Neural varifolds: an aggregate representation for quantifying the geometry of point clouds
Authors: Juheon Lee
Xiaohao Cai
Carola-Bibiane Schönlieb
Simon Masnou
Venue: Accepted by TMLR ()

Abstract: Point clouds are popular 3D representations for real-life objects (such as in LiDAR and Kinect) due to their detailed and compact representation of surface-based geometry. Recent approaches characterise the geometry of point clouds by bringing deep learning based techniques together with geometric fidelity metrics such as optimal transportation costs (e.g., Chamfer and Wasserstein metrics). In this paper, we propose a new surface geometry characterisation within this realm, namely a neural varifold representation of point clouds. Here, the surface is represented as a measure/distribution over both point positions and tangent spaces of point clouds. The varifold representation quantifies not only the surface geometry of point clouds through the manifold-based representation, but also subtle geometric consistencies on the surface due to the combined product space. This study proposes neural varifold algorithms to compute the varifold norm between two point clouds using neural networks on point clouds and their neural tangent kernel representations. The proposed neural varifold is evaluated on three different sought-after tasks -- shape matching, few-shot shape classification, and shape reconstruction. Detailed evaluation and comparison to the state-of-the-art methods demonstrate that the proposed versatile neural varifold is superior in shape matching and few-shot shape classification, and is competitive for shape reconstruction.

Published in Transactions on Machine Learning Research (06/2025)
Neural varifolds: an aggregate representation for quantifying the geometry of point clouds
Juheon Lee" Independent Researcher
juheon.lee. 626 @2wllmmallllmmmmmmmmmmmmmmm com
Xiaohao Cai
X. Caiiasttonnncc uk
School of Electronics and Computer Science, University of Southampton
Carola-Bibiane Schönlieb
chs31 @com. ac. uk
Deportment of Applied Mathematics and Theoretical Physics, University of Combridge
Simon Masnou
masnou@math www.::/mmmaaammiiii--oom -oonn----oo--j-
Institut Comille Jordan, Université Claude Bernard Lyon
Reviewed on OpenReview: https: // openreview....... formn? id- PO2hoA7ulm
Abstract
Point clouds are popular 3D representations for real-life objects (such as in LiDAR and Kinect) due to their detailed and compact representation of surface-based geometry. Recent approaches characterise the geometry of point clouds by bringing deep learning based techniques together with geometric fidelity metrics such optimal transportation costs (e.g., Chamfer and Wasserstein metries). In this paper, we propose new surface geometry characterisation within this realm, namely a neural varifold representation of point clouds. Here, the surface is represented as a measure/ measure/distriiuteee /distributiio over both point positions and tangent spaces of point clouds. The varifold representation quantifies not only the surface geometry of point clouds through the manifold-based representation, but also subtle geometric consistencies the surface due to the combined product space. This study proposes neural varifold algorithms to compute the varifold norm between two point clouds using neural worworks on point clouds and their neural langent kernel represenlations. The proposed neural varifold is evaluated on three different sought-after tasks shape matching, few-shot shape classification, and shape reconstruction. Detailed evaluation and comparison to the state-of- -herrrt methods demonstrate that the proposed versatile neural varifold is superior in shape matching and few-shot shape classification, and is competitive for shape reconstruction. The public code is available at httpps//gttttub..cm////8/// varifoll
1 Introduction
Point clouds are preferred in applications including computer graphics, autonomous driving, robotics, and augmented reality. However, manipulating/ediin ieditinggg/diigggggggiigggiigii point clouds data in its raw form is rather cumbersome. Neural networks have made breakthroughs in wide variety of fields ranging from natural language processing to computer vision. Point cloud data in general lack underlying grid structures. As a result, convolution operations on point cloud data require special techniques including voxelisation (Deng et al.] 2021 Shi el. al. 2020 Choy al. 2019). graph representations (Bruna et al. 2014
Bronstein et al.) 2017 7 Wang et al.) 2019 Geometric deep learning and its variants
or pointwise convolutions (Qi et al., 2017a bj Thomas et. al. 2019).
have addressed technical problems of translating neural networks point cloud data (Bronstein et al. 2017 With advanced graph theory and harmonic analysis, convolutions on point cloud data can be defined in the
*This research my personal endeavor and unrelated my current job.

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Published in Transactions on Machine Learning Research (06/2025)
context of spectral (Bruna et al., 2014 Defferrard et al 2016) or spatial (Monti et al. 2017 Wang al. 2019) domains. Although geometric deep learning on point clouds has successfully achieved top performance in shape classification and segmentation tasks, capturing subtle changes in 3D surface remains challenging due to the unstructured and non-smooth nature of point clouds. A possible direction to learn subtle changes on 3D surface adopls some concepts developed in the field of theoretical geometric analysis. In other words, deep learning architectures might be improved by incorporating theoretical knowledge from geometric analysis. In this work, we introduce concepts borrowed from geometric measure theory, where the representation of shapes measures distributions has been instrumental.
Geometric mcasure theory has been actively investigated by mathematicians: however, its technicality may have hindered its popularity and use in many applications. Geometric measure-theoretic concepts have recently been introduced to measure shape correspondence in non-rigid shape matching (Vaillant & Glaunès
2005 Charon Trouvé 2013 Hsich & Charon 2021 and curvature estimation (Buct et al. 2017 2022). We
introduce the theory of varifolds to improve learning representation of 3D point clouds. An oriented d-varifold is a measure over point positions and oriented langent. k-planes, i.e., a measure the Cartesian product space of I²" and the oriented Grassmannian manifold nt). Varifolds can be viewed generalisations of d-dimensional smooth shapes in Euclidean space R". The varifold structure helps to better diffcrentiate not only the macro macro-geometry of the surface through the manifold-based representation, but also the subtle singularities in the surface due to the combined product space. Varifolds provide representations of general surfaces without parameterization. They not only can represent consistently point clouds that approximate surfaces in 3D, but are also scalable to arbitrary surface discretisation (e.g., meshes). In this study, we use varifolds to analyse and quantify the geometry of point clouds.
Our contributions:
Introduce the notion of neural varifold as learning representation of point clouds. Varifold representation of 3D point clouds coupling space position and tangent planes provide both theoretical and practical analyses of the surface geometry.
Propose two algorithms to compute the varifold norm between two point clouds using neural networks on point clouds and their neural tangent kernel representations. The reproducing kernel Hilbert space of the varifold is computed by the product of two neural tangent kernels of positional and Grassmannian features of point clouds. The neural varifold can take advantage of the expressive power of neural networks as well as the varifold representation of point clouds.
Apply the usage of neural varifold in evaluating shape similarity between point clouds on various tasks including shape matching, few-shot shape classification and shape reconstruction.
2 Related works
Geometric deep learning point clouds. PointNet is the first pioneering work on point clouds. It consists of a set of fully connected layers followed by symmetric functions to aggregate feature representations. In other words, PointNet is neural networks on a graph without edge connections. In order to incorporate local noighbourhood information with PointNet, +ointNet++ (Qi ct al. 2017b) applied PointNet to individual patches of the local neighbourhood, and then stacked them logether. PointCNN (Li et al. 2018 further refined the PointNet framework with hierarchical X-Convv which calculates inner products of X ttrransformation and convolution filters of point clouds. Dynamic graph CNN (DGCNN) (Wang et al.) 2019 adopted the graph neural network framework to incorporate local neighbourhood information by applying convolutions over the graph edges and dynamically updating graph for each layer. Furthermore, the tangent convolution architecture (Tatacchenko et al. 2018)) incorporated 3D surface geometry by projecting point clouds on local tangent planes, and then applying convolution filters.
Shape similarity measurcs. Quantifying the shape similarity between point clouds or other 3D modality has been an active area of research. In the literature, Chamfer distance (CD) and Earth Mover's Distance (EMD) are two most popular forms in the context of point cloud analysis (Achlioptas et. al. 2018
Wu
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et al. 2021 CD is nearest-neighbor--eee based metric. Although it is computationally efficient, its accuracy drops significantly when the densities of the two point clouds differ greatly or when some subsets are much denser than others. EMD, on the other hand, is based on optimal transport theory and involves solving an optimization problem to construct a mapping from one point cloud to another (Achlioptas et al., 2018 Nguyen et al. 2021). Although EMD is more robust. to noise and variations in density, it computationns more expensive. In the context of medical imaging, EMD and CD lack desirable mathematical propertics such as guaranteeing diffeomorphie mappings that are important for preserving the biological structure of organs and tissues. As result, the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework
has gained more attention (Dupuis et al. 1998 Vaillant & Glaunes 2005 Charon & Trouvé 2013). Although
LDDMM provides biologically meaningful and topology-preserving transformations, it is computationally intensive, often requiring the solution of complex partial differential equations or high-dimensional optimization problems. This makes it less efficient and less suitable for general general-purpose
purpose applications.
Varifolds. Geometric measure theory provides various tools for understanding, characterising and analysing surface geometry in various contexts, e.g., currents (Vaillant & Glaunès 2005), varifolds [Charon Trouvé 2013; Buct et al.] 2017] (022) or normal cycles (Roussillnn & Glaunès] (019). Despite their potential use for many applications, few studies have explored real-world applications ions of varifolds in the context of non-rigid surface registration (Charon & Trouvé 2013).
