Matérn Kernels for Tunable Implicit Surface Reconstruction
Authors: Maximilian Weiherer
Bernhard Egger
Venue: ICLR 2025 Poster (2025)
Keywords: surface reconstruction
kernel methods
neural tangent kernel
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: We use and theoretically analyze the family of Matérn kernels for tunable implicit surface reconstruction, demonstrating that they outperform recently employed arc-cosine kernels.

Abstract: We propose to use the family of Matérn kernels for implicit surface reconstruction, building upon the recent success of kernel methods for 3D reconstruction of oriented point clouds. As we show from a theoretical and practical perspective, Matérn kernels have some appealing properties which make them particularly well suited for surface reconstruction---outperforming state-of-the-art methods based on the arc-cosine kernel while being significantly easier to implement, faster to compute, and scalable. Being stationary, we demonstrate that Matérn kernels allow for tunable surface reconstruction in the same way as Fourier feature mappings help coordinate-based MLPs overcome spectral bias. Moreover, we theoretically analyze Matérn kernels' connection to SIREN networks as well as their relation to previously employed arc-cosine kernels. Finally, based on recently introduced Neural Kernel Fields, we present data-dependent Matérn kernels and conclude that especially the Laplace kernel (being part of the Matérn family) is extremely competitive, performing almost on par with state-of-the-art methods in the noise-free case while having a more than five times shorter training time.

Published as a conference paper at ICLR 2025
MATÉRN KERNELS FOR TUNABLE IMPLICIT SURFACE RECONSTRUCTION
Maximilian Weiherer & Bernhard Egger
Department of Computer Science Alexander-Universstäl Universität Erlangen-Nimberg (maximi lian. we wherer, eggerara/d.gggeeeaaaaaea
ABSTRACT
We propose to use the family of Matérn kernels for implicit surface reconstruc- tion, building upon the recent success of kernel methods for 3D reconstruction of oriented point clouds. As we show from theoretical and practical perspec- tive, Matérn kernels have some appealing properties which make them particu- larly well suited for surface pnronstructioonoouppfffffooo ods based on the are-cosine kernel while being significantly easier to implement, faster to compute, and scalable. Being stationary, we demonstrate that Matérn ker- nels allow for lunable surface reconstruction in the same way as Fourier feature mappings help coordinate- MLPs overcome spectral bias. Moreover, oretically analyze Matérn kernels' connection to SIREN networks as well as their relation previously employed arc-cosine kernels. Finally, based on recently introduced Neural Kernel Fields, we present data-dependent Matérn kernels and conclude that especially the Laplace kernel (being part of the Matérn family) is extremely competitive, performing almost par with state-of-the-arr methods in the noise-free case while having a more than five times shorter training time !]
state-of-the- art meth-
the-
INTRODUCTION
Recovering the shape of objects from sparse or only partial observations is challenging task. For- mally, let Ω C Rd and X' {x1, ..,..., I' forms, together with associated per-point normals, dataset D = X' x {ni: 122, Rmm) C Ω x Idd The goal of surface reconstruction is to recover the object's shape from D. The shape of objects may be represented as dense point clouds, polygonal meshes, manifold atlases, voxel grids, or (as the zero-level set of) implicit functions, which is the representation work. Specifically, if f : Rdd _ TR denotes an implicit function, such as a Signed Distance Func- tion (SDF), then, from practical perspective, implicit surface reconstruction aims at finding an optimal solution ff to the following Kernel Ridge Regression (KRR) problem:
C Ω be set of m points in d dimensions which
chose to focus in this
m Σ
f = arg min JEH
+ - n, All
where the function space H = (Q) is usually taken to be Reproducing Kernel Hilbert Space (RKHS) with associated reproducing kernel & Ω Ω - R. and X 0. Once f at hand, the reconstructed object's is surface is implicitly given as S {a f()) = 0} Rd and can be extracted using Marching Cubes (Lorensen & Cline 1987). It is quite easy to observe that, as X - 0, the optimization problem in Eq. (1) turns into the constrained minimization problem,
minimize 1| s.t. f(xx) =0 and V((x)) - n Vi E 21, m} lEZ
which makes it easier to see how the predicted surface behaves away from the input points. As seen from Eq. (2). while & should interpolate the given points X' exactly, the behavior in be- tween and away from those points is solely determined by the induced norm || of the func-
tion space, H. over which we are optimizing. The properties of || are uniquely defined by
available http:////ettttttcooo/eeeeeeeeeee otcaraaeeeetettttaceeeeeeenntttt wwconstructincncccccc

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Published as a conference paper at ICLR 2025
Mateen lecoel spectaal deneity diffeeat values (h=3)
Matêra bernel and epectmi donetty Sor ditiecent valuos (yrw 1/20
0.6
86
0.6
0.5
0.4
64
0.
60
0.5
- +/4 x/7
-1/6
7/4
4/
Figure 1:
Matérn kernels with associated spectral densities. We propose to use the family of Matérn kernels for tunable implicit surface reconstruction, parametrized by a smoothness parameter, > 0, that controls the differentiability of the kernel, and a bandwidth parameter h > 0. Both parameters allow explicit manipulation of the kernel and its spectrum. Importantly, Matérn kernels recover the Laplace kernel for 1/2 and the Gaussian kernel -> xc.
the reproducing kernel, k; this is a direct consequence of the Representer Theorem (Kimeldorf & Wahba 1970 Schölkopf et al. 2001) which states that solutions to Eq. 1 are of the form [(r
Ka. Here, E IR" and K = (KJ, Jig k(x,, m2)) E Rmmm.. This shows that the behavior of the norm and hence, estimated surfaces S. can be explicitly con-
-== (;)( yielding |/
trolled by the kernel function, enabling easy injection of inductive biases (such as smoothness as- sumptions) into the surface reconstruction problem. Even further, if the chosen kernel has adjustable parameters, they can be used to adaptively (on shape-by-shape basis) manipulate the inductive bias.
Although already used in the mid-90s and early 2000s (Savchenko al.) 2001}, only recently, kernel-based methods for 3D implicit surface reconstruction became extremely competitive, with Neural Kernel Surface Reconstruction (NKSR; [Huang et al. (2023) eventually evolving as the new late-of-theeart. While early works mostly focused on (polyharmonic) radial basis functions such as thin-plate splines, recent work (Williams et al.) [021]] 2022) employ the first- order are-cosine kernel. Introduced by Cho & Saul (2009), arc-cosine kernels have been shown to mimic the computation of (two-layer) fully-connected ReLU networks; indeed, if the layer-width tends to infinity, the first-order arc-cosine kernel is identified with the Neural Tangent Kernel (NTK; Jacot et al. (2018) of the network. One important aspect of this kernel is that it does not have adjustable parameters. While this can be advantageous in some situations, also prevents users from tuning their reconstructions. If the arc-cosine kernel fails to accurately recover surface, then there is no possibility (except for using more observations) to improve the result.
1995 Carr et al. 1997.
In this work, we suggest different family of kernels (with parameters v. h > 0) for implicit surface reconstruction: Matérn kernels (Matérn 1986 Stein 1999). see Figures [ and 2 Con- trary to arc-cosine kernels, Matérn kernels are stationary (hence translation invariant), spatially decaying (thus leading to sparse Gram matri- ces), and unify a variety of popular kernel func- tions, including the Laplace and Gaussian kernel. As we will show, Matérn kernels have appeal- ing properties, making them the ideal candidate for
surface reconstruction. From a practical perspective, we demonstrate that sim- ple change of the kernel function from arc-cosine to Matérn leads to a consistently improved re- construction accuracy and significant speedup. From a theoretical perspective, we argue that Matérn kernels allow for tunable surface recon- struction in the same way as Fourier feature map- pings (Tancik et al.) 2020) enable coordinate- based Multi-layer Perceptrons (MLPs) to learn figh-frequency functions in low-dimensional do-
Input points
Arc-cosine
Matérn 3/2
Ground truth
Figure 2: Tunable Matérn kernels lead to bet- ter surface reconstructions than the previ- ously employed arc-cosine kernel. Here, we show surface reconstructions from sparse point clouds of just 1,000 points.
