Matérn Kernels for Tunable Implicit Surface Reconstruction

## Abstract

Abstract We propose to use the family of Matérn kernels for tunable implicit surface reconstruction, building upon the recent success of kernel methods for 3D reconstruction of oriented point clouds.
As we show, both, 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 scaleable.
Being stationary, we demonstrate that the Matérn kernels’ spectrum can be tuned in the same fashion as Fourier feature mappings help coordinate-based MLPs to overcome spectral bias.
Moreover, we theoretically analyze Matérn kernel’s connection to Siren networks as well as its 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. 1 1 1 Code available at: https://github.com/mweiherer/matern-surface-reconstruction .

## 1 Introduction

Recovering the shape of objects from sparse or only partial observations is a challenging task.
Formally, let $\Omega\subset\mathbb{R}^{d}$ and $\mathcal{X}=\{x_{1},x_{2},\dots,x_{m}\}\subset\Omega$ be a set of $m$ points in $d$ dimensions which forms, together with associated per-point normals, a data set $\mathcal{D}=\mathcal{X}\times\{n_{1},n_{2},\dots,n_{m}\}\subset\Omega\times
\mathbb{R}^{d}$.
The goal of surface reconstruction is to recover the objects’ shape from $\mathcal{D}$.
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 we chose to focus on in this work.
Specifically, if $f:\mathbb{R}^{d}\longrightarrow\mathbb{R}$ denotes an implicit function, such as a signed distance function (SDF), then, from a practical perspective, implicit surface reconstruction aims at finding an optimal solution $\hat{f}$ to the following kernel ridge regression (KRR) problem:

$$ $\hat{f}=\operatorname*{arg\,min}_{f\in H}\left\{\sum_{i=1}^{m}|f(x_{i})|^{2}+ \|\nabla f(x_{i})-n_{i}\|^{2}+\lambda\|f\|_{H}^{2}\right\},$ (1) $$

where the function space $H:=H(\Omega)$ is usually taken to be a reproducing kernel Hilbert space (RKHS) with associated reproducing kernel $k:\Omega^{2}\longrightarrow\mathbb{R}$, and $\lambda>0$.
Once $\hat{f}$ at hand, the reconstructed object’s surface is implicitly given as $\mathcal{\hat{S}}=\{x:\hat{f}(x)=0\}\subset\mathbb{R}^{d}$ and can be extracted using Marching Cubes (Lorensen & Cline, ).
It is quite easy to observe that, as $\lambda\rightarrow 0$, the optimization problem in Eq. (1) turns into the following constrained minimization problem

$$ $\operatorname*{minimize}_{f\in H}\|f\|_{H}\quad\text{s.t.}\quad f(x_{i})=0 \quad\text{and}\quad\nabla f(x_{i})=n_{i}\quad\forall i\in\{1,2,\dots,m\}$ (2) $$

which makes it easier to see how $\hat{f}$ (thus the estimated surface $\mathcal{\hat{S}}$), behaves away from the input points, $\mathcal{X}$.
Clearly, while $\mathcal{\hat{S}}$ should interpolate the given points $\mathcal{X}$ exactly, the behavior in between and away from those points is solely determined by the induced norm $\|f\|_{H}$ of the function space, $H$, over which we are optimizing.
The properties of $\|f\|_{H}$ in $H$ are uniquely determined by the reproducing kernel, $k$; this is a direct consequence of the Representer Theorem (Kimeldorf & Wahba, ; Schölkopf et al., ) which states that solutions to Eq. (1) are of the form $f(x)=\sum_{i=1}^{m}\alpha_{i}k(x,x_{i})$, yielding $\|f\|_{H}^{2}=\alpha^{\top}K\alpha$.
Here, $\alpha\in\mathbb{R}^{m}$ and $K=(K)_{ij}=k(x_{i},x_{j})\in\mathbb{R}^{m\times m}$.
This shows that the behavior of the norm in Eq. (2) and hence, of estimated surfaces $\mathcal{\hat{S}}$, can be explicitly controlled by choosing different kernel functions, effectively allowing us to inject certain inductive biases (such as smoothness assumptions) into the surface reconstruction problem.

Figure: ((a))
Refer to caption: x1.png

Even further, if the chosen kernel has adjustable parameters, they can be used to adaptively (on a shape-by-shape basis) manipulate the inductive bias.