3 Varifold representations for point clouds
The notion of varifold was introduced in geometric measure theory in the context of the Plateau problem, that is, finding minimal surface spanning a given closed curve in R (Almgren 1964 Allard 1972: Simon 1983). Intuitively, the concept of varifold extends the notion of differentiable manifold by replacing the requirement for differentiability with a condition of rectifiability, or even weaker conditions (Menne] 2017
This modification enables the representation of more complex surfaces, including those with singularities and even point clouds (Menne 2017; Buet et al. 2017 2022). A general oriented d-varifold in an open set Q C R" is a Radon mcasure on the product space of Q with the oriented Grassmannian G(d, n). For example, given a point cloud { NN in R" and a collection of oriented d-planes (Ti) an oriented d-varifold naturally associated with the pairs (z,, i == 1........ N, is given by V = = 8x, 2 òn.
Various metrics and topologies can be defined in the space of varifolds. The mass distance is a possible choice for metric:
dmass (H. v) sup 11.o G(d,m)
odu - - -axciam, odrl de Co(2 x G(d, n)), llol s $
(1)
However, the mass distance is not well suited for point clouds. For example, the distance between varifolds associated with the Dirac masses 6, and 6n remains bounded away from O as it is always possible to find a test function ø such that [Ø(0) - (ss) = 2, no matter how close s is to O. The I-Wasserstein distance is not a more suitable choice in our context since it cannot compare two varifold measures with different mass. For example, the 1-Wasserstein distance between varifolds associated with Dirac masses (1 + 2)6n and dn is infinite for all E + 0. The bounded Lipschitz distance (or flat distance) has, in contrast, good properties; we refer for more details to Piccoli & Rossi (2016) and the references therein.
Definition 3.1 (Bounded Lipschitz distance). Being a and v two varifolds on a locally compact metric space (X.d), we define
(B, v) =
( (1/. S odp - (axxramm)
Saxcram odvl & E Cd ((d,))), l S 1, $1}.
(2)
Although the bounded Lipschitz distance is a very suitable theoretical tool for comparing varifolds, in practice, there is no straightforward way to evaluate it numerically. Instcad, the Reproducing Kernel Hilbert Space (RKHS) approach provides effective numerical tools for evaluating and comparing varifolds (Charon & Trouve 2013
Hsieh & Charon 2021 In particular, the following proposition is useful.
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Let Kpos and kG be continuous positive definite kernels on R" and G(d,n), respectively. Assume also that for any x E RR" kipos(z,)) Co(RR)). Then kpos 8 ki is a posilive definite kernel on R" x G(d, and the RKHS W associated with Kpos @ kg is continuously embedded in Co x G(d, i.e., there exists cw > 0 such that for any 6 E W, we have ||ol| < <owllollw.
Proposition 3.2. (Hsich & Charon 2021
Let TW: W. CR R x n)) be a continuous embedding given by Proposition 3.2 Varifolds can be viewed, through the adjoint Tw as elements of the dual Hilbert space W and the Hilbert norm of W provides a pseudo-metric (as TW is not injective in general): given two varifolds v,
dw- (1, 1| vlliv- -lllliv- 2/,, IVlliv.
(3)
where
(1, = Jamxxiiamn kpo x') (T, T") du(a, d)de(e' advld
This pseudo-metric provides an effective way to compare varifolds by separating the positional and Grassman- nian components. One can derive bound with respect to dBL if we further assume that W is continuously embedded into (IR" x G(d,n) (Charon & Trouvé 2013
|| vllw- sup od(p v) S cwdBL(u, dEW. Iw 2 x((d,n)
Neural tangent kernel. Recent advances in neural network theory have revealed link between kernel methods and over parameterree neural networks (Jacot et al., 2018 Arora et al. 2019). Given training data pairs {æ;; ) i=1: where a; E Rd and y; E IR, let f(0:x;) be fully connected neural network with L-hidden layers with inputs z; and parameters 0 = (W(") ((0) ...., W((), ( Let d, be the width of the neural network for each layer h. The neural network function can be written recursively as
(x) (w()) ( (x) + 600, g(++1)))) ((F( (x), h
(4)
where (xx) = x and is non-linear activation function.
Assume the weights (h) E pdd+1xds and bias ((h) E Rd at each layer h are initialised with Gaussian distribution ( NO d2)/d)) and ((h) ~ N(0, of), respectively. Consider training a neural network by minimising the least square loss function
((0)
(()
(5)
Suppose the least square loss 1(0) is minimised with an infinitesimally small learning rate, iee, de (v)(tt)) Let u(t)= = a;)(tt) E R be the neural network outputs on all at time t, and y (Bi)ke[Mr) be the desired output. Then ut)) follows the evolution law
du
(()(((()))) y).
(6)
di
where
H(t)g af(0(t):x: af(((t):zz)
(7)
00
00
If the width of the neural network at each layer goes to infinity, ie. d, + cc, with a fixed training set, then (t)) remains unchanged. Under random initialisation of the parameters 0, H() converges probability deterministic kernel H the "neural tangent kernel" (i.e. NTK) (Jacot et al. 2018). Indeed, with few known activation functions yo (e.g. ReL.U), the neural tangent kernel H' can be computed by closed-form
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solution recursively using Gaussian process corresponding covariance function is defined as
Lee et al.
2017
Arora et al.
2019). For each layer h, the
0 (zi, 2z) og
w
(8)
do
[ [I x3))
Σ 1) [2 x,)
(M) (z,, 23y) = -h (T,, x33)
E <R2x2
(9)
Eth 1) (z),
( (x, of + o (u.) (e(u)
(10)
In order to compute the neural tangent kernel, derivative covariance is defined as 11)) (z; k = (u. (u.)) [+(u){p()).
(11)
Then, with e (2,, | (z;, (2; Fy), the neural tangent kernel at each layer (*) can be computed as follows
( x.y)) = (xi, ..y) + O (h- (1-1) (xi, z.)).
(12)
The convergence of (14) v,) to Hig is proved in Theorem 3.1 in Arora et al. 2011
3.1
Neural varifold computation
In this section, we present the kernel representation of varifold on point clouds via neural tangent kernel. We first introduce the neural tangent kernel representation of popular neural networks on point clouds Q
et al.
2017a Arora et al.] 2019) by computing the neural tangent kernel for position and Grassmannian components, individually.
Given the set of â point clouds S (s1 82, .... sn), where each point cloud s; = {P1.P2. ... .P } is a set of points, and â, are respectively the number of point clouds and the number of points in each point cloud. Note that the number of points in cach point cloud need not be the same (e.g., |ss| * |s2]). For simplicity, we assumee below that different point clouds have the same number of points. Consider a PointNet-like architecture that consists of L-hidden layers fully connected neural network shared by all points. For (1,3) E [m] x [mh], the covariance matrix (M)) and neural tangent kernel e (pt+P;) at layer h are defined and computed as in equation I0 and equation 12 Assuming that each point P consists of positional information and surface normal direction, i.e. P; E IR x 82, the varifold representation can be defined with neural tangent kernel theory in two different ways. One way is to follow the Miaron-Trrovè approach Charon & Trouve 2013) by computing the position and Grassmannian kernels scparately. While the original Charon- Trouvé approach uses the radial basis kernel for the positional elements and Cauchy-Binel kernel for the Grassmannian parts, in our cases, we use the neural tangent kernel representation for both the positional and Grassmannian parts. Let p; (x, 23} E R S be a pair of position 2; E IR3 and its surface normal 23 $2, i 1, 1,...,7h m. The neural varifold representation is defined
θ varifold (p:.
P (x),,)) O (2), 23).
(13)
We call the above representation PointNet-NTKL. As shown in Corollary 3.3 below, PointNet-NTKKI is valid Charon-Trouvé type kernel. From the point of view of neural tangent theory, PointNet-NTKi in cquation 13 has two infinile-width neural networks on positional and Grassmannian components separately, and then aggregates information from the neural networks by element wise product of the two neural tangent kernels.
Corollary 3.3. In the limit of resolution going infinity, the neural tangent kernels Opos and O are continuous positive definite kernels on positions and tangent planes, respectively. The varifold kernel varifold pos 0 is positive definite kernel on R x G(d,n) n) and the associated RKHS W is continuously embedded into (R x n)).
The other way to define a varifold representation is by treating each point as a 6-dimensional feature, with abuse of notation, P; = (α), E R In this case, a single neural tangent kernel corresponding to an infinite-witth neural network can be used, i.e.,
θ varifold (p:, (1#;,=)) #1). 23))
(14)
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Dolphin
Cup
1lippo
Bunny
Airplane
Source
Target
CD
EMD
CT
NTK1
NTK2
Figure 1: Shape matching examples with different shape similarity metries, i.e,, CD, EMD, CT, NTK1 and NTK2. Hippo is a shortened term referring to the hippocampus.
We call it PointNee-NTKK. Since PointNed-NTT2 does not compute the positional and Grassmannian kernels separately, it computationally cheaper than PointNet-NTKL. It cannot be associated the limit with Charon-Trouvé type kernel, in contrast with PointNet-NTKI, but remains thcoretically well grounded because the explicit. coupling of positions and normals is a key aspect of the theory of varifolds that provides strong theoretical guarantees (e.g., convergence, compactness, weak regularity, second-order information, etc.). Furthermore, PointNeetNTTK2 falls into the category of neural networks proposed for point clouds (Qi let al. 2017a b) that treat point positions and surface normals as 6-feature vectors, and thus PointNet-NTKK is a natural extension of current neural networks practices for point clouds.