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mains, effectively overcoming spectral bias (the slow convergence towards high-frequency details). In addition, we prove that, as the layer- approaches infinity, Matérn kemelss are identified with the NTK of two-layer SIREN (fully-connected MLPs with sine activations; Siizmaann et al. (2020)) networks-together with Fourier feature mappings the two arguably most influential methods to overcome spectral bias in coordinate-based MI.ss. Lastly, establish a connection between arc- cosine and Matérn kernels by showing that the Laplace kernel (Matérn kernel with y 1/2) and the arc-cosine kernel are equal up an affine transformation when restricted the sphere. In summary, our key contributions are:
We propose use Matérn kernels for tunable implicit surface reconstruction.
We theoretically analyze Matérn kernels, relating them to Fourier feature mappings, SIREN networks, and are-cosine kernels. Moreover, we derive practical insights into how choose the tunable bandwidth parameter based on a new bound of the L2 reconstruction enror.
We propose data-dependent (i.e,, learnable) Matérn kernels by leveraging the Neural Kernel Field (NKF) framework (Williams al. 2022).
Our experimental evaluation reveals that Matérn 1/2 and 3/2 are extremely competitive, outperform- ing the arc-cosine kernel while being significantly easier implement (essentially two lines of stan- dard PyTTrrch code), faster to compute, and scalable. In addition to geometry, we show that Matérn kernels surpass the arc-cosine kernel in reconstructing other high-frequency scene attributes, such as texture. Finally, we demonstrate that learnable Matérn kernels (1) outperform the data-dependent arc-cosine kernel (as implemented in the original NKF framework) while being more than four times faster to train, and (2) perform almost on par with highly sophisticated and well well-engineered NKSR in the noise-free case while having a more than five times shorter training time.
RELATED WORK
We briefly review some relevant literature about 3D implicit surface reconstruction from oriented point clouds, focusing on kernel-based methods. For a more in-depth overview, including neural- network-based reconstruction methods, see surveys by Berger et al. (2017); Huang et al. (202b))
Early kernel-based 3D surface reconstruction methods mostly employ Radial Basis Functions (RBFs) such thin-plate splines (Savchenko et al., 1995), biharmonic RBFs (Carr al., 1997 2001), or Gaussian kernels (Schölkopf et al.] 2005). Nowadays, the most widely used surface recon- struction technique is Screened Poisson Surface Reconstruction (SPSR: Kazhdan & Hoppe (2013) which, however, can itself be viewed as kernel method (Williams et al. (002)). Only recently. Neural Splines (NS: Williams et al. (202I)) proposed use the so-called (first-order) arc-cosine kernel,
kAC(z, (x)
||||||||]
(sin + (T 0) cos 0), where
= cos-
[ (
x
(3)
for implicit surface reconstruction, achieving state-of-the results that outperform classical surface reconstruction techniques and non-linear methods based on neural networks by large margin. This method laid the cornerstone for Neural Kernel Fields (NKFs: Williams et al. (2022)) ) which attempts to make are-cosine kernels learnable by passing input points through task specific neural network before evaluating the kernel function, similar to Deep Kernel Learning (Wilson et al., 2016). Based on SPSR, NeuralGalerkin (Huang al. 2022a) proposed to learn basis functions inferred by sparse convolutional network instead of using a fixed Bézier basis as in SPSR, hence can be seen kernel method in the broader sense. Neural Kernel Surface Reconstruction (NKSR: |Haan et al. 2023)) built upon NKF and NeuralGalerkin and proposed an all-purpose, highly scalable surface reconstruction method that is robust against noise and eventually became state-of-the-an
MATÉRN KERNELS FOR TUNABLE SURFACE RECONSTRUCTION
We propose to use the family of Matérn kernels (Matérn reconstruction, being defined as
1986
Stein
1999) for implicit surface
21-v T(u)
/2VuT h
V2VT Kv h
(x v) = l - - (T(
(4)
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where 0 is smoothness parameter that explicitly controls the differentiability, and h > is known as the shape parameter (or bandwidth) of the kernel. I denotes the Gamma function, and K, is the modified Bessel function of the second kind of order Matémn kemels generalize variety of other kernel functions; the most popular ones can be written in closed form as
1/2: (T) exp
(5)
h
V
V3T
V 3/2 (T) cxp
1
(6)
v5r
V5T
5r
V 5/2 (T) exp
3h2
where 1/ is known as the I Laplace kernel. In the limiting case of v - co, Matérn kernels recover the popular Gaussian kernel:
OT) lim (T( exp
(8)
2h2
For more information about Matérn kernels, please refer Poreu al. (003). Next, we revisit some basic properties of Matérn kernels and place them in the context of surface reconstruction.
3.1 BASIC PROPERTIES
Differentiable. Matérn kernels allow for controlled smoothness, being exactly [v] 1 times dif- ferentiable (in the square sense). Since functions f in an RKHS H inherit the differentiability class of the inducing kernel & due to the reproducing property, the from Eq. ①T reconstructed sur- face S enjoys the same smoothness properties as k. In the context of 3D surface reconstruction, this allows easy injection of inductive biases into the reconstruction problem; for instance, if the roughness or noisiness of the objects to be reconstructed is known in advance, one can adjust the smoothness of the kernel accordingly. This is not possible with the arc-cosine kernel.
Stationary. Matérn kernels are stationary, i.e., they only depend on the difference x - between two points 2, y E Q. If the distance Euclidean, Matérn kernels are also isotropic; they only depend on || yll- As a result, Matérn kernels are rotation and translation invariant, hence being independent the absolute embedding points X' (based on the reproducing property, translating input points does not change the reconstructed surface). Consequently, since the kernel's value only depends on the relative positions, objects scene (or parts of an object) with similar geometric properties will be reconstructed consistently each instance (or part) remain the same. This is in contrast to the arc-cosine kernel, whose value depends the absolute position of points, see Eq. (3); hence, it is not translation invariant (thus non-stationary), and multiple instances of an object in scene will generally not yield the same surface reconstruction. See Appendix A for further discussion.
long the relative distances between points
Spatially decaying, As opposed to arc-cosine kernels, Matérn kernels are spatially decaying; they tend
the distance || y/l becomes large. Consequently, although Matérn kernels technically not compact (or locally supported), because their value decays exponentially fast (Porcu al.) 2023), they can be considered effectively compact. From computational perspective. this is a very desirable property as it leads to sparse Gram matrices when truncated with a special kernel (Gienton 200T). allowing the use of highly efficient and scalable sparse linear solvers. Finally, we note that simply truncating any kernel does not yield valid, positive definite kernel in general.
We now derive new theoretical insights into Matérn kernels, ultimately aiming provide arguments as to why we believe this family of kernels is particularly well suited for surface reconstruction.
3.2 EIGENVALUE DECAY, SPECTRAL BIAS, AND RECONSTRUCTION ERROR
Eigenvalue decay and spectral bias. We begin by showing that Matérn kernels allow for tunable surface reconstruction in the same way Fourier feature mappings help coordinate- overcome spectral bias. As investigated in Tancik et al. (2020), a rapid decrease of the NTK's ' eigen- values implies associated MI.P's slow convergence to high-frequency components of the target
MLPs
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0.05
x= 1/2
6 0.5
0.04
F- 3/2
0.01
P- 5/2
8 5
0.03
P- 11
0.03
anc cosine
0.02
0.02
ñ 50
GDC- cosine
0.01
0.01
0.00
50
1.00
150
200
50
1000
160
200
Figure 3: Eigenvalue decay of Matérn kermels. While we fix and vary on the left, the EDR for V = 1/2 and different values of h is shown the right. Larger values of and h, i.e, smoother kernels, lead to faster eigenvalue decay; hence, slow convergence high-frequency details.
function (up to the point where the network is practically unable to learn these components). Con- sequently, smaller eigenvalue decay rate (EDR), i.e., slower eigenvalue decay, leads to faster convergence to high-frequency content-in the context of implicit surface reconstruction, more de- tailed geometry. To overcome this slow convergence, a phenomenon known spectral bias, Tancik iet al. (2020) use Fourier feature mapping of the form
(x) = [ai cos(2Tb) z), aj sin(2 x), ..),..,,,, c2s x), ag sin(2 x)]
applied to the inputs x Rd before passing them the MLP, effectively transforming the MLP's NTK, YNIK, into a stationary kernel, kNIK(>(<)) (())) = kNIK (ky (z y) =: 1NTK (x y) whose spectrum can be tuned through manipulation of the Fourier basis frequencies. b, E IR and corre- sponding coefficients, 0; IR, of the kernel ( y) Ei=1 af cos(2mb; (a: v)). This shows that y enables explicit control over the NTK's EDR, overcoming spectral bias.
