Curvature Diversity-Driven Deformation and Domain Alignment for Point Cloud
Authors: Mengxi Wu
Hao Huang
Yi Fang
Mohammad Rostami
Venue: Accepted by TMLR ()

Abstract: Unsupervised Domain Adaptation is crucial for point cloud learning due to geometric variations across different generation methods and sensors. To tackle this challenge, we propose Curvature Diversity-Driven Nuclear-Norm Wasserstein Domain Alignment (CDND). We first introduce a Curvature Diversity-driven Deformation Reconstruction (CurvRec) task, enabling the model to extract salient features from semantically rich regions of a given point cloud. We then propose a theoretical framework for Deformation-based Nuclear-norm Wasserstein Discrepancy (D-NWD), extending the Nuclear-norm Wasserstein Discrepancy to original and deformed samples. Our theoretical analysis demonstrates that D-NWD is effective for any deformation method. Empirical experiment results show that our CDND achieves state-of-the-art performance by a noticeable margin over existing approaches.

Published in Transactions on Machine Learning Research (09/2025)
Curvature Diversity-Driven Deformation and Domain Align- ment for Point Cloud
Mengxi Wu
mengrivnsuusc.edu
Deportment of Computer Science University of Southern California
Hao Huang
hh1811 @ny.. edu
Department Computer Science New York University
Yi Fang
yfang Dfang@ryy..co edu
Department of Computer Science New York University
Mohammad Rostami
mrostami@isi.cmm edu
Department of Computer Science Universily of Southern California
Reviewed on OpenReview: https: // openrevie.. net/ forum? id= aPXintFFrGGerrreeeree
Abstract
Unsupervised Domain Adaptation crucial for point cloud learning due to geometric vari- ations across different generation methods and sensors. To tackle this challenge, we pro- pose Curvature Diversity-Drive Nuclear-Norm Wasserstein Domain Alignment (CDND). We first introduce Curvature Diversity-driven Deformation Reconstruction (CurvRec) task, enabling the model to extract salient features from semantically rich re- gions of given point cloud. We then propose theoretical framework for Deformation-based Nuclear-norm Wasserstein Discrepancy (D-NWD), extending the Nuclear-norm Wasscrstein Discrepancy to original and deformed samples. Our theoretical analysis demonstrates that D-NWD effective for any deformation method. Empirical experiment results show that our CDND achieves state-of-tho-art performance by noticcable margin over existing ap- proaches, Our codes available at: https://githu: http:ttuubbcom/WXWX6///CCCC /earssec////Reeececce DNWD
1 Introduction
Adopting deep neural werworks (DNNs) for point cloud representation learning has led to significant success across applications such as robotics Maturana & Scherer (2015 Duan et al. (2021), autonomous vehi- cles Mahjourian et al. (2018); Cui al. (2021). and scene understanding [Zheng et al. (2013); Zhhu et al. (2017). However, point clouds captured under different conditions often exhibit substantial variations, lead- ing to performance degradation when DNNs tested data from domains different from those used during training. This discrepancy, known as the domain gap, is especially problematic in real-world settings. A straightforward solution- ronttion-rttraiingg the model on newly labeled data-is typically impractical due to the high cost of manual annotation. Unsupervised domain adaptation (UDA) addresses this issue by transferring knowledge from labeled source domains to unlabeled target domains. While UDA has been well studied in 2D planar domains (o.g., images), its extension 3D point clouds remains underexplored due challenges from the irregular, unstructured, and unordered nature of 3D data. These properties amplify geometric discrepancies bet ween source and target domains, making direct adaptation methods less effective.

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Published in Transactions on Machine Learning Research (09/2025)
To overcome these challenges, we propose Curvature Diversity-Driven Nuclear-Norm Wasserstein Domain Alignment (CDND). Our approach is motivated by the observation that not all regions in point cloud equally contribute to semantic understanding and geometric variation can reflect semantic richness. Key semantic features- such as object part boundarics and edges- ooften occur in regions with complex geometry. As a result, areas with greater geometric variation lend to carry more meaningful semantic information. Curvature diversity can capture geometric variation by measuring how much curvature changes within a local region. Regions with high curvature diversity reflecting substantial variation in surface orientation and structure- often correspond to semantically rich components such as edges or jointed structures. In contrast, low-diversity regions are typically flatter and less informative, like planar surfaces or object bases.
Our first contribution is the Curvature Diversity-driven Deformation Reconstruction (CurvRec). We intro- duce a principled region selection strategy guided by curvature diversity, which we quantify using entropy a standard statistical measure of uncertainty. Entropy captures how spread out the curvature values are within a region: high entropy indicates diverse curvature and complex geometry, while low entropy suggests goo- metric uniformity. Leveraging this, CurvRec selectively deforms low-entropy (less informative) regions and preserves high-entropy (semantically rich) ones. This contrasts with prior methods such as AAhituue et al. 22021), which apply random deformation, and Zou et al. (2021), which classify fixed high-curvature regions. Our method instead focuses deformation on low-diversity areas, allowing the feature extractor to prioritize learning from semantically rich regions.
Our second contribution is a theoretical framework for the Deformation-based Nuclear-norm Wasserstein Discrepancy (D-NWD). Unlike the conventional NWD, D-NWD incorporates features from both original and deformed samples to align source and target domains. This integration creates a more diverse and robust feature space, improving model generalization under domain shifts. Notably, prior work has not provided a theoretical analysis demonstrating that using the NWD to align features from both original and deformed (or augmented) samples can improve performance on the target domain dataset. Our analysis is the first to demonstrate the effectivencss of this strategy. Our thcorotical framework shows that D-NWD effectively reduces the domain gap between source and larget domains. Furthermore, our analysis illustrates that D-NWD generic and adaptable to any deformation method, not just the one presented in this paper. Our theorctical analysis is non-trivial since incorporating features from deformed samples requires us to establish entirely new bounds and prove that the NWD remains effective in this expanded feature space spanning both original and deformed samples.
Extensive experiments on standard classification and segmentation benchmarks demonstrate that CDND achieves state-of-the-art performance compared to existing approaches.
2 Related Works
UDA for Point Clouds While Unsupervised Domain Adaptation (UDA) has been widely explored for
2D planar image data Ganin & Lempitsky (2015); Mansour et al.
(2008) ; Stan Rostami (2024a) Du et al.)
b2 y Kumari & Singh| (2024), fewer works address UDA for 3D point clouds and non- coanplanar data, where
firect adaptation of 2D techniques proves non-trivial Qin et al. (2019) Achituve et al. (2021) Shen el al.
b022 Zou et al. (2021] Liang et al. 20022;; Chen et al. (2023): Katageri et al. [2024]; Wei et al.
(024); Wu
& Rostamil Pan et al. b; Cardace et al. 2022) Qin el al. Qin et al. (0019) propose PointDAN, combining
local and global domain alignment and introducing the PointDA benchmark for classification. Achituve et
al. Achituve et al. (2021) present rrconstrucioor-foo--fffooatttttt
method using PointMixup Chen et al.
(2020), and introduce the PointSegDA benchmark for segmentation. Zou et al. Zou et al.
(2021) design
two geometry- self-supervised tasks for learning Nomain-invariaan features. Fan et al. Fan et al. (2022)
introduce reliable pseudo-label voting mechanism along with global-local structure modeling. Shen et al. Shen ct al. (2022) employ geometry-aware implicit functions for modeling domain-specific variations. Liang el al. Liang el al. (2022) introduce a masked local 3D structure prediction task to promote feature invariance via spatial context recovery. Cardace et al. [ardacee et al. (2023) leverage self-distillation to iteratively refine pscudo labels. Katageri et al. Katageri et al. (2024) integrate contrastive learning with optimal transport to improve both alignment and discrimination. Chen et al. Chen al. (2023) develop a boundary point prediction task to enhance robustness via geometric boundary cues. Wei et al. Wei et al.
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2024] propose multi-scale part-based representation capturing both local and global part-level information for better adaptation and generalization. These works reflect a growing trend toward integrating geometric priors with adaptation objectives. Our work differs by proposing a principled deformation strategy rooted in curvature diversity, a sophisticated self-supervised task, and a generalized thcoretical framework (D-NWD) applicable to any deformation method, achieving state-oftthe-arl results.
Optimal Transport for Domain Adaptation Although many works on UDA rely on adversarial learn- ing Tzeng et al. (217)); Tang & Jia 2020); Long et al. (2018) Jian & Rostami (0233) to align the distributions
of two domains, many rely on direct distribution matching using metric minimization [Wang et al. (2017):
Li al. (2020): ; Fatras ct al. (2021). The Wasserstein distance from optimal transport theory widely
used in domain adaptation due its ability to caplure geometric relationships between distributions Courtyy
et al. [2016] 2017); Redko et al. (2017); Rostami & Galstyan (2023); Nanamukul et al. (2024). Gautheron
et al. Gautheron et al. (2019) propose Wasserstein-guided representation lcarning and a feature selection
method to lackle domain shift. Sliced Wasserstein discrepancy is used in Lee et al. (2019) Stan & -Rostami
(2024b) to replace the I distance in Maximum Classifier Discrepancy Saito et al. (2018), offering more
meaningful class-level divergence. DeepJDOT Damodaran et al. (2018) maps source-target pairs via a cou- pling matrix, while CGDM Du al.] (2021) minimizes cross-domain gradient discrepancies. Xu et al. Xul et al. 2020) propose a weighted transport method using spatial prototypes for better sample-level alignment. Fatras et al. Fatras et al. (2021) introduce unbalanced optimal transport with mini-batch training for scala- bility. Our D-NWD builds upon NWD Chen et al. 2022), extending it with ieformation-baset supervision and theoretical generality.
3 Proposed Method
We begin by defining the unsupervised domain adaptation (UDA) problem, followed by an overview of our UDA approach, called Curvature Drrvven Nuelear-Norm Wasserstein Domain Alignment (CDND), in Section 3.1 We then present the details of our main contributions: (1) Curvature Diversity-based
Deformation Reconstruction task, described in Scctions 3.2 and 3.3, and (2) thcoretical framework for D-NWD, which extends NWD to incorporate both original and deformed samples, as detailed in Section 3.4 and Note that our theoretical analysis applicable to any deformation method used.