PointNet-NTKI and PoimtNei-NTT2 equation 13 and equation 14 are computing NTK values between two points P; and Pj. The above forms can compute only pointiss-rrelttonshhn in a single point cloud. However, in many point cloud applications, two
point clouds need to be evaluated. Given the set of point clouds S, one needs compute Gram matrix of size ñ in general. In order to reduce the size of the Gram matrix, we aggregate information by summation/avenn averaae in all elements of varifold thus forming an î x ñ matrix, i.e.,
n m, which is computationally prohibitive
O varifold (8 83)
ΣV Σ
t varifold (p; E Si.P3 8z).
(15)
iSm i
Analogous equation 3 two sets of point clouds s; and 8j. The varifold metric can be computed as follows
the varifold representation evrifflld can be used shape similarity metric between
||s: 8j 112 2 Ivarifold
- varifold (8i, 8:) 20 varifoldd (si, 85) (st,
θ variffll (sj, 83).
(16)
Furthermore, the varifold representation can be used for shape classification or any regression with the labels on point clouds data. Given training and test point cloud sets and their label pairs (Xurain 3 Virain) = (($133)).- (st, } and (Xiett V best)) = ((844) (sa. ya) then noural varifold and its norm can reformulated to predict labels using kernel ridge regression, i.e.,
3(+1).
y test
e varifold
(Xest Xtrain train
varifold
(Xusann Xtrain) XI)-1a train
(17)
where X is the regularisation parameter.

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Published in Transactions on Machine Learning Research (06/2025)
4 Experiments
Datasct and experimental setting. We evaluate the varifold kernel repre- sentations and conduct comparisons on three different sought-after tasks: point cloud based shape matching between two different 3D meshes, point cloud based few-shot shape classification, and point cloud based 3D shape reconstruction The details of each experiment setup are available in Appendix A.1. and the high-level pseudo-codes for each lask are available at Appendex A.4 The abla- tion analysis of neural network depth, width, and hyperparameter settings is available in Appendix [A.5] For ease of reference, we below shorten PointNet- NTK1, PointNet-NTT2, Chamfer dis- tance, Charon- Earth Mover's s distance as NTK1, NTK2, CD, CT and EMD, respectively.
Table I:
Results of shape matching deforming the given source shapes into the target shapes using a neural network trained with various shape similarity metrics. Metrics used in columns and rows are to train the neural network and for quantitative evaluation, respectively. Every value indicates the shape matching distance. In particular, the lowest and second lowest values (i.c., the best and the second best) in each row are highlighted in bold and underscored, respectively. Note that here "NTKI and NTK2' are
used as metrics.
Metric
CD
EMD
CT
NTKI
NTK2
CD
2.49F-4
3.39F-4
2.90E-4
2.84F-4
3.04E-4
Dolphin
EMD
7.56E0
3.87E0
4.15E0
4.13F0
4.27E0
CT
3.76F-2
2.94F-2
1.22E-2
1.63E-2
1,95E-2
NTKI
6.56E-3
L89E-3
2.93E-3
4.82E-4
6.34E-4
NTK2
1.72E-2
4.33E-3
9.99E-3
1.34E-3
1.25E-3
CD)
4.55F-3
9.74K-3
4.13F-3
3.26FF-
3.36F-3
EMD
2.03FI
3.53FI
2.06F1
185E1
1.79F1
Cup
CT
6.90F-1
2,85F0
4.07E-
3.29F- 1
3.20E-1
varifold norm and
NTKI
1.72F-2
7.27E-
1.97222
6.07E-3
6.50F-3
NTK2
3.14F-2
3.29E0
4.53E-2
1.34E-2
1.21
Hippocampus
CD
3.49F-1
3.2F-1
2.43E-1
2.67E-1
2.65-1
EMD
2.80E5
2.10
2.25F5
2.09E5
1.96E5
CT
2.27E3
2.92E3
2.32E3
2.19E3
2.15E3
4.1 Shape matching
NTK1
1.84E5
1.0199
59.7E5
4.93E3
9.98E3
NTK2
6,37E4
3.09E9
1.56E6
1.54E3
1.54E3
To evaluate the surface representation using neural varifolds and make compar- ison with existing shape similarity met- rics, synthetic shape matching experi- ments are conducted. We train MLP net- works with hidden layers with width of 64 and 128 units respectively. These net- works use various shape similarity met- rics as loss functions to deform the given source shape into the target shape (more details are available in Appendix A.1.1].
CD
9.32E-3
5.12E-3
3.60E-3
4.40E-3
4.32E-3
Bunny
EMD
2.31E4
4.74E3
3.72E3
3.13E3
3.52E3
CT
2.40E-1
1,25E0
7.51E-2
1.28E-1
1.23E-
NTK1
2.57E-2
1.32D-2
1.83E-3
2.22E-4
2.94E-4
NTK2
3.85E-2
2.68E-2
3,33E-3
8.85E-4
6.43E-4
CD
L.36E-3
4.07E-4
3.72 E-3
3.81E-3
5.90E-3
Airlane
EMD
1.16E4
4.12E2
3.38E2
3.43E2
7.50E2
CT
8.71E-2
3.62EO
-3.58E-4
1.67E-3
3.68E-3
NTKI
2.27E-3
180E-1
0.41E-6
0.31E-6
0.72E-6
NTK2
6.14F
5,13E0
8.69F-6
3.17E-6
2.42F-6
Figure shows five examples of shape matching based on various shape similarity metric losses. The neural network trained with CD captures geometric details well, except for the airplane. For hippocampi, CD oversmooths sharp edges; and for the bunny, it oversmooths the cars. While CD matches airplane wing shapes, it is noisier than CT, NTK1, and NTK2 methods. The EMD-trained network performs well on the dolphin shape but struggles with geometric details and surface consistency for other shapes, likely due to insufficient parameters for the transportation plan. More iterations and a lower convergence threshold make training inefficient. Networks trained with NTK1 and NTK2 metrics penalise broken meshes and surface noise, resulting in better mesh quality. NTK2 oversmooths high-freyuency features on the dolphin, while NTK1 achieves good results. NTK1 and NTK2 show superior shape matching for airplanc fusclage and wings. The network trained with CT gives acceptable results except for the airplane; however, one main disadvantage is that CT's radial basis kernel is sensitive to point cloud density, requiring hyperparameter o adjustments for each pair of point clouds to avoid poor results.
Table I presents the quantitative evaluation of the shape matching task. Each column indicates that the shape matching neural network is trained with a specific shape similarity metric as the loss function. In the case of dolphin, when the evaluation metric is the same as the loss function used to train the network, the network trained with the same evaluation metric achieves the best results. This is natural as the neural network is trained to minimise the loss function. It is worth highlighting that the shape matching network trained with
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the NTK1 loss achieves the second best score for all evaluation metrics except for itself. In other words, NTK1 can capture common characteristics of all shape similarity metrics used to train the network. Furthermore, in the case of shape matching between two different cups, our neural varifold metrics (NTK1 and NTK2) achieve either the best or second best results regardless which shape evaluation metric is used. This indicates that the neural varifold metrics can capture better geometric details as well as surface smoothness for the cup shape than other metrics. In the case of shape matching between the source hippocampus and the target hippocampus, the network trained with CT excels in the CD metric, while the network trained with NTK1 achieves superior results with respect to NTKI and NTK2 metrics. The shape matching network trained with NTK2 outperforms in the FMD, CT and NTK2 metrics. In the case of the bunny, CT shows the best results with respect to CD and CT, while NTK1 shows the best matching results with respect to EMD and NTK1. NTK2, on the other hand, shows the second best results with respect to all metrics except. for itself. In the case of airplane, CT shows the best matching results with respect to CD, EMD and CT. However, the CT metric itself shows the negative value, i.c. unstable. This is mainly because the RBF kernel used in CT is badly scaled. NTKI shows the second best shape matching results with respect to all metries except. for itself. The detailed analysis for the role of the NTK layers shape matching is available in Appendix A.5.2
4.2 Few-shot shape classification
In this section, the proposed NTKs are firstly compared with the current statc-of-the-art few- shot classification methods on the MedelNet440- FS benchmark (Ye et al. 2023). ModelNet400 benchmark (Ye et al. 2023) divided different shape categories in No04c datasets for pre-training the worw with 30 classes and then evaluated few-shot shape classification on 10 classes. The ex- periment was conducted in the standard few-shot learning setup, i.e. N-way K-shot Q-query. The definition of N-way K-shot Q-query is available in Appendix A.1
Table 2: Few-shot shape classífication comparison on the ModelNetAO-PPS classification benchmark in terms of two setups, i.e., 5way-1shot and 5way-5shot. Every value indicates the mean shape classification accuracy with 95% confidence interval. NTK1 (DGCNN) and NTK2 (DGCNN) imply that, instead of point clouds positions and their normals, pointwise features from the pre- pre-trained DGCNN are used for our NTK1 and NTK2.