Although already used in the mid-90s and early 2000s (Savchenko et al., ; Carr et al., ; ), only recently, kernel-based methods for 3D implicit surface reconstruction became extremely competitive, with Neural Kernel Surface Reconstruction (NKSR; Huang et al. ()) eventually evolving as the new state-of-the-art.
While early works mostly focused on (polyharmonic) radial basis functions such as thin-plate splines, recent work (Williams et al., ; ) employ the first-order arc-cosine kernel.
Introduced by Cho & Saul (), 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. ()) of the network.

Figure: Figure 2: Tunable Matérn kernels lead to better surface reconstructions compared to the previously employed arc-cosine kernel. Here, we show surface reconstructions from sparse point clouds of just 1,000 points.
Refer to caption: x3.png

One important aspect of this kernel is that it does not have adjustable parameters.
While this can be advantageous in some situations, it also prevents users from tuning their reconstructions.
Clearly, if the arc-cosine kernel fails to accurately recover a surface, then there is no possibility (except for using more observations) to improve the result.

In this work, we propose to use a different family of kernels (with parameters $\nu,h>0$) for implicit surface reconstruction: Matérn kernels (Matérn, ; Stein, ), see Figures 1 and 2.
Contrary to arc-cosine kernels, Matérn kernels are stationary (hence rotation- and translation-invariant), locally-supported (thus leading to sparse Gram matrices), and unify a variety of popular kernel functions, including the Laplace (for $\nu=1/2$) and Gaussian kernel (as $\nu\rightarrow\infty$).
As we will show, both, from a theoretical and practical perspective, Matérn kernels have some intriguing properties which make them particularly well suited for surface reconstruction.
Most notably, we argue that Matérn kernels allow for tunable surface reconstruction in the same way as Fourier feature mappings (Tancik et al., ) enable coordinate-based Multi-layer Perceptrons (MLPs) to learn high-frequency functions in low-dimensional domains, effectively overcoming spectral bias (the slow convergence towards high-frequency details).
In addition, our theoretical analysis reveals that, as the layer-width approaches infinity, Matérn kernels are identified with the NTK of two-layer Siren (fully-connected MLPs with sine activation function; Sitzmann et al. ()) networks—together with Fourier feature mappings the two arguably most influential methods to overcome spectral bias in coordinate-based MLPs.
Lastly, we establish a connection between arc-cosine and Matérn kernels by showing that the Laplace kernel (i.e., the Matérn kernel with $\nu=1/2$) and the arc-cosine kernel are equal up to an affine transformation when restricted to the hypersphere $\mathbb{S}^{d-1}$.

Our experimental evaluation demonstrates that Matérn 1/2 and 3/2 are extremely competitive, outperforming the arc-cosine kernel while being significantly easier to implement (essentially two lines of standard PyTorch code), faster to compute, and scalable.
Moreover, by leveraging Neural Kernel Fields (Williams et al., ), we show that data-dependent (a.k.a. learnable) Matérn kernels perform almost on par with highly sophisticated and well-engineered NKSR in the noise-free case while being more than five times faster to train.

## 2 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. (); Huang et al. ().

Early approaches for kernel-based 3D surface reconstruction mostly employ radial basis functions (RBFs) such as thin-plate splines (Savchenko et al., ), biharmonic RBFs (Carr et al., ; ), or Gaussian kernels (Schölkopf et al., ).
Nowadays, the probably most widely used surface reconstruction technique is Screened Poisson Surface Reconstruction (SPSR; (Kazhdan & Hoppe, )) which, however, can itself be viewed as kernel method as shown in Williams et al. ().
Only recently, Neural Splines (NS; Williams et al. ()) proposed to use the so-called (first-order) arc-cosine kernel,

$$ $k_{\text{AC}}(x,y)=\frac{\|x\|\|y\|}{\pi}\left(\sin\theta+(\pi-\theta)\cos \theta\right),\quad\text{where}\quad\theta=\cos^{-1}\left(\frac{x^{\top}y}{\|x \|\|y\|}\right)$ (3) $$

for implicit surface reconstruction, achieving state-of-the-art results that outperform classical surface reconstruction techniques as well as non-linear methods based on neural networks by a large margin.
This method laid the cornerstone for the Neural Kernel Field (NKF; Williams et al. ()) framework which attempts to make arc-cosine kernels learnable by passing input points through a task-specific neural network before evaluating the kernel function, similar to Deep Kernel Learning (Wilson et al., ).
Based on SPSR, NeuralGalerkin (Huang et al., ) proposed to learn a set of spatially-varying basis functions inferred by a sparse convolutional network instead of using a fixed Bézier basis as in SPSR, hence can be seen as kernel method in the broader sense.
Neural Kernel Surface Reconstruction (NKSR; Huang et al. ()) built upon NKF and NeuralGalerkin and proposed an all-purpose surface reconstruction method that is highly scaleable and robust against noise, eventually becoming state-of-the-art.