3.1 Problem Formulation
We consider source domain with labeled samples and larget domain with unlabeled samples which has a different data distribution. Our goal is to develop a UDA method to train a model that accurately predicts labels for the target domain using both the source labeled datasct and the target unlabeled datasct. Let S represent the source domain, where X denotes the i-th batch of samples and y their corresponding labels. Similarly, let T represent. the target domain, where X is the i-th batch of samples. The feature space induced by S and T is denoted by Ωn, In addition, we introduce deformed domains Sd and 7d, with their feature space Qa. We assume that P. and Q2 are disjoint subsets of IR", i.c., Jon = 0, with RoUN S R". This assumption is generally valid in practice, as the probability of a deformed sample being exactly identical to original one is negligible. A point cloud from the source domain is denotod as Vg E RR= and from the larget domain is oe E R where the number of points. The corresponding deformed point clouds are denoted by z and z
The pipeline of our CDND is presented in Figure 1 Our model first uses feature extractor E to obtain shape features from both source and larget point clouds. To minimize domain gaps and ensure domain-invariant features, we: (1) use a curvature diversity-driven deformation reconstruction task via a reconstruction de- coder hest, and (2) employ the D-NWD align domains through a classifier C. The aligned features are then used for downstream tasks, í.e., point cloud classification and segmentation. The model is trained using source-labeled and target-unlabeled data.
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datellow
Reconstruction Loss SS..
Source datallow
Target detaflow
SSI.
Bechwordd propapation for fiss Backward propagation for Ko-swo
Alignment Loss LD-NWD
X
Reconstruction Decoder
|c/f lccfflll.
Feature Extractor
f
Label Classifier
&,
&
GRL
Curvature Diversity-Base Based Deformation
Compute curveture diversity
Select V regions and deform
Compute curvature
TPS K-NN
grouping
Figure 1: Pipeline of CDND. The inputs are the source batch X, and larget batch X,- We first deform them into X, and X2 using Curvature Diversity-Based Deformation. Next, X,, X,, X,, and X, are sent into a feature extractor. The features of deformed samples are fed into a reconstruction decoder to reconstruct the deformed regions. For domain alignment, both original and deformed features are sent D-NWD. Aside from the two losses shown in the figure, cross-entropy loss is computed on X, and X, with labels. An NWD loss CKWD on X, and X, is also computed to ensure prediction consistency between the target original and deformed pairs.
3.2 Curvature Diversity-Driven Deformation
To extract domainn-nnvriann features from point clouds, three deformation strategies have been explored in the literature AAchtuvee et al. (202]): volume-based ceature-based, and sample- based, according to the way of dividing point clouds into regions for deformation. Although these strategies use different techniques to select regions for deformation, they all divide a point cloud into regions based on some spatial locations or arrangements and uniformly randomly select regions to be deformed. However, uniformly random selection may not be optimal as regions within a point cloud vary in their semantic richness, í.e., some regions contain more semantic information that is generalizable across the domains, while some regions may be domain- specific. These semantically rich regions are crucial for tasks such classification, distinguishable characteristics. For instance, to differentiate point cloud of a plant from that of a lamp, focusing on the leaves and flowers which have richer semantic information would be more effective than focusing on the flower pot, which is similar to the base a lamp. Thus, deforming regions with richer semannic information causes the point cloud to lose semantic meaning, making difficult for a classifier to classify it. Our idca to identify shared semantic information to improve model generalization in the target domain. To encourage the feature extractor to prioritize regions with rich information, we propose deforming regions that are less semantically rich. Our deformation strategy helps the model learn to extract features from the most informative or salient regions of point cloud.
they have more
To evaluate the richness of semantics, we propose using curvature diversity as a measurement. For the curvature diversity-drine deformation, we adopt the following steps. First, we use Farthest Point Sampling (FPS) Moenning & Dodgson (2003) sample center points as centers of k regions of a given point cloud. Then, for each center point, we use k-Nearest Neighbor (k-NN) to select m nearest points. As a result, each region is represented by center point along with these m nearest points. Next, select the N regions with
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the smallest curvature diversity to deform. To deform these selected regions, replace all the points within these regions with new points. These new points are generated by sampling from Gaussian distribution, where the mean is set to the average position of all the original points in that region, and the variance is set to 0.001. In Figure 1 X, and X represent the deformed samples, and the points shown grayscale are those drawn from the Gaussian distribution.
We compute point curvature using PCA Zou et al. (002I). Specifically, we first select a small neighborhood around each point and apply PCA to determine the principal directions and their eigenvalues. The curvature is then caleulated the number of cigenvalues. Larger variation in curvature indicates a more intricate goometry and more significant shape changes within a region. The fourth lamp sample in the bottom row of Figure illustrates this property: regions with warmer colors represent areas of higher curvature diversity. To measure the diversity or variation of curvature a region, we propose to use entropy of curvature. Entropy effectively captures the variations in curvature values, allowing quantifying the richness of semantics within region. Formally, we use the following lo measure the curvature diversity:
/mmn// liel/E==l [All, where Amia is the smallest eigenvalue of the matrix, and K is
Gai = min (
= max Nii
max
HH GEN
N RJ
(1)
min
H H(C
_. log((S,norm 10010),
morr
Cmax
enin + 10-10"
where, of represents the curvatures of the i-th point in the j-th region of the point cloud which contains Nu: points in total. To standardize these values, first calculate e/in and Cuaxx which are the minimum and maximum values of all curvatures within the j-th region, respectively. Using these values, we then normalize the curvature values to be in [0, 1], denoted as {c/norm) Then, we calculate the curvature diversity ((clom) by applying [ The point-wise curvature values are computed only once before training. During training, we only compute the curval diversity (entropy) for each region, based on these precomputed curvature values. This avoids repeated curvature estimation and keeps the runtime overhead minimal. Since curvature diversity calculation involves simple normalization and entropy over small regions, its cost is negligible compared to the main training process.
3.3 Deformation Reconstruction Loss
After deforming the selected regions, obtain a deformed point cloud zd from the original x. The deformed input nd is processed by the feature extractor F to generate F(a=",, which is then passed to a reconstruction decoder ASSL. to reconstruct r. The self-supervised loss LSSL minimizes the distance between hssu (E(x)) and x. We use the Chamfer distance in LSSt. focusing on the original points in x within the deformed region R and their reconstructions from Id Formally, let I (1.2....) represent the indices of the points in x n R. and we define LSsL as follows:
LSSL = Σ ((xilver, (hest (F(a"))
(2)
(a,ad,))
where oi is the i-th point in the point cloud æ and denotes the Chamfer distance which is defined as:
D(R,, = Σ min
l|e bent
61l2
+
Σ
min |6 |6-a|
(3)
aeRt
aem
BE
where D(R1, R2) measures the discrepancy between point cloud regions R1, R2 C R33 Note that when we reconstruct only the deformed regions, we can also reduce computational burden and improve time efficiency.
'Rigorously speaking, {o/ (omm } an un-normalized distribution without being divided by partition function normal- ixation constant, but does affect our claim of curvature diversity.
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3.4 Domain Alignment via D-NWD
We begin with introducing the 1-Wasserslein distance.
Definition (1ssassrrttcin distance). Adler & Lunz (2018) The 1-Wasserstein distance quantifies the minimal cost of transporting mass between two probability measures defined on the same metric space. Let I and be two probubility mensures over a metric space (Q, d), where d(r, y) denotes the distance between points and y in Q. Then the I-Wasserstein distance is defined as:
Wi(uv)= inf / gET(UN) Jaxa d(x,3) dr(x, v):
(4)
where ((,, is the set of all couplings with marginals and that is,
J dy = p(x), v) = v(y).
The Casennviiic-ubbisteeeee duality further states that the 1- 1-Wasserstein distance can be expressed in dual form
Wi(p, SKk
sup
Eaz [M(x:)] Rezu [M(x)],
(5)
where the supremam is taken over all functions h Ω - R with Lipschitz constant at most KL, i.c.,
[h(x) h(v)l zzy d(x, v)
| == sup
< KL.
(6)
The Nuclear-norm Wasserstein Discrepancy (NWD) Chen al. 2022) belongs to the family of 1- I-Wasserstein distances, with sophisticatedly chosen h's. Below, present the form of h in NWD. Consider a prediction matrix P E b predicted by the classifier C, where is the number of samples in a batch and M is the number of classes. The non-negative self-correlation matrix Z R MxM is computed as Z = PTP. The intra-class correlation Ja is defined as the sum the main diagonal elements of Z, and the inter-class correlation Le is the sum of all the off-diagonal elements of Z:
M
M
La
Zui
L.
Zz
In the source domain, Ia is large, and Ie is relatively small because most samples are correctly classified. Conversely, in the target domain, Ia is small, and La is relatively large due to the large error from the lack of supervised training on the target domain. Hence, Ia - Ie can represent the discrepancy between the two domains, as La - Le is large for the source domain but small for the target domain. Note that I, = | can be represented as the squared Frobenius norm of P, and thus Ia I 2|] P We can rewrite P. C(F2) and P. C(F.), where F, and F; are feature representation batches from the source and target domains, respectively. We find |[C||] gives high scores to the source domain and low scores to the target domain. Thus, we can set h in Eq. 5 to be ||O|| and represent the domain discrepancy as:
Wr(Ps,P))
sup |K
EF [I[C((.)) [I,-,(([((()))]))))
where v, is the probability measure for features of samples in and v4 the probability mcasure for features of samples in T. To enhance prediction diversity, the Frobenius norm can be replaced with the nuclear norm, which maximizes the rank of P while still being bounded by the Frobenius norm Chen et al. (022). Thus, the domain discrepancy can be rewritten as:
WN(Ps, mx)
sup ||||||| SK,
EF (FC[[[]) LE II,,,,,,,II(C((F [[[(()]])]]
(7)
We have ΣM4 Zr, WE (I,...,b) and j {1,-.. M), and thus Chen al.