ModelNet4O-FF
Methods
5way-Ishor
5way-5shot
Prototypical Net Relation Net
69.96 + 0.67
85.51 = 0.52
68.57 + 0.73
82.01 + 0.53
Table shows the shape classification results on two different few-shot classification setups, i.c., Sway-Ishot-1squey, and 1wwy-bshoo--1qqeey. In the case of the 5way-Ishot classification, the cur-
PointBERT
69.41 3.16
86.83 1 2.03
PCIA
82.21 + 0.76
89.42 + 0.53
NTK1
64.94 + 0.84
83.42 + 0.59
NTK2
62.67 + 0.81
81.53 + 0.59
rent state-of-the--rt method PCIA achieves the
NTKI (DGCNN)
69.30 0.76
86.75 - 0.51
NTK2 (DGCNN)
75.23 + 0.71
90.20 1 0.49
best results by around 7% margin in comparison to the second best method NTK2 (pre-trained). In the case of the 5way-5shot classification, NTK2 outperforms PCIA by around 0.8% margin. Note
Point cloud imputs positious and unit normal vectors, ioo. G-Geutuze vectors. Note that the original paper's a reported accuracy for Sway-lsbet and Seray-Sebot ix
81.19% and 59. 30%, respectively.
that PCIA requires to train backbone networks with PCIA modules and needs to fix the size of query. NTKs, on the other hand, can directly use the extracted backbone network features without further training the few-shot layer weights and do meta-learning in any arbitrary N-way K-shot Q-query settings. If NTKs are used without pre-trained backbone features, i.c., directly using positional and normal coordinates, then the results are subpar in comparison to other meta-learning approaches. This understandable as few-shot. architectures built on top of the backbonn fcatures, while NTKs without a pro-trained model, can only access the raw features, and thus cannot take advantages of the powerful feature learning capability of the neural networks. Interestingly, NTK1 outperforms NTK2 without pre-trained features, while NTK2 (DGCNN) outperforms NTK1 (DGCNN). This is because we use the pre-trained DGCNN on point clouds with spatial coordinates (x,yz) as a backbone network for extracting both positional and normal features. Relatively low performance on NTK1 (DGCNN) is mainly because there is no appropriate architecture treating position and normal features separately.
Small-data tasks are common when data limited. In the shape classification experiment, we restrict data availability and assume no pre-trained models, requiring training with 1, 5, 10, or 50 samples. Table 3 shows
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ModelNet classification accuracy with limited samples. Kernel-based approaches excel in small-data tasks. In partieular, with only one sample, kernel methods outperfom finite-width neural networks like PointNet and DGCNN on both 1odelNet10 and ModelNet40, with NTK2 and NTKI achieving the best results, respectively. Interestingly, the CT kernel performs as well as NTK1 and NTK2 on ModelNet10 but drops significantly on ModelNet40.. Similar results occur with five samples: NTK1 and NTK2 achieve 81.3% and 81.7% on ModelNet10, while CT, PointNet, and DGCNN lag by 3.1%, 5.1%, and 5.9%, respectively. On ModelNet440, NTK1 outperforms all other methods more signilicantly than on ModelNet10. As the number of training samples increases, finite-width neural networks significantly improve their performance and ModeINet40. With ten samples, NTK1 and NTK2 achieve around 86.1% accuracy, outperforming other methods on ModelNet10 by 2 3%, although DGONN surpasses NTK and PointNet on ModelNet400. With 50 samples, Net and DGCNN purperfor NTK approaches by about 1% on ModelNet10 and 3 5% on 400 NTK1 and NTK2 show similar performance on ModelNet10 (with 0.3% difference), while NTK1 slightly outperform NTK2 on ModelNet40 by 0.6-1.6%. Notably, NTK1 and NTK2 consistently outperform the CT varifold kernel.
both ModelNet10
Kernel-based learning is known for its quadratic computational complexity. However, NTK1 and NTK2 are com- putationally competitive in both few- shot learning and limited data scenar- ios. For instance, training NTK1 and NTK2 on ModelNet 10 with 5 samples takes 47 and 18 seconds, respectively, compared to 254 and 502 seconds for training PointNet and DGCNN for 250 epochs on a single 3090 GPU. The shape classification performance on the full ModelNet data is available in Appendix A.3 Ablation study re- garding the criteria used to choose the number of layers and different layer width for NTKs is available in Appendix A.5
Table 3: ModelNet classification with limited training samples selectod randomly. Every value indicates the average classification accuracy with standard deviation from 20 times iterations.
Methods
I-sample
5-sample
10-sample
50-sample
ModelNet10
PointNet DGCNN
38.84 6.41
76.57
2.28
84.14 + 1.43
91.42
0.89
33.56 1 4.60
75.81
2.40
83.90 1: 1.70
91.54 1 0.68
CT
59.06 1 4.76
78.64
2.90
83.35 1: 1.57
87.98
0.79
NTKI
59.49 + 4.80
81.34
2.78
86.07 + 1.62
90.18 = 0.93
NTK2
59.64 5.50
81.74 3.15
86.12 1: 1.56
90.10 0.73
Mode|Net-40
PointNet DGCNN
33.11
3.28
63.30
2.12
73.63 1: 1.06
85.43
0.31
36.04
3.22
67.49
77.04 +
88.17
0.57
37.71
3.42
60.43
67.13 + 1.11
77.20
0.54
NTKI NTK2
44.03 1 3.51
69.30 + 1.48
75.81 + 1.23
83.88 + 0.53
42.85 + 3.51
67.81 1.47
74.62 + 1.00
83.26 + 0.42
4.3 Shape reconstruction
Shape reconstruction from point clouds is tested for NTKI, SIREN, neural splines, and NKSR. NTK2 is excluded as it unsuitable for this task. Implementation details are Appendix A.2 Reconstruction quality is evaluated with CD and EMD metrics. Figure 2 shows examples (e.g,, airplane and cabinet) with 2048 points. NTK1 performs better in surface completion and smoothness. Additional visualizations are in Appendix A.6
Quantitatively, Table shows the mean and median of using the CD and EMD for 20 shapes randomly selected from each of the 13 differ- ent shape categorics in the ShapeNet. dataset. For the CD, NTK1 shows the best average reconstruction sults for the airplane, cabinet, car and vessel categories; SIREN shows the best reconstruction results for the chair, display and phone categories; and the neural splines method shows the best reconstruction results for the rest. 6 categories. NTK1 based reconstruction achieves the lowest mean FMD for vessel and cabinet, while
Cabinet Airplane
Ground Truth SIREN Neural Splines NKSR
NTK1
Figure 2: Examples of the shape reconstruction comparison.
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neural splines and SIREN achieve the lowest mean EMD for 7 and 5 categories, respectively. NKSR does not achieve the lowest mean CD and FMD for all the categories. In addition, the shape reconstruction results with different number of points (i.e., 512 and 1024) are available in Appendix A.5.5
SIREN shows the lowest distance for both CD and EMD followed by NTK1. Surprisingly, the neural splines method underperforms both the CD and EMD when we consider all the 13 categories. The performance of NTK1 on shape reconstruction is clearly comparable with these state-of-the-an methods. This might be counter-intuitive as it regularises the kernel with additional normal information, this is probably because there is no straightforward way to assign normals the regular grid coordinates, where the signed distance values are estimated by the kernel regression.
Table 4: ShapeNet 3D mesh reconstruction with 2048 points (mean/ mediann values LE3). NS: Neural Splines.
Metric
Method
Airplone
Beach
Cahinet
Car
Chair
Display
Lamp
Speaker
Rifle
Sofa
Table
Phone
Vessel
CD
SIREN
1.501
1.624
2.430
2.725
1.556
2.193
1.392
7.906
1.212
1.734
1.856
1.478
2.537
4.145
1.304
1.969
2.131
1.828
4.577
1.062
2.798
0.400
1.650
1.576
10.058
2.210
(mean)
NKSK
2.000
2.423
2.520
17.720
5.477
3.622
0.414
2.493
1.547
NTK1
0.644
1.314
1.991
2.107
1.731
1.666
1.131
2.806
0.425
1.65-1
1.586
10.397
1.079
SIREN
0.733
1.384
2.153
2.134
1,230
1.469
0.661
3.304
0.581
1.706
1.670
1.424
1.112
CD
(median)
NS
0.947
1.289
1.799
1,640
1.160
1.413
0.479
2.749
0.347
1.586
1.372
1.600
0.788
NKSR
1.205
1.426
1.797
1,830
1.236
1.565
1.579
2.945
0.326
1.638
1.637
1.305
0.894
NTK1
0.621
1.259
1.828
1.836
1.237
1.499
0.566
2.794
0.352
1.578
1.350
1.358
0.797
SIREN
2.990
3. 26763
1.963
5,208
4.649
4.658
24.068
13.292
2.418
3.688
8.7115
3.237
4,500
EMD
(msaa)
NS
22.004
3.571
4.420
4.694
7.916
9.205
16.786
5.857
1.503
3.706
4.194
17.846
5.957
NKSR
7.153
8.456
8,018
8.190
16.824
31.182
21.
9.984
2.329
5.871
13.658
4.152
4.581
NTK1
3.120
4.153
4.420
4.767
7.350
9.653
23.381
6.236
1.592
3.888
5.259
24.101
3.534
SIREN
2.690
2.938
4.520
3.803
4.411
3.314
2.279
6.240
1.605
3.653
3.782
3.060
2.576
EMD
(median)
N8
6.873
3.068
4.154
3.999
4.7400
4.053
3,802
5,123
1.216
3.543
3.695
3.835
2.210
NKSR
3.732
5.119
4.440
5.313
5.683
3.717
4.927
5.975
1.227
3.641
6.375
3.088
2.771
NTKI
2.864
3.319
1.254
3.917
5,293
3.875
3,258
5.795
1.271
3.738
3,980
3,380
2.074
5 Limitation & Conclusion
The proposed method has few noticeable limitations. First, it is based on the simpler PointNet architecture. Future research should explore its performance with more advanced architectures like graph convolutions or voxelised point clouds. Second, the quadratic computational complexity of the kernel regime poses a challenge for large datasots. Kernel approximation methods, such as Nystrom approximation, could reduce this complexity, and their performance compared exact kernels should be evaluated.