Matérn kernels, being stationary, allow for the same degree of control feature mapping; their spectrum can also be tuned (by varying v and/or h, see Figure D This is in stark contrast to previously employed, non-stationary and parameter-less arc-cosine kernels, whose spectrum is not tunable. To further study the dependence of the EDR on v and h, we use:
the EDR as Fourier
Theorem (Seeger et (2008), Theorem 3). The eigenvalues of Matérn kernels decay polynomi- ally at rate
( 21
with bandwidth parameter h 0, and for finite smoothness 0 < CC.
Figure visualizes the EDR for differrnnt values and along with the EDR of the non-tunable, first-order arc-cosine kernel which equates to 0(s-(1+d))) and has only been very recently proven by [i et al. (2024) for general input domains and distributions. As seen, the smoother the kernel (i.e., the larger v) and the greater ht the faster the eigenvalues decay. This implies less detailed surface reconstruction (with a higher error) for smoother kernels and larger I;; indeed, this is exactly what we observe in practice, see Figure 4 Contrary, by lowering both parameters, one can achieve faster convergence high-frequency geometric details, effectively leading to more nuanced surface reconstruction and a smaller reconstruction error.
Reconstruction error. We proceed by investigating the bandwidth's influence the Lg reconstruc- tion error, which is defined (and can be bounded) as
- = (
1/2
c(utd)/2
dx
(10)
where Cx.s is a constant that only depends on X' and Qt and
f (22
2v (2Tl(=1)2 u+d/2
Ii 20
4/2 (22)
[F(f(w))
(11)
the norm of the RKHS induced by a Matérn kernel with bandwidth h and smoothness v. Moreover,
Ca. (2d ( )/2)), and F[J] denotes the Fourier transform of function f.
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Xtrim 12
sapim 22
0.05
Figure 4: Matérn kernels are tunable. Surface reconstructions can be improved practice by varying the bandwidth parameter, h, effectively tuning the kernels' EDR (sce also Figure 3). How- ever, setting á too small (< 0.5) results in overfitting, while setting h too big (> 10) oversmooths (i.e., underfits) the true surface. This is also reflected in the reconstruction error (measured using Chamfer distance), plotted as a function of h on the right.
For more information on the bound, including details on Cr,a- please see, e.g., Santin & Schaback (2016). By inspecting Eq. (TO), notice that the magnitude of the RKHS norm has considerably high influence on the reconstruction error. Based on this observation and the fact that || depends on h, our goal is now to further study the effect of h the RKHS norm. As first result, we have:
Proposition 2. The RKHS norm of Matérn kernels defined in Eq. (17) can be bounded by
Il S 2220
(Ae Ca + (I.(1)
where h > O, and C22, and O2,y are functions of Fif) that do not depend on
A proof can be found in Appendix B We immediately that If and h - co. Moreover, based on Proposition 2 it is easy to show that the norm, as a function of h, decreases as h h² from the left, and increases as h co, where htt is given
in either cases, h -
d 2v
1/(2++d)
(FI(F
h"
Q
with Q (FIf)
(12)
(see Appendix C for details). Our analysis shows that the re- construction eтror can not be made arbitrarily small by just lowering h; if h is chosen too small, the reconstruction error
0.10
r-
0.08
3/2
6/2
starts increase again, i.e., it overfits the true surface. Con- versely, if h is set too high, the resulting surface is too smooth (underfits the true surface), ultimately leading to increased er- rors. The optimal trade-off is reached if h = h". This is also observed in practice, see Figure 4 As seen from Figure however, the described effect is more pronounced for smoother Matérn kernels (large v), as, at some point, the norm (hence, the reconstruction error) increases rapidly for large h. Our the- oretical analysis reveals several practical insights. First, the reconstruction error is generally more sensitive very small values of h. Second, a good starting point is always = 1.
0.02
0.00
2
9
10
Figure 5: RKHS norm as func- tion of h. We plot the bound from Proposition 2 as a function of h.
3.3 RELATION TO NEURAL NETWORKS AND THE ARC-COSINE KERNEL.
Relation to neural networks. Next, we study the connection between Matérn kernels and neural networks. It is well known that the first-order arc-cosine kernel mimics the computation of two- layer, infinite-width ReLU network f _d R. f(x) m 1/2 Σm [ x + 6,]_ when the bottom layer weights W (W1, ..,.... Wm) ddxm and biases b = (b1, b2, be..... bm) E R" are fixed from initialization and drawn from a standard normal distribution, see, e.g., Cho & Saul 2009): Williams al. (2021). Here, [x]+ = max (0, x)) denotes the ReL.U activation function.

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Under similar assumptions, now show that Matérn kemels mimic the computation of two-layer SIRENS (Sitzmann et al. 2020) if the layer-width approaches infinity. We claim the following:
IRd function, m hidden neurons, and fixed bottom layer weights W = (W1, W2..... Wm) E R and
Theorem 3. Consider a two-layer fully- conleecoddeeeed network
R with sine activation
biases (b,, by.... ..... bm)) E R. Let h, >> and Ca. be defined as in Eq. 11 If w is randomly initialized from
-(V+d/2)
2v Pu(w) = h 2 Ca.
+ (2 (221)
h22
and b; u(, 2T), the NTK of f is Matérn kernel with bandwidth h and smoothness as m + cc.
A proof can be found in Appendix D This result establishes for the first time connection between SIRENS and keree methods. While explicitly shown for Matérn kemells, as we argue in the Appendix D the arguments of Theorem B in fact apply to all stationary kernels, making it powerful tool to study the connection between widely used STRENS and any stationary kernel function.
Matérn kernels vs. arc-cosine kernels. Lastly, we aim to compare Matérn kernels with the previ- ously used first-order arc-cosine kernel. Our analysis based on:
Theorem 4 (Chen & Xu (2021), Theorem 1). When restricted to the hypersphere S the RKHS of the Matérn kernel with smoothness y = 1/2 (the Laplace kernel) include the same set of functions as the RKHS induced by the NTK of fully-comnected (L Z 2)-layer ReLU network.
In other words, Theorem 4shorsssShowss shows that the RKHS of the Laplace kernel (a Matérn kernel with smooth- ness v = 1122) the same as the RKHS induced by the NTK of a fully-connected ReLU network when inputs are restricted to gd-1. Based on Theorem 4
we obtain the following connection be-
tween Laplace and the first-order arc-cosine kernel:
Corollary 5. When restricted to the hypersphere Sd-1, (1) the RKHS of the Matérn kernel with smoothness v = 1/2 (the Laplace kernel) is equal to the RKHS induced by the first-order arc-cosine kernel, implying that (2) the Laplace and arc-cosine kernel are equal up to an affine transformation.
See Appendix E for proof. For general input domains, our empirical results presented next support (at least for (L = 2yyl networks) the widely believed claim that the NTK is not significantly different from standard kernels (Belkin et al. 2018 Hui et al. 2019 Geifman et al. 2220), such as the Laplace kernel. Indeed, we show that the Laplace kernel even outperforms the arc-cosine kernel.
4 EXPERIMENTS AND RESULTS
We systematically evaluated the effectiveness of Matérn kernels in the context of implicit surface reconstruction, presenting results ShapeNet (Chang et al. 2015) and the Surface Reconstruction Benchmark (SRB: Berger et al. (2013) in Section 4.1 Moreover, Section 4.2 demonstrates Matérn kernels' ability reconstruct
textures the Google Scanned Objects (GSO:Downs Downs 2022)) dataset and Objaverse (Deitke et al. 2023). Leveraging Neural Kernel Fields (Williams al.
et al.
2022 we present and evaluate learnable Matérn kernels in Section 4.3
4.1 SURFACE RECONSTRUCTION ON SHAPENET AND SRB
ShapeNet. We compare Matérn kernels in a sparse setting against the arc-cosine kernel on ShapeNet (using train/val/test split provided by Mescheder et al. (2019)). To do so, we randomly sample 1, 0000 surface points with corresponding normals for each shape. We implemented Matérn kernels in PyTorch and took the official CUDA implementation of the arc-cosine kernel from NS, eventually integrated into unified framework to ensure fair comparison. Notably, did not use Nyström sampling as in NS and employed Py Torch's 's direct Cholesky solver to solve the KRR problem in Eq. ① instead of iterative conjugate gradient solver. Moreover, similar to NS, approximate the gradient part in Eq. I with finite differences. See Appendix F for details.