2022

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The Eq. 7 is the formal definition of NWD, where C' denotes the classifier, and || represents the muclear norm. In the NWD paper, empirically, NWD can be approximate by LNWD:
1 B, LNWD = - Σ B
1 B. IC - B,
-
_ C(E))..
where F. V, the features for the i-th source batch and Pi i are the features for the i-th target batch. B3 is the number of training batches in the source domain and B, is the number of training batches in the target domain.
min max LNWD C
(8)
Then, the domain alignment is achieved through a min-max game presented Eq.
Now, we introduce our second component, Deformation-based Nuclear-norm Wasserstein Discrepancy (D- NWD). From the previous sections, the curvature diversity deformation reconstruction helps reduce the domain gap between the source and target domains. To further complete classification or segmentation tasks in the presence of the domain gap, we propose D-NWD to align domains, as inspired by NWD Chen et al.
).22). Our D-NWD objective is defined as:
W Wrels IICII-
sup B B oo.a [I(((F)))l) E EE loed [IC(())))),
(9)
Here, Pause and Vaure are probability measures defined over Ω2 Ωd for the features from samples in the original and deformed source and target domains. We align the probability measure of features from the original and deformed samples in the source domain with that of the target domain. Our motivation is that taking features from deformed samples into account would provide a richer, more robust feature space, reduce overfitting, and increase the model's adaptability to variations inherent in data. This technique differs from using NWD, which aligns Vs and 14 defined over Ω2 the probability measures for the features from samples in the original source and target domains. Empirically, our objective Eq. be approximated by LD-NWD:
B. Σ |C(())1- B.
B. Σ (((())).
1
LDNNWD
(10)
- B.
Fgg Usus represents the features for the i-th source batch and F Vrue represents the features for the i-th target batch. Tn practice, we obtain the original and deformed samples by first sampling from the original domain and then generating the corresponding deformed versions. The alignment is then performed through a min-max game, described in the following:
min max LD-NWD- E
(11)
O
To avoid alternating updates, we employ a Gradient Reverse Layer Ganin et al. (0016). following the approach in Chen et al. 2002)). to make the learned features discriminative and domain-agnostic.
3.5 Overall Loss
In addition to deformation and domain alignment loss defined in Eq. and Eq. I1 LCTS on both the original and the deformed source domain samples for supervised training of the classifior:
we use a cross-entropy loss
1 B. E (.).
LCLS = B. i=1
(12)
where Y are labels for batch A.. Since have access to the ground-truth labels for the larget domain data, it impossible to use the supervised cross-entropy loss as in Eq. 12 on samples from T and T. One straightforward alternative is adopt pseudo-labels Fan et al. 2022): ; Liang et al. (2022): Zou et al.) 20221; b Shen et al. (2022). However, this strategy has the risk that the classifier might mistakenly predict
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target samples as the major classes of the source domain. Instead, use NWD to ensure consistency in predictions between T and Tá. Thus, we define a target domain loss <wD as:
1 Be B.
WWD
2 IC(FD)l-
Σ (C((F))) B.
1
(13)
where Pta Vy denotes the deformed target domain batch and F Vt denotes the original target domain and Eq. 13 together, our overall objective loss is:
batch. Combining Eq. 2 Eq.
12 Eq.
min aLCLs + LLsL. E.hssL.C
(14)
min max 61 CD-NWD + +32
F
where a. 7. 31. B2 can be tuned using the target domain validation set and sctting details can be found in Appendix A.2 Note that LDNNW and WWD serve distinct and complementary purposes in Equation 14 CDD is specifically used to minimize the domain gap between source and target features, while <WD is used to improve prediction consistency between original and deformed target distributions, functioning similarly to cross-entropy loss between pseudo-labels and predicted labels on target samples. We use NWD over pseudo-labeling since the latter reinforce incorrect predictions when the classifier is biased toward majority source domain classes.
4 Theoretical Analysis
In this section, present our theoretical contribution for D-NWD, demonstrating the effectiveness of extending NWD to both original and deformed samples. We provide new bounds in Theorems and 2 along with their non-trivial proofs. Following Ben--a et al. (2006) and Chen et al. (2022), we perform our analysis in a binary classification scenario, which can be casily adapted to multi-class classification through reduction techniques such as one-vs-all Rifkin & Klautau (2004) or one-vs-one Allwein et al. (2000)
approaches. Consider (C R + [o, 1}} as a set of source classifiers within the hypothesis space H. Let u. defined on S2, be the probability measure of original source domain and Usa defined on Qa be the probability measure of deformed source domain. We define V4 and us in similar way. The risk error of classifier C the original source domain is defined as (C) = R,,,, [IC(S) ysll. where ye is the label associated -,,, [[C(f.) dall, where ús is the label associated with fe. Similarly, we define ((CC), as the errors on the larget domain. The optimal classifier is defined as C' = arg ming (C") (C) which minimizes the combined error across VeuLe and vt. Our Theorem demonstrates that the expected target error eC( can be bounded by the D-NWD on Paus" and Psuel, WN WNNVoued (Vougd, VeLue). Building Theorem we derive Theorem 2. Theorem 2 establishes that eC(C") can be bounded by D-NWD on empirical probability measures Dsust and D W WN levee). All proofs can be found in Appendix A.1
with the feature fa. We then define Esus
Theorem 1. Let (o)) Fa, Fo. and Fa, be four probability spaces, where Do and Pi are disjoint and Qo USa R.. With the results of Lemma 1, let (So, USd Vsusa) and (Do U Udg Fa, Fur Mr.re) be two probability spaces with probability measures defined as Vaust 1/2v, + 1/2v,a and Voura = 11/2n + 1/2rra. Specificnlly, when sampling from Veuee, there is equal probabilily of 1 1/2 to sample from or Vts. Similorly, sampling from Vausa gives equal probability of 1/2 to draw from Ve Uga. Consider classifier C E Hi and an ideal classifier C* = arg migging (C') + Ee(C)) satisfying the Ki.-Lipchhizz construint, where Hi is a subspace of the hypolhesis space H. For every classifier C in Hi, the following inequality holds:
cc(C) < 22 (0) 4Kg (N Prues + 17";
(15)
where 17" 25 25 (C*) (C*) is the ideal combined error and is a sufficiently small constant.
Theorem 2. Under the assumption of Theorem 1, Q. and Ua are disjoint and Ut U Sa R Let
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(Ro U Rg Fu, and (No U Fur (aaes) be two probability spaces with Vauid 1/2v= + 1/22ga and Viuee = 1/2u + 1/2ua,, where Us, Mat, Mr, Via each has square-erponential moment. From Lemma 3 and 4, Vause satsifies Ti T1(ss) for some Ts and Viuna satsifies Ti(ee) for some ne- Let (fi) N. and ( be two sample sets of size N₂ and Nt draum ii.d from Vsusa and Vluua, respectively. Dause = J. & and Diuna= = No Ei of are associated empirical probability measures. Then, for any n' > n and 1/ < min (1s, mt), there exists a constant No depending on n' such that for any ā > 0 and min, 2 No max(0 (aax(0-(x'x (x'(000))) 1), with probability at least 6,, the following holds for all C:
N.
1
1 V log 3 ( V.N+VNN
1 (V
Ec(CC) S 25 2u (C) + 4Kr WN(P Pouve) + +1" + 4Kg
(16)
where 17" -2e (C*) + E((Ct) is the ideal combined error and is a sufficiently small constont.
Equation 16 justifies why our method can be effective empirically. Specifically, 'f' are sufficiently small constants for relevant. domains with consistent labels because it is the error corresponding to the ideal is also a small constant when training dataset sizes, N, and N1, are large. Eaue (C) is minimized by a supervised classification loss, since source domain samples have labels. Therefore, the primary objective is to minimize WN (Pxusss (Puuee) our D-NWD on empirical measures Pouse and Peued. Hence, minimizing WN(Psued Pelee) can reduce error on the original target domain T and improve the model's performance on the larget domain. It is important to note that our theoretical analysis is not intended to prove that D-NWD is superior to NWD. A direct comparison between NWD and D-NWD is not feasible, as they apply to different probability measures: Pause, PEUIE for D-NWD and ú. P for NWD, Our theoretical contribution lies in showing that, regardless of the deformation method used, optimizing D- NWD on D and P can elfectively reduce error on samples from T. In other words, D-NWD mitigates the negative effects of domain gaps and enhances performance on the original target domain T. Note that we use equal mixture simplify the implementation. However, this assumption is not essential for the validity of our conclusions. The theoretical results can be generalized to arbitrary sampling ratios by adjusting the mixture weights in the probability measures accordingly. In such cases, the only change would be in the constant factors reflecting the updated contribution from each component in the mixture. The core theoretical framework and its implication ions remain intact. Tn Appendix A.3 we present the experiment results with different ratios.
classifier C The term v 7 log N. [V
a 1:1 ratio-in the probability mcasure (c.g., 3 + }V:" to
5 Experiments
We evaluate our method on the PointDA-100 Qin et al. (2019) dataset, domain adaptation dataset for point cloud classification, and on PointSegDA Achituve et al. (2021), a dataset for point cloud segmentation. For the Po11 dataset, we compare our approach against the slate-of-the-art methods for point cloud domain adaptation, including DANN Ganin et al. (2016), PointDAN Qin et al. (211 , RS Sauder & Achituve et al. 22021) GAST Zou et al. (2021), ImplicitPCDA |Shen et al. (022), and the most recent method with publicly available codes, PCFEA Wang & el al (2025) Note that many recent methods do not provide codes for reproducibility, which limits their inclusion. For the PointSegDA dataset, we compare our method with RS, DefRec+PCM, GAST, ImplicitPCDA, and Adapt-SegMap Tsai et al.] (2018). Unfortunately, no suitable methods from the past 1-2 years with available codes are applicable to this dataset. For both datasets, we also evaluate two upper bounds: Supervised-T, which involves training exclusively on labeled target samples, and Supervised, which uses both labeled source and target samples. Additionally, we assess a lower bound, Unsupervised, which utilizes only labeled source samples.