Despite the aforementioned limitations, the proposed method outperforms well-known approaches in the tasks of shape matching and few-shot shape classification, and shows competitive results in shape reconstruction It is important to note that the tasks we evaluated represent only a small subset of the method's 's potential applications. For example, recent advances in Gaussian splatting explore the use of normal and depth information to enhance reconstruction quality (Turkulainen et al. 2024). The proposed neural varifold framework, which incorporates both spatial and normal information, can be applied in this context and extended to other modalities (e.g., depth, texture) with minor modifications (Hsieh & Charon 2021
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A
Appendix
A.1
Experimental setup
A.1.1
Shape matching
For point cloud based shape matching, MLP networks consisting of 2 hidden layers (with width size of 64 and 128, respectively) were trained for computing displacement between two shapes, such that one can deform the source shape to the target shape. The neural networks were trained with different shape similarity metric losses including neural varifold. Point clouds of the given shapes were extracted by collecting the centre of triangular meshes of the given shapes, and the corresponding normals were computed by cross product of two edges of the meshes. The first example is deforming the source unit sphere into the target dolphin shape; the second is matching two different cup designs; the third is matching between two hippocampi; the fourth is the shape matching bewteen sphere and Stanford bunny; and the fifth is the shape matching between two different designs of airplane. The data is acquired from the PyTorch3D, SRNFmatch and KeOps GitHub repositorics (Ravi al. 2020 Martin Baucr & Hsich 2020: Charlier et al., 2021). This experiment evaluates how well the source shape can be deformed based on the chosen shape similarity measure the loss function. A simple 3-layer MLP network was solely trained with single shape similarity measure loss, with the learning rate fixed to 1E-3 and the Adam optimiser. The network was trained with popular shape similarity measures including the CD (Chamfer distance), EMD (Earth Mover's distance), CT ShharonTTrowye varifold norm), and the proposed neural varifold norms (NTK1 and NTK2). In the case of CD and EMID, we followed the same method used for shape reconstruction. For varifold metrics, we used equation 16 note that it is a squared distance commonly usod for optimisation. For the numerical evaluation as metric in Table [ the square-root of equation 16 was used. To be consistent with shape classification experiments, we chose the 5-layer NTK1 and 9-layer NTK2 to train and evaluate the similarity between two shapes. The detailed analysis for the role of the neural network layers shape matching is available in Appendix[A.5, The final outputs from the networks were evaluated with all of the shape similarity measures used in the experiments.
A.1.2 Few-shot shape classification
The ModeINet dataset (Ye et al. 2023 was used in the case of evaluating few-shot learning capability of neural varifold kernels (NTK1 and NTK2) with popular few-shot learning methods including Prototypical Net (Snell et al. 2017), Relation Net (Sung et al. 2018), PointBERT YYu et al. 2022), and PCTA (Ye et al., 2023 The ModeFNet4O-FS datasct (Ye et al.) 2023) consists of 30 training and 10 unseen classes for training the backbone network and evaluating few-shot shape classification. The implementation of baseline methods and backbone networks is based on Ye et al. (2023). The computation of the neural varifold kernels (NTK1 and NTK2) is basod on the newral tangent library (Novak et al. 2020). In this experiment, two diffcrent versions of NTK1 and NTK2s are used. First of all, NTKI and NTK2 are directly computed from the original point cloud features (i.e., positions and their normals). As f'w-shoo learning is usually based on pre-trained neural networks, NTK1 (DGCNN) and NTK2 (DGCNN) are computed from pointwise feature extracted from the backbonn Dynamic graph convolutional neural network (DGCNN). DGCNN (Wang et al. 2019). used in the experiments, consists of 4 EdgeConv layers (Ye et. al.) 2023). Pointwise features are delined as the concatenation of the convolutional features extracted from all 4 EdgeConv layers of the DGCNN. Furthermore, global features are defined as the max-pooling of the pointwise features.
The evaluation was conducted using the standard few-shot classification setup: N-way K-shot Q-query. Tn this sctup, N-way refers to the number of classes used for training and cvaluation; K-shot indicates the number of samples per class used for training; and Q-query specifies the number of samples per class used for evaluating the classification accuracy. All methods are evaluated in two different few-shot learning scenarios: Sway-1shot-15quey and sway-ishot-1squery It is important to note that the reported accuracy in Table 2 represents the average accuracy and its 95% confidence intervals for 700 test cases (i.e., 700 test cases of N-way K-shot 15query).
In addition, we evaluate the scenario when pre-training data/modols are not available. In this experimental setup, each method was also trained with varying number of training samples per class, ranging from to 50, and evaluated their performance on the full 1o0elNet10/44 validation datasets. The number of
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1024 points and their corresponding normals for each object were sampled from the original meshes of the Princeton ModelNet benchmark (Wu et al. 2015). The proposed neural varifold methods are compared with popular neural networks on point clouds including PointNet (Qi et al., 2017a), DGCNN (Wang et al.) 2019), as well as the kernel method (Charon & Trouvé 2013 The computation of the neural varifold kernels (NTK1 and NTK2) is based on the neurul langent library (Nvakk et al.) 2020). To make the results more consistent, samples were randomly chosen and iterated 20 times with different sceds. Both NTK1 and NTK2 are required to fix the number of layers corresponding to the equivalent finite-width neural networks. NTK1 uses 5 fully connected neural network layers while NTK2 adopts 9 fully connected neural network layers. Each layer consists of MLP, layer normalisation and ReLU activation for both NTK1 and NTK2. The shape classification performance on the full ModelNet data is available in Appendix A.33 The criteria used to choose the number of layers and different layer width for both NTK1 and NTK2 are available in Appendix A.5
A.1.3 Shape reconstruction
Lastly, for shape reconstruction from point clouds, ShapeNet dataset. (Chang et al. 2015 was used. In particular, we followed the data processing and shape reconstruction experiments from Williams et al. (2021), i.e., 20 objects from the individual 13 classes were randomly chosen and used for evaluating the shape reconstruction performance. For each shape, 2048 points were sampled from the surface and used for the reconstruction. Our approach was compared with the state-of-the-art shape reconstruction methods including Neural Splines (Williams et al. ). SIREN (Sitzmann et al. 2020) and neural kernel surface reconstruction (NKSR) (Huang et al.] 2023]. To be consistent with existing point cloud based shape reconstruction literature, CD and EMD were used to evaluate each method. Unlike CD, EMD has a number of different implementations for solving a sub-optimisation problem about the transportation of mass. In this study, we borrowed the EMD implementation code from Liu et al. (2020). In the experiment, we fixed the number of NTK1 work layers as 1. This is because there is no significant performance change when different number of network layers is used. The shape reconstruction using neural varifold is heavily influenced by the approaches from kernel interpolation (Cuomo et al. 2013) and neural splines (Williams et al.) 2021). The implementation details are available in Appendix A.2 In addition, the shape reconstruction results with different number of points (i.e., 512 and 1024) are available in Appendix [A.5.5] The visualisation of the ShapeNet reconstruction performance by all the mothods compared is available in Appendix [A.6]
,
A.2 Kernel based shape reconstruction
Consider a set of surface points X' = f.......................... ;x) and their corresponding normals = {-1.... , sampled on an unknown surface M, i.c., X C M. Using an implicit surface representation, all x in A satisfy f(x) for some suitable function J. The best way to approximate the function f is to generate off-surface points and to interpolate the zero iso-surface. Given ) {y1,
ykk Vy; 0 and the distance parameter 8, we define X3 = (i11- 6z,,-.. ... Ih dzz), X = {u1 + 621," .h + dzn). Vo = (-6,..... -d}, and y = -0. .8 in a similar manner. Taking the set unions & = XUX uxt, 2 zuzuz Z and ÿ = yuy uyf. the training data tuple (Kraing Viraia) = (11), 5) (cf. symbols X urain and Dtest are used for multi-point clouds) can be used to obtain the implicit representation of the surface.
Let us define regular voxel grids Xgrid on which all the extended point clouds & lie. Note that there is no straightforward way to define normal vectors the regular voxel grids, which are required for Noin computation. Here, we assign their normals Zgrid as the unit normal vector to z-axis. Then the signed distance corresponding to the regular grid Xesst = (Agridd Zgrid) } can be computed by kernel regression with neural splines or PeintNel-NTKI kernels K K(Krainn Xirain) and K KKKosst, Xirain), i.e.,
Viest = K (Kesst Mueia) [ (Kurinn Xerain) +
(18)
where Veeain and Viest are the signed distances for the extended point clouds and the regular grids, respectively. With the marching cube algorithm in Lorensen & Cline (1998); Lewiner et al. (2003), the implicit signed distance values the regular grid with any resolution can be reformulated to the mesh representation.
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Table 5: ModelNet classification.
Methods
ModelNet10
ModelNet40
PointNet"[[]
94.4
90.5
PointNet++
94.1
91.9
DGCNN [3]
95.0
92.2
CT
89.0
80.5
NTK1
92.2
87.4
NTK2
92.2
86.5
"Point cloud imputs are positions and unit normal vectors 6-fenture vectors; nate that the original paper's reported accuracy for ModelNet40 is 89.2% with only positions forming 3-feature vectors inputs.