Quantitative results in terms of F-Score (with threshold of 0.01) and Chamfer distance (CD; always reported x 10²) can be found in Table [ kernel as well popular SPSR (Kazhdan & Hoppe
Matérn 1/2 and 3/2 consistently outperform the arc-cosine 2013). In addition, experiments show
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F-Score f
CD +
CD
IID
Matém
0.21
4.43
0.5
0.5
1
2
Matém 3/2
0.18
2.93
Matém 5/2
0.58
22.52
Matérn 1/2
93.6
93.7
93.7
4.05
4.02
4.02
Matém xo
3.83
33.27
Matérn 3/2
94.6
94.8
94.9
4.05
4.00
4.09
NS
0.19
3.19
Matérn 5/2
92.4
93.8
92.9
6.42
5.65
6.91
NS
0.17
2,85
Matérn
59.3
52.4
40.9
57.56
50.59
48.38
SPSR
0.21
4.69
FFN
0.28
4.45
Arc-cosine
92.8
4.67
SIREN
0.19
3.86
SAP
0.21
4.85
NS
90.6
4.74
DIGS
0.18
3.55
VisCo
0.18
2.95
SPSR
84.3
6.26
OG-INR
0.20
4.06
Table I: Results on ShapeNet (left) and SRB (right) for non-learnable kernels. Arc-cosine
kernel and Matérn E 31/2,3/2, oo) are implemented in unified framework and based the exact same parameters. "NS*" denotes the best result we could achieve by running the official implementation of NS, see Appendix G Bold marks best result, underline second best.
Matéra Matérn 3/2
Masēm 1/2
Mutém 3/2
Ground truth
Figure 6: Qualitative results for non-learnable kernels SRB (left) and learned kernels on ShapeNet (right). Please see Appendices a for more qualitative results.
that the bandwidth parameter, h, can used to tune the surface reconstruction (lower the error), while not being too sensitive in practice. Matérn 5/2 and the Gaussian kernel (for v + oo) perform significantly worse, being simply too smooth. Figure and Appendix shows qualitative results.
SRB. Next, we evaluate Matérn kernels on the challenging Surface Re- construction Benchmark which consists five complex shapes simu- lated from incomplete and noisy range scans with up 100,000 points. For this, we implemented a highly scalable version of Matérn ker- nels, closely following the NS framework to solve the KRR problem
Time
Matérn 1/2
9.85
Matérn 3/2
12.47
Matérn 5/2
15.26
in Eq. (1) using Nyström sampling (with 15,000 samples) and the | 2017] solver. We compare against NS and SPSR, as well as kemel-free methods, Fourier Feature Networks (FFNss Tan- cik al. (2020)), SIREN, SAP (Peng al. 202T) DiGS (Ben-Shabat et (002)). VisCo (Pumarola et al.) 2022). and OG-INR (Koneputu- godage et al., 2023). All methods are optimization-buee based (FFN and STREN are overfitted to single shape) and use surface normals. For Matérn kernels, we did modest parameter sweep over h E (0.5, 1,2) and took the reconstruction with the lowest Chamfer distance. Runtime is measured on a single NVIDIA V100.
Matérn oo
12.76
FALKON (Rudi et al.
NS*
19.89
NS SPSR
11.91
1.65
Table 2: Runtime com- parison on SRB. Time in seconds.
Quantitative results obtained by employing Chamfer and Hausdorff distance (HD) can be found in Table I demonstrating that Matérn 3/2 performs on par with NS while being significantly faster to compute (compared what measured for NS), see Table 2 Moreover, Matém 3/2 outperforms all evaluated state-of-the-art kernel-free methods (FFN, SIREN, SAP, DiGS, VisCo, and OG-INRI Figure T (left) and Appendix G provide qualitative results and per-object metrics.
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4.2 TEXTURE RECONSTRUCTION ON GSO AND OBJAVERSE
Lastly, we demonstrate Matérn kernels' ability to rep- resent other high-frequency scene attributes, such as texture. To do so, we randomly sample m surface points with corresponding normals and per-point RGB color values from textured meshes, yielding an ex- tended dataset D' D x C2.,c....., Cm C Ω R² x R for each shape. Then, instead of model- ing SDF as in Eq. [T. we seeking func- tion I' Rdd + I21 that, along with signed dis- tances, also predicts per-point RGB values. Please find more information in Appendix H about how we adapt the KRR problem in Eq. Marching Cubes and trilinearily interpolate RGB values at previously predicted surface points. We present quantitative results in terms of PSNR and LPIPS GSO in Table 3 (metrics evaluated on the texture atlas; see Appendix H for further details). Matérn 3/2 surpasses the arc-cosine kernel in the densely sampled setting using 10,000 surface points, well as in the sparse setting. Notably, we did not tune Matérn kernels' bandwidth parameter for this experiment.
m
Kernel
PSNR t
LPIPS 1
Matérn 3/2
19.61
2.05
10,000
Arc-cosine
19.34
2.07
Matérn 3/2
18.92
2.15
2,500
Arc-cosine
18.88
2.09
Table 3:
Texture reconstruction GSO. Matérn kernels outperform the arc- cosine kernel in dense and sparse settings.
estimate f' Finally, we extract the object's surface using
We also show qualitative results the challenging Objaverse (Deitke et al. 2023) dataset in Figure 7 We chose Objaverse as it includes ex- tremely high-resolution and complex textures. Matérn 3/2 reconstructs high- frequency texture details with great pre- cision, overall yielding visually more pleasant reconstructions than the arc- cosine kemel. Reconstructed textures have fewer artifacts and are generally sharper, demonstrating that Matérn ker- nels can overcome spectral bias much better than the arc-cosine kernel. Note that this perfectly confirms our theoret- ical analysis presented in Section 3.2
Arc-cosine
Matérn 3/2
Ground truth
Figure 7: Texture reconstruction Objaverse. Matérn kernels lead to fewer artifacts and sharper details than the arc-cosine kernel.
4.3 DATA-DEPENDENT MATÉRN KERNELS
We leverage the Neural Kernel Field (NKF) framework introduced by Williams al. (2022) to make Matérn kernels learnable. Specifically, NKF proposed feed points z, I through an input- conditioned neural network 4p Rdd - R before evaluating kernel function, k (z, y; D) = I ((;; D)J, [y, (p(;; D))). We re-implemented the NKF framework since the authors did not provide code. We set 9 = 32 for our experiments. Please see Williams et al. (2022) for details.
experinttn
Sparse surface reconstruction and extreme generalization. We compare learned Matérn kemelss against the original NKF framework (with the arc-cosine kernel) and NKSR (Huang et al. 2023) sample m = 1, 1,000 surface points and corresponding
in sparse setting ShapeNet. Again, normals for each shape and set h = 1 for all Matérn kernels. Additionally, we evaluate Matérn kernels' out-of-category generalization ability in an extreme setting, in which we train only on chairs and evaluate on the other 12 ShapeNet categories. Table 4 reports the results, quantified using intersection-over-umii
(IoU), Chamfer distance, and normal consistency (NC). Learned Matérn 1/2 and 3/2 outperform the arc-cosine kernel while being significantly faster to train. Training NKF on the entire ShapeNet dataset (consisting of approx. 30,000 shapes) takes about six days-allmos twice long for Matérn kernels, which require about three days (measured on single NVIDIA A100). Additionally, although Matérn 1/2 is not able to surpass NKSR, comes very close (95.3 95.6 in IoU) while having shorter training time. Regarding out-of-categon, generalization, observe that Matérn 1/2 performs best, followed by Matérn 3/2, NKE, and NKSR. Please see Figure 6 (right) and Appendix foo qualitative results.