Sievers (2019), DefRec+
Additionally, for the PointDA-10 datasct, we incorporate Self-Paced Self-Training (SPST) into GAST, Im- plicitPCDAS and our method, as SPST is originally included in both GAST and ImplicitPCDA. We exclude SPST for the PointSegDA dataset. SPST typically relies on ranking training samples by difficulty and gradually incorporating harder examples into training. The training samples for point cloud segmentation tasks are points in point clouds. However, mIoU is a global metric that evaluates performance across an entire point cloud, making it challenging to assign difficulty scores to individual points in point cloud. The
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mechanism of SPST mismatches the per-point cloud, rather than per-point, evaluation criterion of mloU. Hyperparameter settings and implementation details can be found in Appendix A.2
5.1 Datasets
We use PointDA-10 and PointSegDA datasets in our experiments. PointDA-11 consists of three do- mains: ShapeNeet-10 Chang et al. (2015), ModelNct-10 Wu et al. (2015), and ScanNet-10 Dai et al. (2017): each sharing ten distinct classes. PointSegDA consists of four domains: ADOBE, FAUST, MIT, and SCAPE. These domains share eight distinct classes of human body parts but vary in point distribution, pose, and scanned humans.
5.2 Training Scheme
We use DGCNN the feature extractor Achituve et al. 2021 for fair comparison. We repeat our periments three times using distinct random seeds for initialization and report the average accuracy and standard deviation. To ensure fair comparison, we maintain the same seed for data shuffling and use the Adam optimizer Kingma & Ba 2014) for optimization.
Models
MS
MS+
SM
SS+
S+M
Sts
Avg
Supervised-T
93.9
78.4
96.2-0.1
78.4
96.2
93.940.2
89.5
Unsupervised
8..
43.
75.5
42.5+1.4
63.8
64.2.0.8
62.2
DANN Ganin al.
2016
75.3
41.5_0.2
62.5_1.4
46.1428
53.3
63.2+1.2
57.0
PointDAN Qin et al.
2019
82.5
47.7=1.0
77.0=0.3
48.5+2.1
55.620.6
67.2 67.222.7
63.1
RS Sauder & Sievers
2019
81.5+1.2
35.2
71.9_1.4
39.840.7
61.043.3
63.643.4
58.8
PCM uve et al.
2021
81.7
51.8_0.3
78.6
54.540.3
73.7
71.141.4
68.6
PCFEA Wang & al
2025
84.4+1.1
47.9_2.9
71.8_1.9
47.1+0.7
71.8+7.4
68. 18.1
65.2
GAST Zou et al. 2022
82.3go.6
53.0
72.6
47.6
64.641.5
66.8.a.6
64.5
GAST+SPST
84.5+0.5
54.1-1.8
80.1
46.744.6
81.5+1.7
66.7
68.9
ImplicitPCDA Shen et al.
2022
79.5
41.7=1.a
72.9
47.542.9
67.645.2
66.440.9
62.6
ImplicitPCDA+SPPTT
81.3+2.1
33.2+13.4
73.2
38.0+4.6
66.9
75.0+2.7
61.3
CDND CDND+SPST
84.1+0.3
58.7+0.8
76.2
55.7=1.0
75.1+1.5
72.041.9
70.3
85.441.1
57.6-1.3
85.0+2.2
54.541.1
82.640.7
74.6+4.4
73.3
Table 1: Performance results (accuracy) on PointDA-10 dataset.
Models
MS
MS+
SM
SS+
S
S+S
Avg
NWD
83.3
46.7.1.7
75.5
48.912.5
63.8
66.7-1.0
64.2
DelRec
83.440.5
46.942.3
74.520.9
46.310.6
67.7
64.02z..
64.0
NeW
83.4
51.2+3.0
74.5
53.743.8
67.742.3
68.5+24
66.5
+ D-NWD
83.4
53.1 +2.3
74.5
54.64110
67.7+2.3
67.4
66.8
CurvRec(S)) High
83.810.9
52.0+1.4
78.0
45.943.8
72.5+1.4
66.721.1
66.5
CurvRec(S)) LS)-Low
83.1
53.0+1.9
74.9
44.7+112
74.8+0.9
65.9
66.1
CurvRec(En)-Fin)
82.941.a
52.1+0.4
77.0=0.3
46.741.
70.9
65.8
65.9
CurvRec
84.1
52.2+1.3
76.2
50.140.3
75.1
66.421.5
67.4
CurvRec PCM
83.0
53.7+1.0
74.0=0.6
54.811.
73.8
76.810.9
69.4
NuWD
84.140.2
54.3+222
76.2=u.0
52.744.1
75.1+1.
70.6
68.8
CDND, 32 =
84.140.3
57.4
76.2
55.7
75.1
69.9
69.7
CDND (DWwRRo-D.NWW))
84.1
58.740.8
76.220.0
55.721.0
75.141.6
72.0
70.3
Table 2: Ablation study results (accuracy) on PointDA-10 dataset.
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Models
FA
FM
FS
MA
MF
MS
AF
AM
AS
SA
SF
SM
AVG
Supervised
80.9
81.8 82.4112
80.9 84.0.1.
82.4 84.0 81.8203 82.4
80.9172 81. 81.8 82.3
Unsupervised
78.5 60.9 66.510.6
26.613.5
33.621.3 69.9 38.512.2 31.221.4
30.0 71.1 68.4 64.5 53.6
AdaptSegMap
70.5.3.4
60.1
65.311.3
49.1
54.0
62.8
1..2_1.33..4u
35.1 70.1
67.
63.8 56.5
RS
78.7 60.7
38.4
10.44...444...m
30.4
36.61o.8 70.7
10.8
73.011.655.0 11.33...11.357.
DeRRec+PCM
78.8 1o.2 100.91..63.6110
48. 18.4 48.6
70.1
46.911.0
33.210.3 10.3 aa 37. 37.610.1 66.3
66.5110
62.6ga.2 56.9
GAST
76.7 12.3 55.011.0 60.311.
52. 14.4
3..2
09.611.2
43.3 25.9 30.8 57.4
16.111. 11. 64.61..5 53.1
ImplicitPCIA
47.5 53.211.0 54.213.4 11.8 34..
44.1 10.9 42.3
11.
70.611.4
55.0 52.4
CDND
81 81.5 61.4 68.6g 67.7 43.6 35.3 10.5 77.5
70.44..
65.1 59.9
Table 3: Performance results (mIoU) on PointSegDA datasct.
5.3 Comparative Results
The results are presented in Table I We use S+ to represent the ScanNet dataset, M to represent ModelNet, and S to represent the ShapeNet dataset. MS, SM, etc. are the abbreviated source and target domain pairs. The CDND model shows significant improvement over the other approaches on the PointDA-10 dataset with the highest average accuracy of 70.3%, outperforming all other models, CDND delivers state-of-the-art performance on five out of six tasks. It excels in tasks with large domain gap, such as MS+ S M, SS+, and S*S. In these tasks, one domain is a synthetic datasct, and the other domain is real-world dataset. This shows its proficiency in handling complex transformations. Especially, CDND scores 58.7% on MS*, outperforming the second-best method by approximately 6%. Additionally, CDND maintains competitive accuracy in tasks with a small domain gap, such as MS and SM, with scores of 84.1% and 76.2%, respectively. With SPST, the performance is further improved, as CDND+SPST achieves 73.3%, outperforming GAST+SPST by 4.4% and ImplicitPCDA+ also outperforms both GAST+SPST and ImplicitPCIA+ across various tasks highlights its ability to adapt to diverse domain challenges, making it promising choice for point cloud classification in the UDA setting.
Results on PointDA.
by 12%. Note that plain CDND on average. The performance of CDND
Results on PointSegDA. The results are presented in Table 3 We use A to represent the ADOBE dataset, F to represent the FAUST dataset, M to represent the MIT dataset, and S to represent the SCAPE dataset. AF, FA, etc. are the abbreviated source and target domain pairs. CDNN achieves the highest average score of 59.9%, which surpasses the sccond-best method, RS, by a margin of 2.0%, which significant terms of mIoU on the segmentation lask. Its superior performance is partieularly evident in MA and SA tasks; in the MA task, CDND achieves mloU of 68.6%, outperforming RS by 9%. Similarly, in the SA task, CDND secures mIoU of 77.5%, which is around 7% higher than RS. These results showcase its adaptability and learning capability. Additionally, in the FA task, CDND achieves a score of 81.5%, even slightly surpassing the supervised baseline. In other tasks, i.e., FM, AS, and SM tasks, CDND either matches or comes very close to the top-performing models, validating its status as a consistently high-performing model. In Appendix A.3 we have visualization results to highlight the qualitative differences.
Overall, the performance improvement of our method, while consistent, is less pronounced compared to the results on PointDA-10. This is primarily due to the nature of the segmentation task, which involves fine- grained, per-point predictions. Our curvature drversity-driven deformation focuses on regional semantics and is more directly beneficial for object-level classification tasks. It is worth noting that other state-of-the-ar mothods also report smaller margins on PointSegDA. Nevertheless, our method still achioves the highest average mloU, demonstrating its effectiveness segmentation settings.
5.4 Ablation Study
To dernonstrate the effectiveness of each component of CDND, we conduct ablative studies on the Poin dataset. There are several ways to evaluate curvature diversity. While standard deviation is commonly used to evaluate the diversity of data points, we propose using entropy. We compare our entropy-based approach (CurvRec(En)) with a standard deviation-bart method (CuryRec(S)). To validate our hypothesis that focusing on low curvature diversity regions can improve performance, we investigate the impact of deform-
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Source Features
Target Features Before Adaptation
Target Features After Adaptation
10-
10
10
Figure 2: UMAP visnalizations depict pro-activation data representations for the MS* task, with different colors denoting different classes. The center plot shows the target domain test data representations generated from a model trained on the source dataset without any adaptation. The left and right plots show the source and target domain data reprosentations after adaptation using CDND.
ing areas with both high (CurVRe((En)+HHiyl,, Hiig and low (CurvRec, CurvRec(S)+Low) curvature diversity.