A.3 Shape classification with the full ModelNet dataset
The overall shape classification accuracy with neural varifold and the comparison with state-of-the-art methods on both ModelNet10 and ModelNet40 are given in Table 5 where the entire training data is used. The table shows that the finite-iidth neural network based shape classification methods (i.e., PointNet, +ointNet+++ and DGCNN) in general outperform the kernel based approaches, i.e., CT, NTK1 and NTK2. DGCNN shows the best accuracy on both ModelNet10 and ModclNet40 among the methods compared. In the case of kernel based methods, NTK1 outperforms both NTK2 and CT. The results are largely expected since the infinite-width neural networks with either NTK NNGP kernel representations underperform in comparison with the equivalent finite-width neural networks (Lee et al. 2020) when sufficient training samples are available. The computational complexity of kernel-based approaches is quadratic. With the ModelNet 10 dataset containing 4899 samples, NTK1 and NTK2 respectively require approximately 12 hours and 6 hours of training time, whereas PointNet and DGCNN achieve similar accuracy with nearly 1 hour of training time using the entire dataset.
Table Shape classification performance of NTK1 and NTK2 with different number of neural network layers adopted in MLP and Conv 1D ModelNet440.
Number Layers
PointNet-NTKi (5-sample)
PointNet-NTKK (5-sample)
l-layer MLP
67.70 +-t 1.66
64.70 = 1.34
3-layer MLP
69.06 1.57
66.79 1.50
5-layer MLP
69.29 + 1.48
67.34 1:: 1.45
7-layer MLP
69.29 1.43
67.64 +:: 1.47
9-layer MLP
69.21 i 1.48
67.81 1 1.47
l-ayerr Conv 1D
66.06 1.71
63.20 + 1.30
3 layer ConvlD
68.82 : 1.62
66.88 = 1.52
5-layer Conv1D
69.09 + 1.51
67.42 + 1.45
7-layer CD
68.87 1 1.53
67.77 1 1.41
9 9-layer CD
68.68 + 1.46
67.89 + 1.47
A.4 Pseudo-code for PointNet-NTK NTK Computation and Its Applications in Shape Matching, Classification, and Reconstruction
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Algorithm PointNet-NTT Computations
Require: &, = (1) (Z 2m bm $, = [[21, ". f2mm 2m 11, > 0
if PointNet-NTKI then
X, (21,422-"0
Jm 0g, 3mm
Z, 2 {=1 22,
, {2), 82 .., m
pos + Algorithm
2
X, X, N)
nor Amgorithm
2 Z, N)
varifold + @pos nor
else if PointNet-NTKK2 then
CONCAT [Tyh: 2m) CONCAT (Bris 2 θ varifold + Algorithm (P. P,N)
P - (CO 21),
P + (CONCAT($), &
end if
return varifold
Hemark: As an example, X e Rxx is formed by concatenating all E {21, (21,02,"" 2m
Algorithm NTK Corresponding to N-layer mnfinitewidtth MLP with ReLU Activation"
Require: X, A >
Initialise θ (0) (0)
-
d (d 10 .....) diag(XX XxT),
d x (d
....) diag( X X
for h <- 1 to N do
w (k-1) + [.
(1-11)
/
(1-1) g(n-1)
d (h
= 1,2,
{1-11) (h d
(--1)
d
g(h-1)), where (Fo) = arccosw (h-1)
x
1
{(A) (1(1111) d
d
kh-
1)
where
x
(Fi)i.) I V
0h
1)
th
1)
(w) (w/)) (/11)) + (x arccos w (h-11) Jw! 11-1))
(h) < (h) + θ (1- (h-1)
(h) (())
1 d
th
(h-1)
2
x
end for
return θ (h)
"Although Algorithm
2 assumes the NTK representation corresponding to N-layer MLP with ReLU activation Cho
& Saul 2009 Lee al. 2019 Novak al.) 2020 , several popular neural network layers have their corresponding
closed-fon NTK representations NNvak et al. 2020 Lee et al. 2020 Hron et al. 2020).
Algorithm Shape Matching
Require: f(=:0) S, T, Temax >> O, Riter 0
while Rmax > flater do
Us E s| vertices S
ds f(us: 0) displacements between S and T
S + new source shape with deformed vertices vi ds
zg. 2g + sample surface points and corresponding normals from $
#T, =T sample surface points and corresponding normals from T
$g + {zg:
87 - [r]
Compute ||ss - ST I varifold in equation 16 using Algorithm Backpropagate and update 0
S+S
Nitter Rittr +1
end while
Remark: Here f(: 2-layer MLP neural network, is the weights of the neural network S and T the source and target shapes, respectively.
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Algorithm Shape Classification
Require: X itainn {81, 82, ,1),, Vtaain = {/ 122 ... , Restt = {8141, 81421 ;88): N > 0, where 9; = P2, ... . p.
.2
for to do
for s 1 to do
θ varifold (si, + Algorithm
(8i, 8j, N)
Aggregate points O yarifflld trait((..))
issm Zism @varifold(p) (p: E Pt E Es))
end for end for for
t 1+ to f do
for
++++10 to á do
θ varifold (si, 83) + Algorithm
(8i, 85, N)
Aggregate points end for end for ypred
θ pariifold
Zigm Ejsn
varifold (p; 8t, 85)
bestra,))
+ Ovarifold rest X (xrai) ( varifold train
Crain, X train)) A1) yy train
Algorithm Shape Reconstruction"
Require: X {01,-.. .xx) .. 2x..J' ty .k d, Xgrid, Zgrid, N
Ensure: Vy; 0 and 8 >>0
X xt f {I1 -21, ..., 2b 8zu}, {u1 + 02 2x + 5zx}
t (-,,.... 07 (5, ... .0)
2, 2.J
X XUX U
2 U 2 Z,yUU US
Krainn Pirain + (2, 2
varifold (Merain; X f- A Algorithm Xtrain: Mernin N)
Nestt + θ varifold (Nestt Xerain) Algorithm 1 (Keett Xerain: N)
(Kgrid Zgrid]
pred varifold (Restt Xursi) varifold (Xesinn Xirain) + A1] VYrrain
S Srecon +
Marching cube algorithm (Lewiner et al. 2003) (Keest predd
Please refer to more detailed explanation of terms and equations in Appendix A.2
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A.5
Ablation analysis
A.5.1 Neural varifolds with different number of neural network layers
This section shows the shape classification results based on different number of neural network layers. In this experiment, we randomly choose samples per class on the training set of ModelNet40 and evaluate on its validation set. As shown in Soction [ we iterate the experiments 20 times with different random sceds. The key concept of the PointNet (Qi al. 2017a) is the permutation invariant convolution operations on point clouds. For example, MLP or Conv1D with width convolution window is permutation invariance. Tn this experiment, the ModelNet40 dataset. As shown in Table 6 the classification accuracy of NTK1 with Conv1D operation is lower in comparison with the ones with MLP layers. In particular, 5-layer and 7-layer MLPs show similar performance with the NTK1 architccture, i.c., 69.29% classification accuracy. In order to reduce the computational cost, we recommend fixing the number of layers NTK1 to In the case of NTK2, its performance increases as more layers are being added for it with both MLP and Conv1D operations. Furthermore, NTK2 with ConvlD operation shows slightly higher classification accuracy in comparison with the ones with MLP layers. The percentage of the performance improvement becomes lower as the number of layers increases. In particular, 9-layer MLP versus 7-layer MLP for NTK2 only brings 0.2% improvement; therefore, it is computationally inefficient to increase the number of layers anymore. Although NTK2 with 9-layer Conv1D achieves 0.08% higher accuracy than the one with 9-layer MLP, NTK2 with 9-layer MLP rather than Conv1D is used for the rest of the experiments in order to make the architecture consistent with the NTK1.
choose different number of either MLP or ConvID layers, and check how it performs
A.5.2 Shape matching with different number of neural network layers
Table 7: Ablation analysis for shape matching with respect to different number of neural network layers within NTK psueo-metrics. The number inside of the brackets (-) indicates the number of layers used for computing the NTK pseudo-metrics.
Metric
NTKI (1)
NTKI (5)
NTK1 (9)
CD
2.82E-1
2.67E-1
2.99E-1 1
EMD
2,43E5
2.09E5
2.46E5
CT
2.19E3
2.17E3
2.17E3
NTKI
7.74E3
4.93E3
4.90E3
NTK2
2.56E3
1.54E3
1.92E3
Metric
NTK2 (1)
NTK2 (5)
NTK2 (9)
CD
2.59E-1
2.61E-1
2.64E-1
EMD
2.14E5
2.32E5
1.93E5
CT
2.15E3
2.17E3
2.15ES
NTK1
9.57E3
8.70E3
9.98E3
NTK2
1.28E3
1.41F3
1.53E3
In this section, the behavior of the NTK pseudo-metrice with respect to different number of layers is evaluated. Note that the neural network width is not considered in this scenario as all pseudo-metrics are computed analytically (i.e., (einite-width). In this study, simple shape matching networks were trained solely by NTK psuedo-metrics with different number of layers. Table 7 shows that the shape matching network trained with the 5-layer NTKI metric achieves the best score with respect to CD, EMD, CT and NTK2 metrics, while the one with the 9-layer NTK1 metric achieves the best score with respect to CT and NTK1 metrics. This is in accordance with the ablation analysis for shape classification, where 5-layer NTKI achieves the best classification accuracy in the ModelNet10 dataset. In comparison, NTK2 shows mixed signal. The shape matching network trained with the 1-layer NTK2 metric achieves the best outcome with respect to Chanfer, CT and NTK2 metrics, while the one trained with the 9-layer NTK2 achieves the best results with respect to EMD and CT metrics. The network trained with 5-layer NTK2 shows the best result with respect to the NTK1 metrie. This not exactly in accordance with respect shape classification with the NTK2 metric, where the shape classification accuracy improves the number of layers increases. However, training neural
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network always involves some non-deterministic nature; therefore, it is yet difficult to conchude whether the number of neural network layers is important for improving the shape matching quality or not.