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Published as a conference paper at ICLR 2025
IoU1 f
CD |
NC1
Train only on chairs, test on all
Matérn 1/2
95.3
2.58
95.6
IoU 1
CD 4
NC t
Time/epoch
Matérn 3/2
94.9
2.70
95.3
Matérn 1/2
93.4
3,06
94.3
7.71 min
Matérn 5/2
93.3
3.07
94.9
Matérn 3/2
92.8
3.34
94.2
8.26 min
Matérn oc
92.1
3.39
94.2
Matérn 5/2
90.5
4.11
93.9
8.56 min
NKF*
94.7
2.70
95.2
Matérn oo
84.7
6.70
92.6
7.74 min
NKF 94.7
2.65
95,4
NKF*
92.8
3.30
94.1
31.98 min
NKSR
95.6
2.34
95.4
NKSR
89.6
2.70
94.1
41.66 min
Table 4: Results on ShapeNet for learned kernels. "NKF*" denotes a re-implemented variant of NKF. NKF* and Matérn v E {1/2, 3/2, 20) are based on the same framework and parameters; they differ only in the employed kernel. Runtime is measured on a single NVIDIA A100 with batch size of one to ensure fair comparison. Bold marks best result, underline second best.
noise
I|
Small noise 0.0025)
Big 0.005)
IoU f
CD L
II
IoU 1
CD 1
IoU1
CD.
0.5
0.5
2
0.5
1
0.5
1
0.5
2
0.5
Mutérn 1/2
93.3
93.6
93.5
2,88
2.85
2.87
91.9
92.1
92.1
3.03
3.04
3.07
89.5
89.6
89.6
3.37
3.36
3.39
Matém 3/2
92.7
92.7
92.3
3.08
3.20
3.25
89.7
89.8
89.7
3.48
3.69
3.54
85.8
85.5
86.3
4.23
4.52
4.15
Matér 5/2
91.8
90.5
3.58
87.6
4.48
84.3
5.03
5.11
Matém
86.4
85.8
86.8
5,80
5.07
4.75
81.3
83.3
81.0
7.27
5.62
6.45
76.7
80.7
79.3
8.27
5.90
6.67
NKF
93.2
2.97
1|
92.0
3.12
89.5
3.39
NKSR
91.1
2.65
II
90.1
2.98
88.4
3.41
Table 5: Robustness against noise. We compare Matérn kernels' robustness against noise on subset of the ShapeNet dataset using different noise levels, E {0, 0.0025, 0.005] and bandwidths, E {0.5, 1, 2). Bold marks best result, underline second best.
Robustness against noise. We evaluate Matérn kemel''s robustness against different noise levels. a E {0, 0.0025, 0.005). on a subset of the ShapeNet dataset which includes approximately 1,700 shapes. To construct the dataset, we downsampled each ShapeNet category to include only 5% of the shapes. Table 5 presents the results. NKSR, being specifically optimized to deal with noisy inputs, degrades the least with increasing noise level (3% in IoU from no to big noise versus 4.3% for Matérn 1/2). Moreover, Matérn 1/2 is slightly more robust against noise than NKF. Generally, varying the bandwidth, h, helps increase the robustness against noise.
Convergence speed. Finally, we analyze Matérn kernels' conver- gence speed in comparison to the arc-cosine kernel as implemented in NKF, see inset figure on the right. As exemplified, Matérn 1/2 already converges after just 100 epochs, while the arc-cosine kernel requires twice as many epochs, also never being able to reach the same top accuracy. In general, we see that the smoother the kernel, the slower the convergence- an observation that perfectly matches our theoretical analysis in Section 3.2 Matérn 1/2 converges the fastest, followed by Matérn 3/2, 5/2, and the Gaussian kernel.
30
66
80
20
100
200
390
400
opodte
5 CONCLUSION
In this work, we have proposed to use the family of Matérn kernels for implicit surface reconstruction and showed that it consistently outperforms the recently employed first-order arc-cosine kernel- both, in a non-learnable as well as learnable regime-while being significantly easier to implement, faster to compute and train, and highly scalable. We demonstrated that Matérn kernels lead to tunable surface reconstruction, and, based on a new bound of the L2 reconstruction error, derive practical insights into how choose their tunable bandwidth parameter. Moreover, presented an in-depth theoretical analysis, analyzing Matérn kernels' relation to widely used Fourier feature mappings, SIREN networks, and arc-cosine kernels.
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ACKNOWLEDGMENTS
We thank Francis Williams for valuable discussions and his support in re-implementing the NKF framework and Lukas Meyer for his help during NKSR evaluation. This work was funded by the Genman Federal Ministry of Education and Research (BMBF), FKZ: 011S22082 (IRRW). The au- thors are responsible for the content of this publication. The authors gratefully acknowledge the sci- entific support and HPC resources provided by the Erlangen National High Performance Computing Center (NHR @ FAU) of the Friedrich- Alexander-Universital Frlangen-Nürnherg (FAU) under the NHR project b1 IRRW. NHR funding is provided by federal and Bavarian state authoritics. NHR FAU hardware is partially funded by the German Research Foundation (DFG) 440719683
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A ON THE STATIONARITY OF MATÉRN KERNELS
As noted in Section 3.1 of the main paper, Matérn kernels are stationary: they only depend on the difference
between two points æ: y E S2 hence heing translation invariant. This is because k(x - y) = k( +t (y + t)) for E Rd and all stationary kernels, k. In contrast, the arc-cosine kernel
translation invariant as its value depends on the absolute positions, I, y, see Eq. (3) of the main paper (i.e., the arc-cosine kernel can not be written as function of æ y). In fact, the arc-cosine kernel non-stationary. In the following and with the help of Figure [8] we will further discuss why stationary kernels, such as Matérn kemells, might be better choice for kernel-bass implicit surface reconstruction than non-stationary kernels, such as arc-cosine kernel.
First, stationary kernels are naturally translation invariant. Consequently, they are independent of the absolute embedding of the input points points-trannsatiin translating them does not change the reconstructed surface as the kernel's value only depends on the relative distances between points. This is highly beneficial from a practical perspective since no additional steps (such as centering the point cloud) have to be taken before surface reconstruction. Being non-stationary, the arc-cosine kernel behaves differently. we would just use the kernel as in Eq. (3) the main paper, the reconstructed surface would vary depending the actual location of the input points in space. This is a highly undesirable property when it comes to surface reconstruction. To make the arc-cosine kernel translation invariant, input points must be centered before reconstruction (thus eliminating the effects of translation). A centered arc-cosine kernel, denoted as kAC, reads:
|| Ellly øll
((r T ||
(a)
(sin 0 (sT (0) cos where f cos"
π
Here, E Id denotes the centroid of the input point cloud. AcC is obviously translation invariant because | + (e+ t)llly + (c+ t)ll || alllyy- a and (x + (c+ (C++)))++++ tt (c+ t)) (x 2) T (y- a) for all & Rd. This follows because translating the input points also shifts its centroid. In conclusion, while stationary Matérn kernels can be directly used for kernel-based surface reconstruction, additional pre-processing steps are necessary for the arc-cosine kemel.
Second, stationary kernels lead locally consistent surface reconstructions. Since a stationary kernel's 's value only depends the relative positions, objects in scene (or parts of objects) with similar geometric properties will be reconstructed consistently as long as the relative distances be- tween points each instance (or part) remain the same. This is crucial and very desirable property for surface reconstruction which not shared by the arc-cosine kernel (because is non- non-stationary).
Reconssuuced scene
Input points
Arc-cosine
Figure 8: Matérn kernels are translation invariant and lead to locally consistent surface re- constructions. Here, we show that two identical chairs in scene are reconstructed differently depending on the absolute position of the input points when using the non-stationary arc-cosine ker- nel (especially noticeable the chair's legs). In contrast, stationary Matém kernels lead locally consistent surface reconstructions, being independent of the actual embedding (i.e., both chairs have the exact same reconstructed surface).
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B PROOF OF PROPOSITION 2
Proposition 2. The RKHS norm of Matérn kernels as defined in Eq. (77) can be bounded by
I1 I 1220
$12"0 (Ro44 (FU(F + C2 (FL)))
where h >> O, and Cj,p and C2 are functions of F(J) that do not depend on h.
Proof. Based simple bound
u+d/2
2v (2T1|=|1)²
1
=
+
(F()(W) ddw
h
Rd
2v
24d/2
(FI)](W) + [se (22T(W))2e) u+d/2
(w))
h2
R
2v
x+dj2
[ ((2T((/()))
v+d/2
Fl
=
h2
Rd
Rd
=CC(F
x+//2
2y
((FI)))CLLLFT
122
we get
|| Illli
2120
(12x)
-1 ((222)
v+d/2
S 1020
(2x)) Ca.