Effectiveness of CurvRec. We compare against CurvRec(En) variants with CurvRoc(S) variants
Table 2 We observe distinet difference between CurvRec(En)-Hiiie and CurvRec. In contrast, there is much less distinction between CurvRec(S)- -Sig and CurvReeSS)-Low. This observation suggests that entropy is superior method for evaluating curvature diversity in regions. Notably, all CurvRec measures outperform DefRec, regardless of whether the focus is on high or low curvature diversity. However, CurvRec outperforms CurvRoo(En)-Higge showing that is is more effective to deform low curvature regions. Compared to the plain NWD, all CurvRec variants perform better than plain NWD. Specifically, CurvRec surpasses DefRec and NWD by approximately 3%. Though CurvRec demonstrates better performance overall, it does not outperform our proposed CDND. CDND (CurvRec +D-NWD), outperforms all CurvRec variants, DefRee variants, and plain NWD. This highlights the effectiveness of our D-NWD loss. Compared to CurvRec only, integrating with D-NWD improves average performance by 2.9%, with specific gains of 6.5% MS+ and 5.6% on S
Effectivencss of D-NWD. To illustrate the effectiveness our D-NWD, we first compare CDND with two alternatives: CurvRec+ which replaces D-NWD with PCM (PointMixup), and CurVRec+NWD. On average, CDND outperforms both methods. Specifically, compared to Cur,Rec+PPM, CDND achieves improvements of approximately 5% on MS+ and 1% 2% on SS+ MS, and SM. When compared to NurV CDND surpasses 4.4% on MS* ; 3% on SS+, and 1.4% on S+S. To further demonstrate the generalizability of D-NWD across different deformation methods, we include results for DelRec+ D-NWD and DefRec+NWD. Compared to plain DefRec, DefRec+D-NWWF achieves an overall improvement of 2.8%, with notable gains of 6.2% on MS+ 3.4% on S+S, and 8.3% on SS+ Moreover, DefRecDD-NWD consistently outperforms DefRec+NWD on MS+ and S*S, as well as on average, although the margin relatively small. Note that our theoretical analysis proves that D-NWD effectively reduces domain discrepancy across various deformation methods, but the degree to which outperforms NWD depends on the deformation quality. With well-designed deformations method like our proposed CurvRec, D-NWD NWD significantly. While D-NWD is crucial for tasks with large domain gaps (e.g., MS+ S+S, SS+), its advantages are less significant in tasks with smaller domain gaps (c.g., MS, SM, S+M), where CurvRec or DefRec alone already perform strongly. In these cases, plain CurvRec or DefRec performs better, so we retain their performance for MS, SM, and S M. Additionally, we test the use LNWD without adding CTWD enhance consistency in the predictions of original target domain samples and deformed target domain samples, which corresponds CDND with 82 0. CDND with B2 shows only a minor performance decrease compared to CDND, indicating that D-NWD is the primary contributor to good performance.
outperform
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5.5 Analytic Experiments
We conduct analytical experiments to gain deeper insights into the effectiveness of our approach. Specifically, assess how CDND impacts the distribution of the target domain in the classifier's output space for the challenging ModelNet to ScanNet task (MS*); ModelNet is a synthotic datasct, while ScanNet is real- world dataset, making the domain shift between them particularly challenging, We used UMAP to visualize and compare data representations of validation data from the source domain and test data from the target domain, both before and after applying CDND. Figure 2 shows each point as a data representation in the classifier's output space before softmax activation, with different colors denoting different classes. The middle illustrates that, prior to adaptation, the classifier struggles with the target domain data, points from different classes are heavily intermixed. However, after applying CDND, the class boundaries become more distinct, and the distribution of larget domain representations aligns well with that of the source domain. This improvement is visible in the left and right plots of Figure 2 where the arrangement of points shows a more distinct pattern across both domains. In other words, we see that the feature space becomes domsin-agnostic.. This visualization demonstrates CDND's efficacy in reducing domain shift- induced performance degradation and enhancing class distinction. More UMAP visualization analysis of other baselines can be found in Appendix A.3
plot in Figure 2
6 Conclusion
We developed a new unsupervised domain adaptation approach for point cloud data. Our method inte- grates curvature diversity-based dcformation with Deformation-based D-NWD to mitigate target domain performance degradation. Our theoretical analysis of D-NWD shows it minimizes domain error. Additionally, we show that D-NWD can be applied to any deformation method. Experiments indicate that our approach surpasses SOTA methods on two point cloud benchmarks. Ablation studies con- firm that both components of CDND are necessary for optimal performance. Future works include scenarios when source domain data is not. accessible due to privacy or security concerns, or when the domains share only a subset of their classes.
upper bound for target
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Kuniaki Saito, Kohei Watanabe, Yoshitaka Ushiku, and Tatsuya Harada. Maximum classifier discrepancy for unsupervised domain adaptation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 3723-3732, 2018.
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Xin Wei, Xiang Gu, and Jian Sun. Multi-scale part-based feature representation for 3d domain generalization and adaptation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2024.
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A Appendix
A.1 Proofs for Theorems
In this section, we first prove Theorem 1, which serves as the foundation for Theorem Our proofs are structured as follows: we begin by proving Lemma 1, which supports a key assumption in Theorem 1. Next, we present the proof of Theorem After proving Theorem 1, we prove Lemma 4 and conclude with the proof of Theorem 2.
Definition 2 (Probability Spacce). Durrett (2019) A probabilily space is a triple (2,F, Ω represents the sample space, the set of all possible outcomes. FF represents the set of events and is a a-algebra, which a nonempty collection of subsets of Q. F is closed under complements and countable unions. v represents probability measure on the measurable space FI, It a function F [0, 1] that assigns to each evend A E F real value u(A) (the probabilily of A). salisfies the following three arioms:
Non Nogntiiati For every A E F, v(A) v() = 0.
Normalization: v(S)= 1.
o-adiitivity (Countable Additivity): For any countable sequence of pairwise disjoint events A1,A2, A2, Ag.... E F. (where A; Ag = 0 for i + j),
8
v
A
=
v(A))
Lemma 1. Let (9,,FFi, FI, and F2, be two probability spaces, where 92, Sh are two disjoint sample spaces. Let P1.P2 E [0, 1] be conslants such that pı + P2 1. Let (Rz, Fa) be measurable space, where F is the o-algebra on Ωg = Ω U U2. Then, the measure defined on the measurable space (Sz,, Fz) as:
va(A) (An(()) (A
YAA
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is probability measure on (Dz, Fs).
Proof. Since and V/ are probability measures, they satisfy Di (B) 2 0 for all B E Fi and p²(C)) 2 0 for all CE F2. For any set Fs, we have:
= ((A))=pri(Ann))) + +222
Given that P1,22 2 and ( 2 0 and 02 (A n 22)) 2 0, follows that v(A) 2 0. Thus, is non-negative. Then, we need to show that v( U = Consider:
v U22) Pivi (( (22) n n (2 122) lg2))
Since (Ω, U n Ωn and (Ω) U (2) n f22 02 and / (21) and p2 (2(2) 1, have:
yy (f U (22) P1 1 + P2 P1 P2 =
Thus, Vs is normalized. Let { be countable collection of pairwise disjoint sets in Fs. By definition of V3,
0 U [ PI41
((UA) nla + P222
((jA)
1/3
(==1
Since the A; are pairwise disjoint, (Ueh1 Ar) n = U (A,)1)), and similarly for G2. Using the a- additivity of 1/1 and V2:
8 U(4)8)
Pıd
P1 (A,)),
:=1
U(A)
(2)))
P2
P2
i=1
Thus,
20 U i=1
A Pr Σ + Σ
(2, Σ (nnr(Ann))+puuAAA
1/3
Since v²(Ai) PIv (A, (A ) + P2 (22), we get:
U4: 4: Σ (3())
Thus, Va satisfics o-additivity. Since V/ satisfies non-negativity, normalization, and o-additivity, by definition, Va is a valid probability measure. Now, for the following Lemmas and Theorems, we definc:
e. (C,,, Es -
E (C1, C2) =E [Ic](f.) C2(f))
where C1, C2 are two classifiers. We define Cx and Equra C2) in the same manner.
Lemma (Lemma Chen et al. (2022) ) Let v,vl be two probability measures on (2, Let d(x,y) be the distance between I ~ V and ~ v'. WN represents the NWD. Given a family of classifiers C E Hi and a ideal classifier C* H4 satisfying the -ipsschct constroint, where Hi a subspace of H, the following holds for every C²C* Hi.
|e(C, - E'CC, s 2K, WN((),
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where is the ervor v and E' is the error on v'.
Theorem 1. Let Us) Fa, (Bo, where Qo and Pdd are disjoint and Qo USd C R". With the results of Lemma 1, let (2o Ula. Fu. Fu, Vouse) and (Ro U U&g Fur be lao probabilily spaces with probubility measures defined as Vaus" = 1/22, + 1/2u,a and Veurd = 1/204 + 1/2uga. Specifically, when sampling from Veure, there an equal probability of 1/2 to sample from 1s or Mes. Similarly, sampling from Vause gives an equal probability of 1/2 to draw from Vs or vga. Consider clossifier C E H1 and ideal classifier C* arg minc Eauge(C) + Ec(C)) satisfying the K Li-Iisschitz constraint, where Hi is a subspace of the hypothesis space H. For every classifier C in H1, the following inequality holds:
and FR, Fa. V(2) be four probability spaces,
Ec(C) S 28 (C) + 4Kr, WN(Uause Maves) n²;
where 1f" (C + (x(C)) is the ideal combined error and is sufficiently small constant.
Proof. Let Z be an indicator random variable that indicates whether the sample fa is drawn from V/ or uua:
7 0 if the sample is from uta.