## 3 Background: Matérn Kernels

We propose to use the family of Matérn kernels (Matérn, ; Stein, ) instead of the arc-cosine kernel for implicit surface reconstruction, being defined as:

$$ $k_{\nu}(x,y)=\Phi_{\nu}(\|x-y\|)=\Phi_{\nu}(\tau)=\frac{2^{1-\nu}}{\Gamma(\nu) }\left(\frac{\sqrt{2\nu}\tau}{h}\right)^{\nu}K_{\nu}\left(\frac{\sqrt{2\nu} \tau}{h}\right),$ (4) $$

where $\nu>0$ is a smoothness parameter that explicitly controls the differentiability, and $h>0$ is known as the shape parameter (or bandwidth) of the kernel.
$\Gamma$ denotes the Gamma function, and $K_{\nu}$ is the modified Bessel function of the second kind of order $\nu$.
For $\nu=p+\frac{1}{2}$, $p\in\mathbb{N}^{+}$ (i.e., if $\nu$ is a half-integer), Eq. (4) simplifies to a product of an exponential and polynomial of degree $p$, that is

$$ $\Phi_{p+\frac{1}{2}}(\tau)=\exp\left(-\frac{\sqrt{2p+1}\tau}{h}\right)\frac{p! }{(2p)!}\sum_{i=0}^{p}\frac{(p+i)!}{i!(p-i)!}\left(\frac{2\sqrt{2p+1}\tau}{h} \right)^{p-i}.$ (5) $$

Matérn kernels generalize a variety of other kernels; using Eq. (5), the most popular ones can be written in closed-form as:

$$ $\displaystyle p$ $\displaystyle=0:\Phi_{\text{1/2}}(\tau)=\exp\left(-\frac{\tau}{h}\right),$ (6) $\displaystyle p$ $\displaystyle=1:\Phi_{\text{3/2}}(\tau)=\exp\left(-\frac{\sqrt{3}\tau}{h} \right)\left(1+\frac{\sqrt{3}\tau}{h}\right),$ (7) $\displaystyle p$ $\displaystyle=2:\Phi_{\text{5/2}}(\tau)=\exp\left(-\frac{\sqrt{5}\tau}{h} \right)\left(1+\frac{\sqrt{5}\tau}{h}+\frac{5\tau^{2}}{3h^{2}}\right),$ (8) $$

where $\Phi_{\text{1/2}}$ is known as the Laplace kernel.
In the limiting case of $\nu\rightarrow\infty$, Matérn kernels recover the popular Gaussian kernel:

$$ $\Phi_{\infty}(\tau):=\lim_{\nu\rightarrow\infty}\Phi_{\nu}(\tau)=\exp\left(- \frac{\tau^{2}}{2h^{2}}\right).$ (9) $$

The spectral density of Matérn kernels is given as the inverse Fourier transform of $\Phi_{\nu}$, that is

$$ $p_{\nu}(\omega)=\mathcal{F}^{-1}[\Phi_{\nu}]=h^{-2\nu}C_{d,\nu}\left(\frac{2 \nu}{h^{2}}+(2\pi\|\omega\|)^{2}\right)^{-(\nu+d/2)}$ (10) $$

which is due to the fact that Matérn kernels are stationary (or shift-invariant—they only depend on the difference $x-y$ between two points, $x,y\in\Omega$); hence, kernel and associated spectral density are Fourier duals (a consequence of Bochner’s Theorem).
Moreover, $C_{d,\nu}:=(2^{d}\pi^{d/2}\Gamma(\nu+d/2)(2\nu)^{\nu})/\Gamma(\nu)$, and $\mathcal{F}[f]$ denotes the Fourier transform of a function $f$.