C(FUI)+Ca
112
2
1,204d (20)
(2T) (2T)
C'FFI))+ ((2))(///))))) (aL(F(L
(FV1
(F15)
22
h22+4 C. Cdp + (F,(()
which concludes the proof.
C DERIVATION OF htt IN EQ.
12
Taking derivative of the bound presented in Proposition
2 w.r.t. h yields
d
2020 (1ertd (L, (EL
1
1 hdtt
(F 20td (FI())
dh
Since d, O, the leading factor 1 1/h never becomes zero, so we must have
d C. (F(l)) 1/(2v+d)
2F dC), dC),
- n'
2v C2 (F(ff)
which shows what we have stated in Eq. 12 of the main paper.
D PROOF OF THEOREM
The proof of Theorem based on Bochner's theorem (in harmonic analysis) which reads:
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Theorem (Bochner). A continuous function lo R - R with k(0) is positive definite if and only if there exists a finite positive Borel measure on IR' such that
k(T)
e ddp(w).
In essence, Bochner's theorem states that l and are Fourier duals; given a stationary kernel k, one can obtain j (better known as spectral densiry if normalized) by applying the inverse Fourier transform to k. On the other hand, given a spectral density μs, one can obtain the corresponding kernel function by applying the Fourier transform to p. Based on Bochner's theorem,
k(T) k(z v)
J
di (=
Enwn [cos(w] (x --))]
because k is real and for even spectral densities. On the other hand, Rahimi & Recht (2007) showed k(T) k(æ y) = (wwnlccs(w" (x [d)l
(Bons/cos(u" (xr -0)1 +
(x + v) 26)
=0(+)
-uu-nub--W0002/0ossss (x y)) + cos(w _ (x + y) + 26)] = -B=-p.b-2((0..m) cos(w" + b) cos(w T b)],
where (+) follows because b is uniformly distributed in j0, 2n] (hence, the inner expectation w.r.t. b becomes zero).
We will now prove Theorem 3 copied below for convenience:
Theorem Consider a two-layer fully-connected network R2d
TR with sine activation
function, m hidden neurons, and fixed bottom layer weights W = (wi, W2.-... -.. Wm) Tgdx and
biases br..b.... E Th. Let h, >> and Ca,y be defined as Eq. 177 1f wg
randomly initialized from
-(++d/2)
(w) = h 2 Caa
22 122
+ (2T (211)())))
and b, ~ U(0, 2n),, the NTK of f is Matérn kernel with bandwidth h and smoothness as m + cc.
Proof. Let f :dd be two-layer fully-connected network with sine activation function, i.e.,
2
f(x) =
Us sin((n [w) b;)
with bottom-layer weights W (wi, W2 Wm) E ddxm biases b (b1, ......, bm) R" and top-layer weights e = (M1, ....... Um) R". Assume that W and are fixed (that is, we only allow the second layer to be trained), and initialized according to some distribution. Choose
(2
2v h2
-(v+d/2)
W: po(w) h 2
+ (2m((()))2
(13)
for h O and by ((, 2T) for all 21, ...., n).
Then, based on the fact that oz,
V sisin(w x bi)/ Vm, it easy that the NTK of is given by
ANIK KNIK(I, = Σ O., f (x)),,
2 sin (w: x + by) sin(w, y + by)
m
m
cos cos(w:) (.) x by) cos(u, y + b.) 2 cos(w, (x 3) 2b.)
m71
1
1
2 cos( [w. x b.) cos(w, + b.)
2 cos(w;) (w: (z + v) + 26z).
m
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As + co, finally obtain
kNTK(I, (2, y) ((w,b) cocosw x b) cos(w TTy + 6)] - -2 (wc,b) [cos(w" (x y) + 26)
=0(+)
(++)
Here, (+) follows because is uniformly distributed, and (**) since w is distributed according to Pu which is, in fact, the spectral density of ki, see, e.g,, Eq. (10) in Kanagawa et al. (2018).
In fact, Theorem can be generalized in that the NTK of a SIREN is generally any stationary kernel:
Remark 7. The NTK of a SIREN can be associated with any stationary kernel if the bottom-layer weights in Eq. 13 are initialized according the corresponding kernel's spectral density.
This is a direct consequence of Bochner's 's theorem.
E PROOF OF COROLLARY 5
We first re-state the following result, needed to prove the second part of Corollary 5
Theorem (Saitoh & Sawano (016). Theorem 2.17). Let Let kn and lzz be two positive definite kernels. Denote by H1, H2 the RKHSs induced by kı and 222 Then,
H1 C H2 + x2k2 - k1 is positive definite for y >> O.
We now ready prove Corollary which is copied below for convenience:
Corollary 5. When restricted to the hypersphere Sd-1, (1) the RKHS of the Matérn kernel with smoothness v = 1/2 (the Laplace kernel) equal to the RKHS induced by the first-order arc-cosine kernel, implying that (2) the Laplace and arc-cosine kernel are equal up affine transformation.
Proof. (1) Follows from Theorem by noting that the NTK of a fully-connected (L 2)-layer ReLU network is the first-order are-cosine kernel, see, e.g., Cho & Saul (2009); Williams et al. 2021). (2) follows from (I) by noting that H1r C НAC when restricted to S so, based on
Theorem k kAC k1z2 valid kernel for 7 > 0. Re-arranging the equation, we have
= + b, where a = 1/72, bk
This shows that the Laplace and arc-cosine kernel are indeed related by an affine transformation when restricted to gd-1, concluding the proof.
F ADDITIONAL DETAILS FOR SHAPENET EXPERIMENT
In this section, provide implementation details and additional qualitative results on the ShapeNet experiment as presented Section 4.1 of the main paper.
F.1 IMPLEMENTATION DETAILS
Following common practice (see, egg., Williams
(022)), ), use finite differences ap- of the main paper. Specifically, denote X' X+ X where x+ {=1 + cn1, + eng...., + cnm) and X'
proximate the gradient part of the KRR problem in Eq.
{z) en .. eR. for a fixed > 0. The Representer Theorem (Kimeldorf
Wahba 1970 Schölkopf al. 2001) tells us that the solution to Eq. (T) is the form
2m ogk(a,
f(x)
Σ
which is linear coefficients a E
I 2m given by
(K + > where K = (K)i k(b1,2z)) 2m22m
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Am
pointe
Aresoteinc
Matcm 1/2
Mation 3/2
Ground trul
Figure 9: Additional qualitative results on ShapeNet for non-learnable kernels. We compare the arc-cosine kernel against Matérn 1/2 and 3/2 for different values of h.
and
y1 - ( te if E X+, if E for i = 1, 1,2,.... ,m2
Here, 2m denotes the identity matrix. We employ PyTorch's ' built-in Cholesky solver to numerically stable solve for and set 0.005 for all experiments.
F.2 FURTHER QUALITATIVE RESULTS
We show additional qualitative results on ShapeNet in Figure
G FURTHER DETAILS ON THE SURFACE RECONSTRUCTION BENCHMARK
This section details the experimental setting that has been used to Neural Splines (NS; Williams et al. )021)) on SRB in Section 4.1 of the main paper. Moreover, we present per-object metrics and additional qualitative results.
G.1 EXPERIMENTAL SETTING
For NS, we used the official implementation provided here: httts:://itthu: com/ leaass/neural-spl.e
Similar to the original paper, we used 15,000 Nyström samples and did the same parameter sweep over the regularization parameter, À (0. 10, 12 10-11 10-10). For the rest of the parameters (not mentioned in the paper), de- fault values provided in the repository have been used, except for the grid size which we set to 512. We used exactly the same setting for Matérn kernels, except that we did a modest param- eter sweep over the bandwidth parameter, E (0.5,1.2) 2). We utilized the SRB data from here: https://github. com/ Twi 11illliiiimm aamiammm/ai....mmm wtwaaamssdeeeeeeee deep- www.pp-e--omttrtirr.......