Z = 1 if the sample is from
By the Law of Total Expectation, have:
E fi-v [IC(F) - (f-(ll
E f rund [10(f)) (r(() Z= 0 = 0) E + frve fo -r)( = 1|
Substituting P(Z = 0) = and P(Z = 11 P1, pt,
Equte (C,te(C, (C,C*)= Po fr uoed [I((f)) - C (fill 7 Pi wodd
2 1].
Recognize that E fe Send II((f)) - 0a (fii)l 2 = 1] is the expectation when fo drawn from Vt.
C²C = B Bi [lc(f)) -' 2 11.
Combining these, we get:
C* (c,)e(C,C")) = PolE j, ~2 wo CC z = +] Pie,(CCC C*).
Then,
(C,
Pu E
Pi fore sund [IC(f)) z Ec C*).
P1
Since po feme cr Z 0,
En (C,C)) (( pi
-
1
Substituting P1 1/2, we obtain:
2εLurd 2
Based on Lemma 2, we have:
" -"t))" -Enur( s 2K, Meree)
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By triangular incquality,
e((C) s + Ee((C"
EsU 60, 0*) S Esune + Esuse (C*).
Then, we derive:
cc(C) S ((() + Ec(C"
(e((C)) + 23 ALL (C", C)
Ex(C") + 2E s (C, (*) + 2Equea(C, -2 (C, (C, C*)
<E((C)) + 2E su C, (*) + 4Kg Wy
<e((C") + 28 2 (C) + 28 + 4K, WN(Maued,
28 (C) + 4Kr. (N ((ue)) +1f"
Definition (Ia-Transportation Cost Information Inequality). Djellout et al. (2004) Given
> 0, probability measure v on a measurable space (Ω, satisfies Ti(1) of the inequality
2 Wi(W, 1) v V (((
where
H(vlv)= = / flog dv' dv/ dy
holds for any probability measure on (2, where W1 represents the Wasserstein distance.
Lemma (Corollary 2.6 in Bolley & Villani (2005) For probability mensure V on measurable space F), the following stalements are equivalent:
satisfies Ti()) inequality for some n that can be explicitly found.
has square-crponential moment, i.e., there crists >> O such that
J exp(od(<x y) du(x) is finăte
for any E Q. Here, d is a measurable distance over Ω.
Lemma Let F1,1), and (12,, v2) be two probability spaces, where Qn and f2 are disjoint. Let P1. P2 [0, 1] be constants such that Pt P2 1. Define measure V3 on measurable space Fs),
where Ω3 22 l2:
v2(A) = ()n(())))) + (A YAE
Assume that V1 and V2 admit square- caureeeepooentiall moments: for some constants a1 0r > O, and for all yn E Q1 y/2 522
[ exp (aidi(z, dr)(x) 00, / exp (a2d2(2, 92) duz(x) 0c,
where d1 Ω x 22 + oo) and d2 G2 x S2 + 20) are distance functions. Then, is prob- ability measure on (1z, Fz), and there exists a distance function d : 9t x Q3 - [0, 00) and a constant a E (0, min(ain, such that v3 admits a square- exporentapo moment: for all y E Ja.
f 93
exp exp (ad(x, v) dvz(x) < oc.
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Proof. First, we define d: fg U2g + 00):
(1(2, v)
if x, y E S2 if xr y E l2 if x f1 fh (or vice versa)
d( v)
dz(
C
where is a finite constant chosen to ensure d is a metric on Ω23 can be expressed as:
d(z, v) = d1 (Ix y)l,,, + d2(x,, 2, + C1 WE or zE02.yE0,
where 1 is the indicator function. Accodring to Theorem 1.9 (d) in Rudin 1987 the indicator functions are measurable because Q x 22,, S2 x f2, and (S) x 92) (St x Q1) are all measurable sels in Ωg x Ωg d1 and d2 are also measurable functions by assumption. Therefore, is a sum of products of measurable functions. Hence, d is measurable. For any y in f21,
/ Ds
exp(ad(æ, dus(x))
exp(adı (z. 2) (2) + +p2 / exp(ad(æ, v)²) (x)(2)
P1
Jn (x) + exp(a(3) duz(x)
S
(2) + exp(aC²) < co.
P1 nn
For any in 92
/
exp(ad(æ, y)²) dvz(x)
exp(ad(x, y) duj (x) + / exp(ads(1, (x, y)²) duz 22
$ P1 (
explaC²) (x) + p2 exp(a2d2(z, y) dvz(x)
= =p =xpex + p2 / exp(a2d2 v) duz(z) < oo.
This proves that has square-exponential moment for some 0 < < (2)
Lemma 5. (Thcorem 1.1 of Bolley et al. (2007) Let v be a probability measure on (2,F) where Ω S R satisfies Ti()) inequality. Let i N EN dys be ils associated empirical measure defined on sample set [ of size N drawn i.i.d from v. Then for any n' > n and n/' < n, there exists some constant Ng depending on and some mmuoreeexpoeent moment of v such that for any e > 0 and N 2 No (ax(t)(n'+2), 1), the following holds:
PW > c exp (- of N22)
(-2
Theorem 2. Under the assumption of Theorem 1, Qo and Da are disjoint and Qo U Qa C R". Let (Lo U Qd, Fa, Vause) and (St, Da, vece)) be two probability spaces with Vaust 1/2v, + 1/2uga and Vauid 1/2v, + 1/2uga, where Va, Via each has square-erpomential moment. From Lemma 3 and 4, Vause satsifies Ti (na) for some πs and Vasaa satsifies Ti(n)) for some nt- Let (i and (f be two sample sets of size N, and No drown i.ddd from Veus and Viurd, respectively. Dause N. EN df and
PIuL N. EN. 09 associated empirical probability measures. Then, for any n' > and < min(n. m))
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there erists constant No depending r' such that for any d > 0 and min(Ne, No) 2 No max(6 with probability at least 1 - 6, the following holds for all C::
(oaxs((000''') (x'+2), ,2),2)). 1),
2 1 (VN.+VA V log ( V...VN.
Ex(C) S 28 sude (C) + 4Kr Wx Wx Povc)) nÿ +44
(17)
where 17" 2 (C*) (c(C") is the ideal combined error and is sufficiently small constont.
Proof. Based on Theorem 1,
Ec(C) S 25 (C) 4Kg Ww
Wyr(Paee (Voued, Mlues) + +n".
As part of broader class of Wasserstein distances, WN satisfies the axioms of a distance Villani ct al. 2009h. Hence, WN satisfies the triangle inequality:
Ex (C)) < uu (C) + 4Kr Ww((ausd PaLIE)) ) + 4Kr, WN(Pause Meuea) ) + 17"
au (C) + 4Kg WN Dausa) + 4K2 WN (Pausa Deura) + 4KL WN (Pruur Vrura) + 11"
From Lemma 5,
WW(Ususd,Paun S
//
log
8
N
2
I
WNresuee Pluee) S
log
V N₂
17'
Wy belongs to the family of W-Wasccrttein distance. By the symmotry property of distance,
1
WP Vaued) = W WNNMsuee, Deuaa) S
1.
log
N
2
(1)
V V
WN (Dreett Pourd) (Veuret
V r/
log
N₂
Substituting back, we have:
2
log 8 (V (V.. + V V
1
(C( < 2s (C)+ W +17" + +K,, V Vi
N.
A.2 Implementation Details
Our code based on the open-source implemenntttio of the DeRRec+PCM. We trained our three CDND models with seeds {1, 2, 3} on A100 GPUs. For the PointSegDA dataset, we fixed the learning rate to be 0.001 and conducted grid search optimize the hyperparameters a, 7, 81, and B2 for each task. The specific hyperparameter values can be found in Table 5. Similarly, for the PointDA dataset, the hyperparameters are listed in Table 4. Training on the PointDA dataset. takes approximately 10 hours, and around 8 hours required for source-only training without any adaptation method. This results in a high computational cost for hyperparameter tuning. Therefore, we do not tune the hyperparameters extensively. Similarly, for GAST and ImplicitPCDA, we use the hyperparameters provided in their spurce code for the PointDA dataset.
However, GAST and ImplicitPCDA have not boen tested on the PointSogDA datasct before. When im- plementing GAST, we conduct grid search on the PointSegDA dataset, exploring values of 0.1, 0.2, 0.5, and 1.0 for both frot and Lioe- For ImplicitPCDA, considering values of 0.1, 0.2, 0.5, and 1.0 for LM- Please refer to the original papers for the definitions of Crot, Lioe, and LM-
perform grid search the PointSegDA dataset,
For our method, we set the number of regions to be deformed to for PointDA and 10 for PointSegDA. Each region contains 55 points. The total number of regions is set 20 for the PointDA dataset and 40 for the PointSegDA dataset. For DefRec+PCM and CurvRec+PCM, we follow the hyperparameter settings used in the open-source implementation of DefRec and PCM (PointMixup).
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Hyperparameter
Values
Hyperparameter
Values
Learning Rate
0.001,
0.0001 (S*M, MS)
Learning Rate
0.001
0.5
1.0 [0.05, 0.1, 0.2, 0.5, 1.0] [0.0, 0.05, 0.1, 0.2, 0.5, 1.0] [0.0, 0.2] 150
0.5
31
[0.0, 1.0]
81
B2
0.2
32
# Epochs
200
# of Epochs
Table 4: Hyperparameters for PointDA.
Table 5: Hyperparameters for PointSegDA.
Challenges of Applying SPST with mloU
The mloU metric is defined as:
M
mloU =
1
Σ
TPm
M m=1
TPm m + FPm + FNN
M number of classes
TP. m true positive for class
where:
FPm false positive for class m
FN false negative for class m
SPST typically relies on ranking training samples by difficulty and gradually incorporating harder examples into training. The training samples for point cloud segmentation tasks are points in point clouds. However, mloU is a global metric that evaluates performance across an entire point cloud, making it challenging to assign difficulty scores to individual points a point cloud. The mechanism of SPST mismatches the per-point cloud, rather than per-point, evaluation criterion of mloU.