The RKHS induced by Matérn kernels is quite well understood (see, e.g., (Kanagawa et al., )) and can be formulated as $H_{\nu}=\{f\in L_{2}(\mathbb{R}^{d})\cap C(\mathbb{R}^{d}):\|f\|_{H_{\nu}}^{2}
<\infty\}$, where the RKHS norm is

$$ $\|f\|_{H_{\nu}}^{2}=h^{2\nu}\left((2\pi)^{d/2}C_{d,\nu}\right)^{-1}\int_{ \mathbb{R}^{d}}\left(\frac{2\nu}{h^{2}}+\left(2\pi\|\omega\|\right)^{2}\right) ^{\nu+d/2}|\mathcal{F}[f](\omega)|^{2}\mathrm{d}\omega.$ (11) $$

For more information about Matérn kernels, we refer the interested reader to Porcu et al. () for a recent review.
Next, we revisit some basic properties of Matérn kernels.

### 3.1 Basic Properties

Differentiable.
Matérn kernels allow for controlled smoothness, being exactly $\lceil\nu\rceil-1$ times differentiable (in the mean-square sense); for instance, while the Matérn 1/2 kernel is zero times differentiable, the Gaussian kernel (for $\nu\rightarrow\infty$) is infinitely often differentiable.

Stationary.
Matérn kernels are stationary, i.e., they only depend on the difference $x-y$ between two points $x,y\in\Omega$.
If the distance is Euclidean, Matérn kernels are also isotropic (or homogeneous); they only depend on $\|x-y\|$.
As a result, Matérn kernels are rotation and translation invariant, hence being independent of the absolute embedding of $\mathcal{X}$.
We note that, while the previously used arc-cosine kernel is also rotation invariant, it is not translation invariant.

Locally supported.
As opposed to arc-cosine kernels, Matérn kernels are locally supported (or compact); that is, they tend to zero as the distance $\|x-y\|$ becomes large.
This is a very desirable property, especially from a computational perspective, as it leads to sparse Gram matrices when multiplied with the kernel $\Phi^{\prime}_{\nu^{\prime}}(\tau)=\max\left\{0,\left(1-\tau/h^{\prime}\right)
^{\nu^{\prime}}\right\}$ for suitable $\nu^{\prime}$ and $h^{\prime}$, effectively allowing the use of highly efficient and scalable sparse linear solvers.
Please see Genton () on how to choose suitable values for $\nu^{\prime}$ and the cut-off distance, $h^{\prime}$.

Figure: ((a))
Refer to caption: x4.png

## 4 Matérn Kernels for Tunable Surface Reconstruction

We now derive new theoretical insights into Matérn kernels, ultimately aiming to provide arguments as to why we believe this family of kernels is particularly well suited for surface reconstruction.

### 4.1 Eigenvalue Decay, Spectral Bias, and Reconstruction Error

Eigenvalue decay and spectral bias.
We begin by analyzing Matérn kernel’s eigenvalue decay rate (EDR) and show that it can be tuned by varying the smoothness parameter $\nu$ and/or bandwidth $h$.
As investigated in Tancik et al. () for coordinate-based MLPs, a rapid decrease of the associated NTK’s eigenvalues implies slow convergence to high-frequency components of the target function (up to the point where the network is practically unable to learn these components).
Consequently, a smaller EDR, i.e., a slower eigenvalue decay, leads to faster convergence to high-frequency content; hence, in the context of implicit surface reconstruction, more detailed geometry.
To overcome this slow convergence, a phenomenon known as spectral bias, Tancik et al. () propose to use a (random) Fourier feature mapping of the form $\gamma(x)=[\cos(2\pi Bx),\sin(2\pi Bx)]$ applied to the inputs $x\in\mathbb{R}^{d}$ before passing them to the MLP, effectively transforming the MLP’s NTK, $k_{\text{NTK}}$, into a stationary kernel with tunable bandwidth parameter, $k_{\text{NTK}}(\gamma(x)^{\top}\gamma(y))=k_{\text{NTK}}(k(x-y))=:k^{\prime}_{
\text{NTK}}(x-y)$.
When $B\in\mathbb{R}^{q\times d}$ is drawn from a standard normal distribution (i.e., zero mean and unit variance), $k$ is an approximation of the Gaussian kernel (Rahimi & Recht, ).
This allows explicit manipulation of the kernels’ spectrum, overcoming spectral bias.