Despite our best efforts, we were unable to reproduce the results for the NS kernel reported in the original paper. We run on four different GPUs, including NVIDIA RTX A2000, A40, A100, and V100 (which has been used by the authors of Neural Splines), and tested different configurations of the following parameters (see code); grid_sinn ze, eps, cg-max-iters eg-stop-three outer-layee wayee-vvriinne
In Table [ of the main paper, we report the best result we could achieve V100, which is 0.19 for Chamfer and 3.19 for the Hausdorff distance; Williams et al.
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Published as a conference paper at ICLR 2025
Anchor
Daratech
DC
Gargoyle
Lord Quas
CD
HD
CD
HD
CD
HD
CD 4
HD 4
CD 4
HD L
Matérn 1/2
0.31
6.97
0.25
6.11
0.18
2.62
0.19
5.14
0.12
1.32
Matérn 3/2
0.25
5.08
0.23
4.90
0.15
1.24
0.16
2.54
0.12
0.90
Matérn 5/2
0.93
28.33
0.87
32.17
0.34
15.98
0.55
22.52
0.21
9.49
Matérn oo
2.88
29.48
5.80
45.94
0.35
35.70
3.84
30.70
3.13
24.55
NS
0.27
5.38
0.23
4.67
0.15
1,41
0.17
3.49
0.12
0.99
NS
0.22
4.65
0.21
4.35
0.14
1.35
0.16
3.20
0.12
0.69
SPSR
0.33
7.62
0.26
6.62
0.17
2.79
0.18
4.60
0.12
1.83
FFN
0.31
4.49
0.34
5.97
0.20
2.87
0.22
5.04
0.35
3,90
SIREN
0.32
8.19
0.21
4.30
0.15
2.18
0.17
4.64
0.17
0.82
SAP
0.34
8.83
0.22
3.09
0.17
3.30
0.18
5.54
0.13
3.49
DIGS
0.28
5.71
0.21
5.02
0.15
2.13
0.16
3.81
0.12
1.10
VisCo
0.21
3.00
0.26
4.06
0.15
2.22
0.17
4.40
0.12
1.06
OG-INR 0.29
7.56
0.23
2.89
0.17
2.68
0.19
5.01
0.13
2.14
Table 6: Per-object quantitative results on SRB. "NS*" denotes the best result we could achieve Bold marks best result, underline
by running the official implementation of NS, see Section G.I second best.
2021 reported 0.17 and 2.85 for Chamfer and Hausdorff distance, respectively. In order to achieve this, we had to lower outer-layee vereer-variseee ance form 10 per default to 10-5
To measure runtime, we again used the same setting as in the original NS paper: 15,000 Nyström samples, and À = 10-11, -11, Due to the absence further information about the experimental setting in the original paper, used default values provided in the repository for the rest of the parameters. Runtime was measured on a single NVIDIA V100, similar to Williams et al. (202I).
G.2 ADDITIONAL RESULTS
Per-object metrics shown in T and more qualitative results in Figure 10
H ADDITIONAL DETAILS FOR TEXTURE RECONSTRUCTION EXPERIMENT
This section provides implementation details and further information about the evaluation protocol used for the texture reconstruction experiment presented in Section 4.2 of the main paper.
H.1 IMPLEMENTATION DETAILS
Recall from Section 4.2 of the main paper that we are given dataset D = D x (c1, .2...., ,mu C order also reconstruct texture, we are seeking a function f' IRd Rd. predicting signed distances as well as RGB values. Using notation introduced Section
Q Rd R33 with per-point RGB colors fci,c2,.... Cm
we compute E 222x4 as
a' (K + \ where y' = =y 222x
and ē = [c with (c), 02, (2.,, Cm) m Then, predict the function f' as
f'x)= ock(,,
where af E R4 denotes the i-th row of a'. Clearly, f'(x) = (())) e g with fi denoting the singed distance x, and f1, f/s, f4 are the component functions that represent RGB colors at a position z.
To extract textured surface mesh from the predicted four din--iimnsional volume, two-stage process. First, we apply Marching Cubes (Lorensen & Cline 1987) to f(, and then trilinearily interpolate RGB colors at the previously predicted surface points.
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NS
Matém 1/2
Matérn 3/2
Figure 10: Additional qualitative results on SRB for non-learnable kernels. We compare Neural
Splines (NS: Williams et (202)) to Matérn 1/2 and 3/2.
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Published as a conference paper at ICLR 2025
H.2 EVALUATION PROTOCOL
Our quantitative evaluation on the Google Scanned Objects (GSO; Downs et al. (2022)) dataset closely follows Mitchel et al. (2024), in which authors compute image-based evaluation metrics the texture atlas to quantity reconstruction ability. Specifically, we first sample m surface points along with corresponding normals and per-point RGB colors from shapes GSO. Next, we solve the KRR problem in Eq. ( as always. Then, instead of predicting RGB colors on the estimated geometry (which generally differs between various kernel functions), we predict (i.e,, trilinearily interpolate) colors on the surface points of the original mesh. This allows us to leverage the existing UV parametrization of the original mesh to generate texture atlas with colors predicted from KRR. Once the texture atlas at hand, we compare
the original texture atlas using PSNR and LPIPS.
We used ten objects from GSO with different complexity. The following assets have been used:
Jansport_schooo..acckoo.... Sttp:/pppaaaauuueeeeeeeeee Streak
Nov.Dess.gns.Bool..tttta......
Predito.12.TRX.FF..N
Reebok....CTSCH..HARR....M 0
rd RuBabbi-BBateer.aakkttBBbee
Schleich Lion Action Figure
Weisshai Grrerrrr...... White.Shark
They can downloaded from:
I ADDITIONAL RESULTS FOR DATA-DEPENDENT KERNELS
We show additional qualitative results for learned kernels on ShapeNet in Figure
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Published as a conference paper at ICLR 2025
NKF
NKSR
Matém 1/2
Matér 3/2
Ground truth
Figure 11: Additional qualitative results ShapeNet for learned kernels. We compare Neural Kernel Fields (NKFs; Willliams et al. (202)) and Neural Kemel Surface Reconstruction (NKSR; Huang et al. )023)) against learnable Matérn 1/2 and 3/2.