A.3 Additional Results
We also added visualization comparisons on ModelNet to ScanNet task in Figure 3 featuring three methods: GAST and DefRec (which achieved second and third-best scores on the ModelNet ScanNet task) and PCFEA (the most recently published method). Our approach achieves the most distinct and well-separated class clusters (reprcsented by distinet colors) the target domain test set, while PCFEA and GAST show considerable class mixing, and DefRec shows light clustering of some groups, lacking clear separation.
We present segmentation visualization results in Figure the PointSegLA dataset. Different segmentation parts are highlighted using different colors. We selected three domains- MA, MF, and SA-to demonstrate the effectiveness of our CDND method, which achieves state-of-the-arl performance with a significant mar- gin over the second-best method. In the figure, the last row labeled "GTT" shows the ground truth. Our CDND results are shown in the third row. Although CDND achieves 2% improvement in mloU compared to RS Sauder & Sievers (2019), we include this comparison to highlight the qualitative differences in seg- mentation. The "Unsupervised" row represents results without any domain adaptation technique. From Figure ! it is evident that CDND produces segmentation results most closely aligned with the ground truth, outperforming both RS and the unsupervised baseline. Additionally, It is worth noting that other methods also report small increases in mToU margins on PointSegDA dataset.
We also conduct experiments evaluate the effects of different Gaussian variances and the total number of regions in CurvRec. These experiments performed on the FAUST to ADOBE task of the PointSegDA dataset, and the results
presented in Table 6. Specifically, {10, 25, 40, 55, 70, 85, 100}, and the Gaussian variance, o² E f0.0001, 0.001, 0.01, 0.1}. The number of points per region, Ne, is calculated as 2048 points ensure that & x N₂ 2 2048. This condition is important because we want to maximize coverage of the input point cloud idcally assigning all 2048 points to regions. Without this constraint, 20488 could be less than 2048, leaving some points uncovered. By slightly increasing the number of points per region, we ensure complete near-complete coverage. As shown in the table, the optimal performance is achieved when the
vary the total number of regions, k e
k + +4, where 2048 is the number of points per sample. The additional 4
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variance is set 0.001 and the number of regions is 40, which aligns with our experimental settings. The results also indicate that model performance is relatively insensitive to the number of regions but is more sensitive to the choice of variance. In particular, large variances (e.g., 0.1) lead to more uniform deformations and may negatively affect performance.
0² \ (k, (10, 208)
(25, 85)
(40, 55)
(55, 41)
(70, 33)
(85, 28)
(100, 24)
0.0001
80.5
80.3+2.a
79.8+1.9
79.6+1.2
79.4+1.0
80.6
80.1
0.001
80.8
80.7+1.8
81.5+2..0
80.941.7
79.2+2.3
78.942.11
79.1+2.0
0.01
78.6#2.9
80.3+2.3
81.0#1.9
79.3+4.0
7..9
79.643..
77.5+4.0
0.1
69.0g2.8
68.3
68.226.6
67.244.
67 67.944.9
69.9 69.944.5
67.0g5.2
Table 6: Results for varying o², number of regions k, and points per region Ng.
Another important hyperparameter the number of regions selected for deformation. In Table 7, we present results for the FAUST to ADOBE task of the Peint dataset, where the total number of regions is fixed at k 40 and the Gaussian variance is set to x² 0.001. We vary the number of deformed regions N (expressed as the ratio N/k) to observe its effect performance. The results show that performance is not sensitive to the number of deformed regions. The optimal performance is achieved when the ratio is set to 0.25, which aligns with our experimental settings.
Ratio (N/k)
0.10
0.25
0.30
0.45
FA
80.3
81.5
2.00
81.1
79.5
Table 7: Results for varying the number of deformed regions N, expressed as the ratio N/k.
In the theoretical section, we note that the results can be generalized to arbitrary sampling ratios by adjusting the mixture weights in the underlying probability measures. While it theoretically challenging to prove that a specific sampling ratio between original and deformed samples (e-g., 1:1, 2:3, etc.) universally optimal, our experimental results demonstrate that our chosen setting yields high performance. We conduct experiments on the FAUST ABODE task of the PointSegDA dataset. As shown in Table 8, we achieve the highest mloU with the 1:1 and 4:3 ratios. This supports the intuition that balanced and sufficiently diverse mix of original and deformed samples helps the model generalize more effectively across domains. It is also worth noting that our method is not highly sensitive to changes in the sampling ratio.
We also conducted additional experiments to analyze the impact of hyperparameters on curvature estimation. Curvature estimation depends on clean local gcometry and can be influenced by real-world scanning artifacts. To evaluate, we report classification accuracy on the ModelNet to ShapeNet task from the PointDA dataset, and segmentation performance (mIoU) the FAUST to ADOBE task from the PointSegTA dataset. Table 9 presents the results of varying the number of neighbors used in curvature computation. The performance remains relatively stable (though the segmentation task is more sensitive) across different neighborhood sizes, indicating that curvature estimation is not significantly sensitive to this parameter. Notably, our experiment
Ratio
1:1
1:2
1:3
1:4
2:1
2:3
2:4
3:1
3:2
3:4
4:1
1:2
4:3
FA 81.512. 80.1.11
79.3.1.1
79.5
79.510.9
79.3
79.513.9 76.712.8 79.912.0 79.543.9 9 77.2
77.1a3.0 81.7
Table 8: Results for various originalddefrmmed sample ratios.
Neighborhood
10
20
40
CDND (32)
Gaussian Variance 0.0001 0.01
0.1
Clean
MS (Accuracy)
84.1 83.720.4 83. 93.62n.4 84.1
MS (Accuracy) 83.2 83.540.4 82.741.s 83.6
FA (mIoU)
77.7129 76.723.5 78.412. 81.5
FA (mIoU)
79.9 75.81s.1 56.4119 78.4
Table 9: Results for various neighborhoods.
Table 10: Results for adding Gaussian noise.
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Source Features
Target Features Before Adaptation
Target Features Adaptation
20
33
PCFEA
-46
Source Features
Target Features Before Adaptation
Target Features After Adaptation
75
GAST
-10
Source Features
Target Features Before Adaptation
Target Features After Adaptation
DefRec
-10
-10
Source Features
Target Features Before Adaptation
Target Features After Adaptation
13
15
Ours
-30
15
20
-30
20 25
Figure Umap results of other baselines on PointDA-10 dataset.
setting of 32 neighbors (labeled CDND) yields the best overall performance for both lasks. Regarding the do not arbitrarily choose how many to use the number is determined
number of principal components, by the dimensionality of the input point cloud data. In our case, the input points lie in 3D space, so the covariance matrix is 3 x 3, yielding three eigenvalues and corresponding eigenvectors.
In Table 10, we introduce Gaussian noise to the point coordinates to evaluate the robustness of curvature estimation. For these experiments, the number of neighbors is fixed 40. The classification task (ModelNet to ShapeNet) demonstrates robustness noise, as the accuracy remains relatively consistent even with increasing noise variance. However, the segmentation task (FAUST to ADOBE) is more sensitive- its
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MA
SA
MF
Unsupervised
RS
Ours
GT
Figure 4: Segmentation visualization results of PointSegDA dataset.
performance degrades significantly with higher noise levels. This increased sensitivity in segmentation arises because segmentation tasks require more precise, point-level geometric details assign accurate labels each point. Noise distorts these fino-grained features, making harder for the model to distinguish between regions, especially in complex structures. In contrast, classification tasks benefit. from more global shape features and can tolerate localized noise to greater extent.
Method
Unsupervised
DANN
PointDAN
RS
DefRte PCM
PCFEA
GAST
ImplicitPCDA
CDND
Time (hours)
8.0
9.0
9.2
35.9
10.4
8.3
18.9
9.8
10.4
Table 11: Training time (in hours) of different methods.
Table 11 presents the training time (in hours) for various baseline methods along with our proposed approach, CDND. As observed, our method does not introduce a significant computational overhead compared to the "Unsupervised" baseline (without any domain adaptation method), requiring extra 2.4 additional hours. This demonstrates decent efficiency of CDND) especially when contrasted with methods like RS and GAST,
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Published in Transactions on Machine Learning Research (09/2025)
Lamp
Table
Plant
Entropy
Std
Figure 5: Visualization comparisons between curvature diversity metrics (entropy/standdrr deviation).
which have significantly higher training costs. Furthermore, CDND's training time is still comparable to DANN, PointDAN, and DefRec+PPMM Overall, CDND is practically viable without significantly sacrificing efficiency.
To better illustrate the advantages of using entropy evaluate curvature diversity, we provide visual compar- isons between entropy- and standard deviation curvature diversity measures on ModelNet-10 dataset in Figure In these visnalizations, warmer colors indicate regions with higher semantic richness and cur- vature diversity. We observe that the entropy-based measure more effectively highlights structurally and semantically important regions, such as the joints between object components. For example, in the lamp, both the connection between the cap and the supporting rod, as well as the joint between the rod and the base, are clearly highlighted in red by entropy, indicating high curvature diversity. Similarly, in the table, the interfaces between the tabletop and its supporting legs are also captured with warm colors, emphasiz- ing their semantic importance. In contrast, the standard deviation measure fails consistently highlight these joint regions. Furthermore, in the plant example, most of the leaves and flower structures- wwhich are semantically rich and geometrically intricate- are highlighted by entropy with warm tones. However, these same regions are mostly shown in cooler colors (e.g,, blue/green) under the standard deviation measure, indicating a failure to recognize their informative structure. Overall, these visualizations demonstrate that entropy provides a reliable and semantically aligned indicator of curvature diversity.
Additionally, in Table 2, (urv)ec(Enn)-Hinn performs worse than CurvRoc(S)-High. This result also supports the superiority of entropy in identifying semantically rich regions. Entropy is more effective than standard deviation in capturing areas with high curvature diversity and semantic importance. In the (un)) High experiment, we intentionally deform the parts identified having the highest curvature diversity -.e.. those that are semantically rich. Since entropy more accurately identifies these mcaningful regions, deforming them results in a greater loss of critical geometric information, which negatively impacts model performance. which uses standard deviation to select parts for deformation, may
On the other hand, CurvRec(S)- incorrectly classify geometrically simple or less informative regions as "important." As result, fewer truly semantically rich regions are deformed, and more informative structures are preserved. This allows the model to continue learning from these preserved rich regions, leading to better performance compared to
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Published in Transactions on Machine Learning Research (09/2025)
CurvRec(En)-Haan Therefore, the fact that CurvRec(En)- leads to a larger performance drop actually reinforces our claim: entropy is more accurate in identifying semantically rich parts, and its deformation has more significant impact on the model.