A.5.3 Shape classification with hyperparameters ow and ob
As discussed in the main Section 3 on the NTK, the computation depends on two hyperparameterss representing the weight and bias variances in the infinite-width limit, denoted by ow and ob, respectively. In this section, we evaluate the shape classification performance of 9-layer MLP-based NTK2 by varying each hyperparameter independently. When varying ow. we fix σ = 0.05, and when varying db, we fix ow =.
Table 8: Shape classification performance of 9-layer NTK2 with varying weight variance ow, fixed dy = 0.05.
dw
NTK2 Accuracy (5-sample)
0.50
81.72 3.16
1.00
81.73 = 3.16
1.50
81.76 + 3.15
2.00
81.75 = 3.16
3.00
81.76 = 3.15
Table 9: Shape classification performance of 9-layer NTK2 with varying bias variance db, fixed ow 1.
d
NTK2 Accuracy (5-sample)
0.00
81.76 1 3.15
0.05
81.73 3.16
0.10
81.72 = 3.16
0.20
81.48 = 3.10
0.50
77.59 = 3.01
As shown in Tables and there is no significant variation in performance with changes in the hyperparameters. Although minor improvements are observed with different hyperparameter configurations, throughout all experiments in this paper, we adopt the standard parameterization (a = 1.0, σ 0.05) commonly used in the NTK literature (Lee et al. 2020 Novak et al. 2020). The theoretical interpretations of these NTK hyperparameters have been studied in the context of neural network generalizabilie and trainability (iioo et al. 2020). From a computational perspective, the weight variance hyperparameter ow acts as a multiplicative constant of 1.0 at each layer, while the bias variance a contributes by adding a constant term to all inputs.
A.5.4 Shape classification with different neural network width
In this section, we analyse how the neural network width can impact on shape classification using the 9-layer based NTK2 by varying the width settings from 128, 512, 1024 and 2048 infinitewwidth configurations. We trained the model on 5 randomly sampled point clouds per class from the ModelNet training set. The evaluation was carried out on the 1odelNet10 validation set. This process was repeated five times with different random seeds, and the average shape classification accuracy was computed. Notably, NTK1 was excluded from this experiment due to the absence of a finite-width neural network layer corresponding to the elementwise product between two neural tangent kernels of infinite-width neural networks. The results presented Table 10 demonstrate that the analytical NTK (infinitowwddth NTK) outperforms the empirical NTK computed from the corresponding finite-width neural network with a fixed width size. Furthermore, computing empirical NTK with respect to different length of parameters is known to be expensive as the empirical NTK is expressed as the outer product of the Jacobians of the output of the neural network with respect to the paramcters. The details of the computational complexity and potential accelcration have been studied in Novak et al. (2022). However, if the finite-width neural networks are Irained with the standard way instead of using empirical NTKs on a large dataset (e.g., CIFAR-10), then finite-wdith neural networks can outperform the neural tangent regime with performance significant margins (Lee et al., 2020 Arora et al., 2211)). In other words, there still large gap understanding regarding training dynamics between the finite-width neural networks and their empirical neural kernel representations.
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Table 10: Shape classification performance of 9-layer NTK2 with different neural network width.
Width for each layer
NTK2 (5-sample)
128-width
78.74 + 3.30
512-width
80.08 1 3.02
1024-width
79.97 + 3.24
2048-width
80.46 + 3.13
infinite-width
81.74 + 3.16
Table 11: ShapeNet 3D mesh reconstruction with 1024 points (mean/ values x E3).
Metric
Method
Airplane
Bench
Cabinet
Car
Chaic
Display
Lamp
Speaker
Kitle
Soda
Table
Phone
Vessel
SIREN
0.936
1.499
3.134
5.363
2.492
3.635
2.536
4.100
2.134
3.660
2.284
1.674
1.339
CD
(menn)
Neural Splines
11.610
1.905
2.264
2.440
2. 2,983
4.770
1.418
3.437
0.439
1.924
3.936
9.022
2.255
NKSR
1.898
3.506
6.224
2.286
3.584
46,997
9.229
4.138
0.005
2.029
3.213
2.243
1.285
NOT
1.584
1.742
2,274
2.494
2.655
5.337
1.465
3.947
0.456
1.870
2.029
12.138
1.341
CD
STREN
0.756
1.272
2.466
2.305
1.281
1.385
1.6
3.411
0.487
1.706
1.601
1:300
1.040
(mnodian)
Neural Splines
8.171
1.562
1.830
2.058
2.152
1.548
0.698
3.071
0.359
1.657
1.715
1.594
0.879
NKSR
1,900
2.245
1.799
2.190
2.116
1.880
2.317
3.488
0.107
1.697
1.695
1.3:15
0.956
NTKI
0.820
1.701
1,933
1.995
1.522
1,719
0.733
3.045
0.366
1.719
1.643
1.658
1.016
EMD
SIREN
2.183
3.679
6,385
10.712
5.932
7.527
12.850
8.714
3.164
7.633
4.992
3.645
3.265
Neural Splines
60.566
6.540
5,338
5.380
15.933
8,882
22.745
6.137
1.878
4.335
11.733
18.019
6.367
NKSR
12.939
11.990
16.684
7.571
21.706
14.190
32.236
12:486
3.613
4,930
14.917
6.609
6.715
NoT
6.704
5.984
5,301
5.907
14.868
11.507
29.595
8.070
1.773
4.596
11.606
24.903
3.841
SIREN
1.982
3.211
5,232
4.699
5.678
3.567
2.916
5.548
1.351
3.801
3.122
3.415
2.352
EMD
(inedian)
Neural Splinnes
35.458
4.713
4.745
4.779
11.570
3.915
5.719
4.575
3.630
5.011
4.828
2.276
NKSR
11.317
6.933
5.035
5.432
9.807
8.507
7.871
8.397
1.765
3.524
8.140
3,400
2.354
PointNet-NTKI
3,716
4.639
5,030
4.598
7.613
4.062
9.168
5.456
1.364
4.105
4.257
4.710
2.209
Table 12: ShapeNet 3D mesh reconstruction with 512 points (mean/mediaa) values x x1E3).
Metric
Method
Airplane
Bench
Cabinet
Car
Chair
Display
Lamp
Speaker
Riße
Sofa
Table
Phone
Vessel
SIREN
1.385
1.992
14.975
4.323
2.813
3.094
7,874
5.426
3.731
3.582
10.423
2.524
2.278
Neural Splines
21.410
3.752
2.818
2.985
5.217
5.089
2.050
4.393
0.565
2.228
5.953
8.721
2.699
(mueau)
NKSR
3,974
3.545
2.594
5.348
9.859
5,259
17.419
2.059
6.636
1.540
PoIT
2.454
2.674
2.565
3.233
3.793
6.087
2.193
4.045
0.550
2,252
2.702
14.349
2.090
SIREN
0.715
1.678
3.635
3.122
1.914
1.672
1.540
4.707
1.158
2,256
1.746
1.497
1.130
CD
(inodian)
Neural Splines
21.040
2.466
1.935
2.369
3.347
2,058
1.023
3.361
0.385
2.411
1.226
NKSR
2.627
3,336
1.891
2.015
3.752
NA
4.427
3.753
0.906
1.833
3.355
1.111
0.856
NTTI
1,243
2,246
2.106
2.316
2.473
1.968
1.346
3.330
0.387
1.890
1.963
2,013
1.309
SIREN
3.411
5.833
24.41M
9,460
7.366
6.558
26.828
13.584
5.224
6.457
16.578
4.764
4.831
EMD
Spiiins
120.415
11.740
7.478
6.057
26.382
11.466
30.216
8.686
3.048
5.128
25.433
19.057
8.431
(meea)
NKSR
21,59
21.190
11.435
9.346
30.485
NA
36,010
18.147
13.115
5.226
21.257
4.701
8.605
Not-tNet-VTKi
13.826
9.217
5.614
11.548
16.465
13.501
35.540
8.334
2.436
6.010
15.663
27. 025
5.897
EMD
SIREN
1.964
5.036
8.656
6.643
5.553
3.650
14.281
14.499
2.296
4.682
3.735
3.779
3.012
(modiaan)
Neural Splines
115.527
9.698
4.679
4.863
20.006
4.476
10.8834
5.405
1.548
4.234
8.2115
4.742
3.147
NKSR
25.231
14.795
4.105
669
16.082
NA
727
8.635
3.132
1.147
9.839
3.595
2.650
NTTI
9.863
4.758
6.822
9,296
5.683
1.626
4.497
7.455
3.313
indicates thas method fuils to recomstruck few shapes the given claso.
A.5.5 Shape reconstruction with different point cloud sizes
In this section, we compare shape reconstruction results with different point cloud sizes, i.e., 512, 1024 and 2048 points. As indicated in Tables 4 I and [2] NTK1 and neural splines show that the quality of the reconstructions is degraded as the number of points decreases. For NKSR, its reconstruction quality becomes worse as the number of point clouds decrcases for most categorics, but few categories (i.c., cabinct and vessel) show the opposite trend. In the case of SIREN, the convergence of the SIREN network plays more important role for the shape reconstruction quality. For example, the shape reconstruction results by SIREN on the airplane category show that the shape reconstruction with 1024 points is better than that with 2048 points. This is due to the non-deterministic nature of DNN librarics, i.c., is difficult to control the convergence of the SIREN network with our current, experimental setting 10a epochs. Note that the SIREN reconstruction
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Published in Transactions on Machine Learning Research (06/2025)
is computationally much more expensive (around 20~30 minutes) than either the NTK1, neural splines the NKSR approach (around 1~5 seconds).