22


Reviews:
[{"review_id": "x8yf43KncW", "reviewer": "Reviewer_nQr2", "summary": "This paper enumerates the use of a class of kernels that can be constructed with two interpretable parameters: bandwidth and smoothness. Specific choices of these parameters lead to well-known kernels like Laplace and Gaussian kernels. \nThis paper elaborates both on theory and experimental evaluation. Specifically, one can derive certain expressions for bounds on the eigenvalue decay rate (EDR), and reconstruction error. More interestingly the authors also draw a connection to neural networks and show  that Matern kernels mimic the computation of two-layer SIRENs if the layer width approaches infinity. \n\nOn the practical side, a range of experiments is provided on surface reconstruction on ShapeNet and SRB (SHape Reconstruction Benchmark). The experiments show favorable metrics for certain fixed and learnable Matern Kernel choices. The comparison is drawn to prior methods like NKF, SPSR, and Neural Splines on challenging metrics like performance with noise, and convergence speed.", "questions": "figure on page 6 needs labels and a quick caption", "limitations": "", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- Overall I found the paper compiled quite well and tells an story of how Matern Kernels can be interesting theoretically and practically useful. The exposition focussing on this class of kernels is quite comprehensive.\n- I particularly liked the connection to SIRENS\n- I found the experiments to be well made. Reconstructions and evaluations are indeed favorable in comparison to their competing methods", "weaknesses": "- Despite a well-made submission, I found it hard to appreciate a strong take-home message. A lot of the material on Matern Kernels appears to be well-known from statistics and I don't find it surprising that they can be tuned well in practice by modulating the bandwidth and smoothness parameters. In this sense, I am a bit disappointed about a direct novel result (other than the relation connecting to SIRENS)\n- I did not fully understand and appreciate the excessive focus on bettering the arc-cosine kernel", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "BkM1ljXWcK", "reviewer": "Reviewer_JfMa", "summary": "The paper promotes the use of the Matérn kernel family for implicit surface reconstruction, emphasizing the kernel’s adaptability through two parameters that control its tendency to fit high-frequency details. These claims are substantiated through both theoretical analysis and empirical evidence. The approach is validated on standard surface reconstruction benchmarks.", "questions": "I would appreciate any response to the questions stated above.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "The paper is well-organized and clearly written, with a sufficient amount of background material to make it self-contained.\n\nThe incorporation of Matérn kernels appears to be a valuable contribution, supported by theoretical analysis on their ability to capture high-frequency details effectively.\n\nBoth quantitative and qualitative comparisons are provided, highlighting key properties of the proposed method.", "weaknesses": "Unclear Benefits of Matérn Kernel Properties\n\nSection 3.1 describes Matérn kernel properties, yet it’s unclear how differentiability and stationarity specifically benefit surface reconstruction. More specifically,\n  Differentiability: Should the differentiability of the kernel be related to the (unknown) ground truth function of the surface reconstruction? what are some typical cases/assumptions, for which the kernel differentiability class is appropriate?\nStationarity: How does stationarity improve reconstruction in this context? Some empirical evidence would help clarify its usefulness. Additionally, given the local support property, wouldn’t we expect symmetric parts, such as the chair legs in Figure 6, to yield consistent reconstruction results?\n\nThe Role of Local Support\n\nSince the product of two kernels remains a kernel, could any kernel be multiplied by a locally supported kernel to achieve local support? Why is this a specific property of the Matern kernel? NKSR, for example, multiplies the dot product kernel by a locally supported kernel. Additionally, would using a kernel with a tunable support radius (like NKSR) allow for similar control over high-frequency detail? If so, the added value of using a two-parameter Matérn kernel remains unclear.\n\nComparison Quality\n\nComparison with Arc-Sine Kernels. The chosen baseline is the arc-sine kernel. Why not compare it to the dot-product kernel with local support, as used in NKSR?\n\nComparison with Gaussian Kernel. In Table 1, the Gaussian kernel $(\\nu = \\inf)$ is presented as inferior. However, how does a two-parameter kernel like  $\\nu \\exp(-\\tau^2/h)$  perform? What specific benefits does a Matérn kernel offer over this one? Is there an advantage in tuning Matérn kernel parameters over this form?\n\nUnclear Connection to SIREN\n\nTheorem 3 seems more like a property of SIREN than the Matérn kernel. It appears that the spectral density of any (stationary) kernel could satisfy the conditions in line 329, so the theorem might hold more generally than just for Matérn kernels. Could the relevance of this theorem to the Matérn kernel specifically be clarified?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "xPYCKJizyd", "reviewer": "Reviewer_57Rh", "summary": "This paper proposes using the family of Matérn kernels for implicit surface reconstruction, targeting the 3D reconstruction of oriented point clouds. Leveraging the recent advances in kernel-based reconstruction, the authors demonstrate that Matérn kernels possess theoretical and practical advantages that make them highly suitable for surface reconstruction tasks. They argue that these kernels surpass state-of-the-art methods based on the arc-cosine kernel, offering improved simplicity, computational efficiency, and scalability.\n\nKey contributions include:\n1.Highlighting the Matérn kernels’ stationary properties, which allow tunable spectral properties similar to Fourier feature mappings, addressing spectral bias issues in coordinate-based MLPs.\n2.Providing theoretical insights into the Matérn kernel’s relationship with SIREN networks and its distinctions from arc-cosine kernels previously used in surface reconstruction.\n3.Introducing data-dependent Matérn kernels based on Neural Kernel Fields and showing that the Laplace kernel (a member of the Matérn family) performs competitively with state-of-the-art methods in noise-free settings while significantly reducing training time.", "questions": "(1) SIREN is a work from 2020, and there have been many recent improvements based on SIREN. The authors should consider these in their comparative experiments to showcase the advantages of Matern kernels in implicit reconstruction.\n(2) In the sparse surface reconstruction, in addition to quantitative results, including visualizations of the sampled points would make the effects more intuitive.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "(1) The paper presents a tunable implicit surface reconstruction method based on Matérn kernels. The tunability of the method allows users to adjust parameters based on specific application needs, optimizing reconstruction performance in different scenarios, which is crucial in practical applications.\n(2) Matérn kernels are widely used in statistics and machine learning, and the paper effectively leverages their favorite mathematical properties, making the proposed method more rigorous and interpretable theoretically.\n(3) The paper is well-structured and logically coherent, making it easier for readers to understand the methods and results. The theoretical proofs and experiments in this paper are relatively sufficient.", "weaknesses": "(1) Lacks comparison with the latest implicit surface reconstruction algorithms.\n(2) Some figures are missing ground truth.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "XFAfiGpyK1", "reviewer": "Reviewer_QSTk", "summary": "This work proposes to use Matern kernels -- a unified family of kernel functions -- for 3D reconstruction of shapes from sparsly sampled points. A key property of these kernels is tunability, in that the sharpness of these kernels can be controlled. The authors claim this can help to overcome the known implicit bias in MLPs against the reconstruction of high-frequency features. The contributions of this paper are mostly techincal. The authors provide a theoretical investigation of Matern Kernels, including several toy experiments in support of derivations.", "questions": "In the eigenvalue decay experiments, the authors should state explicity they are considering the eigenvalues of the kernel K(x, y). Assuming I understand this correctly, how many points are used and how are they distributed in this experiment? What shapes (if any) do they represent? \n\nCan h* in equation 13 be computed or estimated from the sampled points so as to give an exact way to recover the optimal h value given $\\nu$?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "Overall, the authors investigation into the suitability of Matern kernels for surface reconstruction in section 4 is the standout. \n\nWhile the experimental details are unclear, the analysis of the eigenvalue decay exposes the mechanics of how the kernels can address spectral bias and supports the authors claims.\n\nThe discussion of the reconstruction error is compelling, and the highlight of the paper. The authors cleverly derive an upper bound on the reconstruction error in terms of the kernel parameters (h, $\\nu$), and expose a canonical threshold (Equation 13).\n\nTo the best of my knowledge, the mathematical formulations are sound, though I did not check them closely.", "weaknesses": "1). Despite the contribution of the paper being technical, the way the math is introduced and discussed is unintuitive and confusing. Section 3 contains significant mathematical \"filler\" e.g. equations (5-8) could be moved to the appendix.  The RKHS norm is introduced early without context. Only a page later, in Equation (12), does its purpose become clear in supplying the upper bound for the reconstruction loss. Even then, it is not clear that it is necessary to reproduce the explicit form of the RHKS norm in the main text (this can be moved to the supplement).  I question the necessity of section 3, as none of the math introduced is the authors' own work and makes it less clear what their contributions are. I think most of it can be removed, and combined with section 4, with the appropriate math introduced in the main text only as it relates to Eqs. 12-13 and the rest left to the supplement. \n\nAdditionally, I'm on the fence about whether 4.2 should be kept in the main paper or moved to the supplement. These insights are nice to know, but don't necessarily have any effect on the implementation, unlike the discussion in 4.1.\n\nThe extra space gained by condensing the above should give the authors more room to both A). make clear exactly what their own contributions are (e.g. the derivations of the upper bound, etc.) and B). provide more room for experiments and discussion.\n\n2). The biggest weakness of the paper are the experimental results. Overall, the authors proposed approach performs only comparably relative to competing methods and does not distinguish itself. This is unfortunate, because the insights in section 4.1 are very nice and had me rooting for the method, but they don't seem to provide a significant increase in performance over existing methods. Overall, the proposed method is not substantially different from prior methods like NKF & NKSR in that it is using just a different choice kernel for surface reconstruction. In general, this is OK, and the authors provide a very nice theoretical investigation into the kernel properties, but in such a case its necessary to show unambiguous, generally uniform improvements over previous similar methods to justify the approach. \n\nAnother important question left unanswered is scalability. Experiments are shown in which simple surfaces are reconstructed from 1000 points. How does the kernel reconstruction perform from sparse samples of 10K or 100K points? I am concerned that the Cholesky factorization is not scalable in these cases, though this does not seem fundamental to the method and the CG could be used. It should be noted that JAX has a very scalable, efficient, and differentiable implementation of the CG solve for sparse systems, and this may be of interest to the authors.\n\nIn a similar vein, no results are shown comparing performance with sampling irregularity. How does the method perform with uniformly distributed samples vs varying degrees of non-uniformly sampled points?\n\nTogether, scalability and robustness under sampling irregularity are two potential axes for comparison against other methods that could allow the proposed method to more obviously stand out. Additionally, what about reconstructing a complex texture on a surface (e.g. from an Objaverse mesh)? A complex texture is itself a high-frequency signal that could help to distinguish the authors approach.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)