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Reviews:
[{"review_id": "8xbQ3LcPGY", "reviewer": "Reviewer_WVuD", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n++ Curvature entropy as a proxy for semantic richness is intuitive and effective.\n++ The paper provides solid mathematical analysis showing generalization bounds under D-NWD.\n++ The ablation studies are well-executed and insightful, verifying the effectiveness of each component (CurvRec, D-NWD, SPST).\n\nWeaknesses:\n-- The writing clarity needs improvement. The introduction is too brief and lacks a thorough explanation of the motivations behind the proposed modules. For example, it is unclear why curvature is assumed to correlate with semantic richness, and why entropy is chosen as the metric for curvature diversity. These choices should be better justified to strengthen the intuition behind the method.\n-- Although Table 2 shows that CurvRec(S)-Low and CurvRec differ in performance, indicating the effectiveness of using entropy to measure curvature diversity, this empirical finding lacks theoretical or visual support. Moreover, the authors do not explain why CurvRec(En)-High underperforms CurvRec(S)-High, which raises questions about the consistency and reliability of the entropy-based selection mechanism.\n-- The Related Works section is missing discussion of some recent advances in UDA for point clouds, such as methods like SD and GLRV, which should be included for a more comprehensive and up-to-date review.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes a novel method called the Curvature Diversity-Driven Nuclear-Norm Wasserstein Domain Alignment (CDND) for Unsupervised Domain Adaptation in 3D point cloud classification and segmentation. The main contributions are two-folds: (1) a deformation reconstruction task that selectively deforms regions with low curvature diversity, thereby encouraging the model to focus on extracting features from semantically rich regions; and (2) an extension of the Nuclear-norm Wasserstein Discrepancy (NWD) that aligns feature distributions of both original and deformed samples, enhancing domain alignment under geometric variations.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Expand the introduction to clearly explain the motivation for the proposed modules. For example, provide justification for choosing entropy as the metric to quantify curvature diversity.\n2. Although Section 3.2 briefly discusses the relationship between curvature and semantic richness, it is recommended to provide a more detailed explanation in the introduction.\n3. Clarify and Support Empirical Findings: Provide additional theoretical analysis or visualizations to support the use of entropy over standard deviation in curvature measurement. Also, offer an explanation for the observation that CurvRec(En)-High performs worse than CurvRec(S)-High, as this could reveal limitations or sensitivities in the entropy-based region selection strategy.\n4. Update Related Work: Include discussion of more recent point cloud UDA methods, such as SD and GLRV, in the Related Works section to improve the completeness and relevance of the literature review.\n5. Self-Supervised Global-Local Structure Modeling for Point Cloud Domain Adaptation with Reliable Voted Pseudo Labels (GLRV, 2022)\nSelf-Distillation for Unsupervised 3D Domain Adaptation(SD, 2023)", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "Cc0Z7YHrIW", "reviewer": "Reviewer_q26F", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strengths:\n\n1. The use of entropy-based curvature diversity for selective deformation is novel, enabling the preservation of semantically rich regions.\n\n2. The paper provides both theoretical and empirical justification for targeting low-curvature regions for deformation, in contrast to prior random or volume-based methods.\n\n3. It extends the Wasserstein distance to account for alignment between deformed and original features, and derives a theoretical bound on the target-domain classification error.\n\n4. The paper is well-written, presents a clear workflow, and includes sufficient implementation details to support reproducibility.\n\n### Weaknesses:\n\n1. The analysis of curvature estimation is insufficient. In particular, ablation studies on parameters, such as the neighborhood size k and the number of principal components used in PCA, are missing. These parameters are important to the proposed method, as the accuracy of curvature estimation significantly influences domain adaptation performance.\n\n2. The curvature estimation relies on clean local geometry, but the impact of real-world scanning artifacts (e.g., noise or occlusion) on its robustness is not discussed.\n\n3. The joint optimization of curvature estimation, deformation generation, and Wasserstein distance alignment introduces considerable complexity compared to other point cloud UDA methods. The reported training time (~10 hours on an A100 GPU) may hinder scalability and practical deployment.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "In this paper, the authors propose Curvature Diversity-Driven Nuclear-norm Wasserstein Domain Alignment (CDND), a novel UDA framework for point clouds. The contributions are twofold: (1) a practically effective method for modeling curvature variation, and (2) a theoretical lower bound on target-domain classification accuracy derived using the Wasserstein distance.\n\n### Contributions\n\nTo be specific, their contributions can be listed below:\n\n1. The method focuses deformation on flat (low-curvature) regions while preserving semantically informative (high-curvature) regions to maintain domain-invariant features.\n\n2. Curvature variation is measured using entropy, allowing the identification and deformation of low-diversity regions.\n\n3. A novel alignment strategy leverages both original and deformed features to enhance domain robustness through nuclear-norm Wasserstein distance.\n\n4. A lower bound on target-domain classification accuracy is derived based on the NWD, providing theoretical insight into the alignment effectiveness.\n\n### New Knowledge:\n\n1. Connecting curvature diversity to deformation for domain invariance.\n\n2. The Wasserstein alignment with deformed features reduces domain gap.\n\n3. The combined deformation and WD in a trainable pipeline.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Please include a table or plot showing the effect of different neighborhood sizes on accuracy and mIoU, as curvature estimation heavily depends on this parameter.\n\n2. In Theorem 1, the assumption of equal probability measures for the two spaces lacks justification. Please either provide a proof that equal weighting minimizes the theoretical bound or include ablation results evaluating different weighting ratios.\n\n3. Add a runtime analysis table comparing the computational overhead of the proposed method with baseline approaches to clarify its scalability and training cost.\n\n4. Include a figure comparing original and deformed point clouds, with low-curvature regions highlighted to illustrate where and how the deformation is applied.\n\n5. Validate the robustness of curvature estimation under real-world imperfections by introducing Gaussian noise to point coordinates and reporting the impact on performance.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "LqnyLifSXG", "reviewer": "Reviewer_1AYa", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths: \n1. Curvature-driven region selection is intuitive and empirically effective; entropy-based diversity outperforms standard-deviation baselines by ~3 points mAcc (Table 2) .\n\n2. Theoretical analysis: Theorem 1-2 extend NWD bounds to the mixed distribution $\\nu_{s\\cup sd},\\nu_{t\\cup td} $.\n\n3. Comprehensive experiments on two benchmarks; CDND improves Avg Acc from 68.9 (GAST + SPST) to 73.3 (Table 1) and Avg mIoU from 57.9 (RS) to 59.9 (Table 3) .\n\nWeaknesses:\n\n1. Writing clarity: Several overloaded symbols (see §2-B).  \n\n2. Complexity analysis: CurvRec adds entropy computation each epoch.  Authors claim “negligible”; please report actual overhead (ms per batch) and memory cost.\n\n3. Ablation on deformation N: How sensitive is performance to the number of regions N and Gaussian variance? Some parameters are not set\n\n4. Segmentation results: Improvement on PointSegDA is smaller (≤2 mIoU).  Provide qualitative visualizations or per-part IoU to show where CurvRec helps.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes CDND, a two–part UDA framework that (i) selects low-entropy, low-curvature “unimportant” regions for Gaussian deformation (CurvRec) and (ii) aligns the joint distribution of original + deformed features across domains via a Deformation-based Nuclear-norm Wasserstein Discrepancy (D-NWD).  Experiments on PointDA-10 (classification) and PointSegDA (segmentation) show consistent improvements over previous methods.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1.\tEquation for $I_e$ is unclear.\nThe explanation for I_e suggests “sum around both indices i and j,” but the equation instead shows a sum over off‐diagonal entries. Please rephrase to clearly state that $I_e = \\sum_{i \\neq j} Z_{ij}$\nwithout the confusing “around both” wording.\n\n2.\tRepeated symbols.\nSeveral symbols are reused in ways that create confusion. For instance, K is first introduced as the number of eigenvalues, but then also appears as a Lipschitz constant. Likewise, C denotes a classifier, but later \\|C\\|F is used, which is ambiguous — is this the classifier weight or its output matrix? Please use distinct symbols for these, such as K\\lambda for the eigenvalue count, and something like \\mathcal{C} or P for the classifier output.\n\n3.\tUndefined distributions $\\nu_s$ and $\\nu_t$.\nThese are introduced in Eq. (10), but never precisely defined. I can only guess that they are uniform empirical measures on batches, but this should be clearly stated in the text. Precise definitions are essential since the later theoretical results depend on the support of these measures.\n\n4.\tMotivation for combining D-NWD with PointMixup is not strong.\nD-NWD is used to align the domain distributions, while PointMixup is a data augmentation. Since both involve deformations, the paper should explain more explicitly why PointMixup alone is insufficient, and why an extra D-NWD term is necessary. At the moment, this feels ad hoc. An ablation or a stronger theoretical argument would improve confidence.\n\n5.\tMissing hyperparameter settings. These should be clearly listed, perhaps in an appendix.\n\nSeveral details are missing for reproducibility. In particular:\n\n\\begin{enumerate}\n\\item how many curvature regions k are actually selected\n\n\\item how many center points are sampled for D-NWD\n\n\\item the values or search ranges of $\\gamma$, $alpha$ for mixup, the D-NWD loss weight\n\n\\item optimizer parameters such as learning rate, training epochs\n\\end{enumerate}", "broader_impact_concerns": "The authors do not provide a detailed broader impact statement. Since this paper focuses on domain adaptation for 3D point cloud data in tasks like classification and segmentation, its techniques could be deployed in safety‐critical systems (e.g., autonomous driving, robotics navigation) where failure in domain generalization may cause unintended consequences.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)