comprehensive analysis of various cost functions and concludes that InfoCD is the most suitable for completion tasks. This conclusion is primarily supported by experimental validation rather than additional mathematical proofs. The paper acknowledges that the UOT-UPC training process may exhibit instability, resembling the mode collapse phenomenon observed in GAN training. However, no theoretical analysis is provided to explain why UOT-UPC training might be unstable.", "experimental_designs_or_analyses": "The proposed method was evaluated on the USSPA and PCN datasets, where it demonstrated superior performance compared to previous state-of-the-art methods. Extensive ablation studies were conducted to thoroughly analyze the impact of different cost functions, further validating the effectiveness of the proposed approach.", "relation_to_broader_scientific_literature": "Prior unpaired completion methods (Cycle4 (Wen et al., 2021), USSPA (Ma et al., 2023)) use heuristic-driven adversarial training. The paper reformulates unpaired point cloud completion as an UOT problem which bridges OT theory and unpaired point cloud completion, addressing limitations in existing heuristic-driven methods.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "**Strengths:**\n\n- First work using UOT for unpaired point cloud completion.\n- Strong experimental validation across multiple datasets.\n- Addresses class imbalance problem, a major issue in real-world data.\n\n**Weaknesses:**\n\n- Training instability due to adversarial training aspects of OT.", "other_comments_or_suggestions": "The results of the proposed approach should be added to Figure 11.", "questions_for_authors": "Why is InfoCD the most suitable for completion tasks? Please provide a theoretical explanation. MSNet and SpareNet previously used EMD to calculate the distance between point sets. Would EMD based on optimal transport be better than InfoCD?", "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:

Meta-review: This paper presents a clear and original contribution by introducing Unbalanced Optimal Transport (UOT) to the task of unpaired point cloud completion, a problem of practical significance in 3D vision where paired data is rarely available. The proposed model, UOT-UPC, reformulates completion as a UOT problem and demonstrates strong robustness to class imbalance—an important and realistic challenge in this domain. Reviewers highlighted several strengths: the novelty of applying UOT in this context, the strong empirical performance on both synthetic and real datasets, and the comprehensive ablation studies—particularly on the impact of the cost function choice. The authors also clearly motivated their use of the InfoCD cost and supported it with both quantitative results and theoretical intuition. The paper is well-structured, and the authors were responsive during the rebuttal phase, addressing all technical and presentation-related suggestions (including clarity improvements and additional experimental comparisons). Overall, this work advances the state of the art in unpaired point cloud completion with a well-motivated, well-executed approach grounded in optimal transport theory. I recommend acceptance.
Decision: Accept (poster)