A.6 Visualisation of ShapeNet reconstruction results
In this section, present additional visualisations of shape reconstruction outcomes obtained through three baseline methods (i.e., SIREN, neural splines, and NKSR), along with the proposed NTKI method, across 13 categories of ShapeNet benchmarks. Five shape reconstruction results are illustrated for each category. Specifically, Figure 3 showcases examples from the Airplane, Bench, and Cabinet categories. Figure f exhibits five instances of shape reconstruction outcomes for the Car, Chair, and Display categories. MMvving on to Figure [] it displays examples from the Lamp, Speaker, and Rifle categories. Similarly, Figure demmnsttraess five instances of shape reconstruction results for the Sofa, Table, and Phone categories. Finally, Figure focuses the shape reconstruction results for the Vessel category.
22

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Published in Transactions on Machine Learning Research (06/2025)
Airplane
Bench
Cabinet
Ground Truth
SIREN
Neural Splines
NKSR
NTK1
Figure Visualisation shape reconstruction results from STREN, Neural Splines, NKSR NTK1 for the Airplane, Bench and Cabinet categories.
23

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Published in Transactions on Machine Learning Research (06/2025)
Car
Chair
+
Display
Ground Truth
SIREN
Neural Splines
NKSR
NTK1
Figure 4: Visualisation of shape reconstruction results from STREN, Neural Splines, NKSR and NTK1 for the Car, Chair and Display categories.
24

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Published in Transactions on Machine Learning Research (06/2025)
Lamp
Speaker
Riffe
Cround Truth
SIREN
Neural Splines
NKSR
NTK1
Figure 5: Visnalisation of shape reconstruction results from SIREN, Neural Splines, NKSR and NTK1 for the Lamp, Speaker and Rifle categories.
25

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Published in Transactions on Machine Learning Research (06/2025)
Sofa
Table
Phone
Ground Truth
STREN
Neural Splines
NKSR
NTK1
Figure 6: Visualisation of shape reconstruction results from SIREN, Neural Splines, NKSR and NTK1 for the Sofa, Table and Phone categories.
26

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Published in Transactions on Machine Learning Research (06/2025)
Vessel
Ground Truth
SIREN
Neural Splines
NKSR
NTK1
Figure 7: Visualisation of shape reconstruction results from SIREN, Neural Splines, NKSR and NTKI for the Vessel category.
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Reviews:
[{"review_id": "eBNtTfmn0k", "reviewer": "Reviewer_UZ4u", "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- Integrating geometric measure theory with deep learning is novel and shows promise for robust geometric characterization.\n- The authors provide clear mathematical foundations and practical algorithms, making the neural varifold concept well-grounded and applicable.\n- The method is thoroughly validated across multiple tasks, demonstrating versatility and effectiveness, particularly in shape matching and few-shot classification scenarios.\n- The thorough comparison of Chamfer distance, Earth Mover’s distance, etc., highlights the benefits and potential use-cases of neural varifolds clearly.\n\n**Weaknesses**\n- Kernel-based methods have quadratic complexity. The authors acknowledge this, but more practical methods or approximations to handle large-scale data effectively are not explored in depth.\n- The neural varifold method relies heavily on normal vectors, which may limit its applicability in scenarios where accurate normal information is unavailable or hard to compute.\n- The neural architecture used (PointNet-based) is relatively simple. The exploration of more advanced neural architectures (e.g., Graph CNNs or transformer-based models) is absent, potentially restricting performance improvements.", "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 introduces a novel representation called neural varifolds, which leverages deep neural networks and concepts from geometric measure theory to characterize the geometry of point clouds. The authors propose representing surfaces as distributions over positions and tangent spaces, combining spatial geometry with geometric consistency. Two algorithms for computing varifold norms using neural networks and their neural tangent kernel representations are presented. Extensive experimental results demonstrate the effectiveness of neural varifolds in shape matching and few-shot shape classification. It also has competitive performance in shape reconstruction tasks compared to state-of-the-art methods.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "**Questions**\n- Have the authors explored methods to reduce the computational complexity of neural varifolds, such as kernel approximations?\n- How robust is the proposed neural varifold method to noisy or incomplete normal information? Could the authors provide insights or experiments on the sensitivity of the proposed method?\n- What motivated the choice of PointNet-based architectures, and would we have substantial performance gains from using more complex architectures (e.g., graph-based neural networks or transformers)?", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "xqkeo1dmEQ", "reviewer": "Reviewer_qt59", "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 introduction of neural varifolds is a novel and promising approach to representing and analyzing point cloud data, particularly in capturing both macro and subtle geometric features. The paper is well-grounded in geometric measure theory, providing a solid theoretical foundation for the proposed methods.\n\n2. The neural varifold representation is shown to be effective across multiple tasks, including shape matching, few-shot classification, and shape reconstruction, demonstrating its versatility.\n\n\n**Weaknesses:**\n\n1. The running time should be evaluated for the proposed neural varifold approach, which could be a limitation for large data. While kernel approximation methods are mentioned, their performance is not evaluated in the paper.\n\n2. The proposed method is based on the simpler PointNet architecture. The paper suggests that future work could explore more advanced architectures like graph convolutions or voxelized point clouds, but this is not addressed in the current work. It is encouraged that to evaluate the more divese network architectures and release the code for community testing.\n\n3. As a measurement of quantifying the geometry of point cloud, the parameter sensity is a very essential part. It is good if the neural varifold is robust for different cases. The paper mentions that the Charon-Trouve varifold norm is sensitive to point cloud density and requires hyperparameter adjustments for each pair of point clouds. It is unclear if the proposed neural varifold methods share similar sensitivity.", "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 introduces a novel approach to quantifying the geometry of point clouds using neural varifolds, which combine deep learning techniques with geometric fidelity metrics. The key contributions are:\n\n1. The authors propose a new representation for point clouds that captures both point positions and tangent spaces, leveraging the concept of varifolds from geometric measure theory. \n\n2. Two algorithms are proposed to compute the varifold norm between two point clouds using neural networks and neural tangent kernel representations. \n\n3. The neural varifold representation is evaluated on three tasks: shape matching, few-shot shape classification, and shape reconstruction. The results demonstrate that the proposed method outperforms state-of-the-art methods in shape matching and few-shot shape classification, and is competitive in shape reconstruction.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. The authors should evaluate the performance of kernel approximation methods (e.g., Nystrom approximation) to address the computational complexity issue. This would strengthen the paper by providing a practical solution for scaling the method to larger datasets. The time complexity should be fully evaluated.\n\n2. The authors should explore the performance of the neural varifold representation with more advanced architectures, such as graph convolutions or voxelized point clouds. This would demonstrate the generalizability of the proposed method.\n\n3. The authors should provide a more detailed analysis of the sensitivity of the proposed neural varifold methods to hyperparameters, particularly in comparison to the Charon-Trouve varifold norm.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "iZCZaW7Krf", "reviewer": "Reviewer_QcnL", "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": "Strength:\n1. The paper is well-written, with clear structure and illustration.\n2. The experiments are thorough and comprehensive. Additional results, ablation studies and visualizations are provided.\n\nWeakness:\n1. The architecture assumption in section 3 (Neural tangent kernel, page 4) is pretty strong. It is not clear whether this representation could be extended to more advanced/complicated architectures.\n2. Comparing to evaluation metrics, like CD and EMD, additional information (surface normal) is needed. In the real-world application, these information may not be available/expensive to acquire.\n3. The related work on different representations/evaluation metrics are not throughly discussed.", "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 introduces a neural varifold representation of 3d point cloud. The authors propose neural varifold algorithms to compute the varifold norm between point clouds using neural networks and neural tangent kernel. Extensive experiments are conducted for the proposed neural varifold on three point cloud tasks to demonstrate the effectiveness of the proposed representations.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. The $p_\\hat{i}$ in equation (13) and equation (14) is different, but with same notation. It would be better to use different notation to distinguish them.\n2. Equation (16) does not make sense. Might have typos with $s_i$ and $s_j$.\n3. In section 4 - Dataset and experimental setting, it would be better if the authors could introduce more on shape similarity metrics, and highlight the difference with the proposed one. Could also introduce this part in the Related Work section.\n4. In Table1, NTK1 and NTK2 are used as metrics, while in Table 2, they are used as methods. It would be better if the authors could distinguish them to avoid confusion.\n5. It would be better if the authors to include a Conclusion section to highlight the contributions and takeaways.", "broader_impact_concerns": "No obvious concerns on the ethical implications of this work.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The reviewers all acknowledged the contributions and value of this work. They nonetheless expressed come concerns, particularly regarding the computational complexity of the method and the use of relatively simple architectures. The authors addressed these points in their responses and commented on them in the paper. Ultimately, all reviewers recommended acceptance in their final recommendation.

Although the AE believes that most of the important points raised in the reviews were incorporated in the revised version of the manuscript, it seems that it was not the case for the question of Reviewer qt59 on sensitivity to hyper-parameters. As the discussion of hyper-parameter values is important for reproducibility, the AE requests the authors to discuss this in the final version of the paper.
Decision: Accept (poster)