Matérn kernels, being stationary, allow for the same degree of control over the EDR as a Fourier feature mapping; their spectrum can also be tuned (by varying $\nu$ and/or $h$, see Figure 1).
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 $\nu$ and $h$, we use:

###### Theorem 1 ( Seeger et al. ( 2008 ) , Theorem 3) .

The eigenvalues of Matérn kernels decay polynomially at rate

$$ $\Theta\left(h^{-2\nu}s^{-(1+2\nu/d)}\right)$ $$

with bandwidth parameter $h>0$, and for finite smoothness $0<\nu<\infty$.

Figure 3 visualizes the EDR for different values of $\nu$ and $h$ along with the EDR of the non-tunable, first-order arc-cosine kernel which equates to $\Theta(s^{-(1+d)/d})$ and has only been very recently proven by Li et al. () for general input domains and distributions.
As seen, the smoother the kernel (i.e., the higher $\nu$) and the higher $h$, the faster the eigenvalues decay.
This implies less detailed surface reconstruction (with a higher error) for smoother kernels and greater $h$; indeed, that is exactly what we observe in practice, see Figure 4.
Contrary, by lowering both parameters, one can achieve faster convergence to high-frequency geometric details, effectively leading to more nuanced surface reconstruction and a smaller reconstruction error.
Interestingly, the first-order arc-cosine kernel and Matérn 1/2 have the same EDR if $h=1$.

Figure: Figure 4: Matérn kernels are tunable. Surface reconstructions can be improved in practice by varying the bandwidth parameter, $h$, effectively tuning the kernels’ EDR (see also Figure 3). However, setting $h$ too small ($<0.5$) results in over-fitting, while setting $h$ too big ($>10$) over-smooths (i.e., under-fits) the true surface. This is also reflected in the reconstruction error (measured using Chamfer distance), plotted as a function of $h$ on the right.
Refer to caption: x6.png

Reconstruction error.
We proceed by investigating the bandwidth’s influence on the reconstruction error, measured by the mean-squared error which is defined (and can be bounded) as:

$$ $\displaystyle\|f-\hat{f}\|_{L_{2}}=\left(\int_{\Omega}\left(f(x)-\hat{f}(x) \right)^{2}\mathrm{d}x\right)^{1/2}\leq C_{\mathcal{X},\Omega}^{(\nu+d)/2}\|f \|_{H_{\nu}},$ (12) $$

where $C_{\mathcal{X},\Omega}$ is a constant that only depends on $\mathcal{X}$ and $\Omega$ and for $\nu<\infty$.
For more information on the bound, including details on $C_{\mathcal{X},\Omega}$, please see, e.g., Santin & Schaback ().
By inspecting Eq. (12), we notice that the magnitude of the RKHS norm has a considerably high influence on the reconstruction error.
Based on this observation and the fact that $\|f\|_{H_{\nu}}$ depends on $h$, our goal is now to further study the effect of $h$ on the RKHS norm.
As a first result, we have:

###### Proposition 2 .

The RKHS norm of Matérn kernels as defined in Eq. (11) can be bounded by

$$ $\|f\|^{2}_{H_{\nu}}\leq h^{2\nu}\left(\frac{1}{h^{2\nu+d}}C_{d,\nu}^{1}( \mathcal{F}[f])+C^{2}_{d,\nu}(\mathcal{F}[f])\right),$ $$

where $h>0$, and $C^{1}_{d,\nu}$ and $C^{2}_{d,\nu}$ are functions of $\mathcal{F}[f]$ that do not depend on $h$.

A proof can be found in Appendix A.
We immediately see that $\|f\|^{2}_{H_{\nu}}\rightarrow\infty$ in either cases, $h\rightarrow 0$ and $h\rightarrow\infty$.
Moreover, based on Proposition 2, it is easy to show that the norm, as a function of $h$, decreases as $h\rightarrow h^{*}$ from the left, and increases as $h\rightarrow\infty$, where $h^{*}$ is given as

$$ $h^{*}=\left(\frac{d}{2\nu}Q\right)^{1/(2\nu+d)}\quad\text{with}\quad Q=\frac{C ^{1}_{d,\nu}(\mathcal{F}[f])}{C^{2}_{d,\nu}(\mathcal{F}[f])}$ (13) $$

(see Appendix B for a derivation).
This can also be seen from the inset figure on the right, where we plot the RKHS norm with respect to the obtained bound as a function of $h$.

[Uncaptioned image]: x7.png

Interestingly, our analysis reveals that the reconstruction error can not be made arbitrarily small by just lowering $h$; if the bandwidth $h$ is chosen too small, the reconstruction error starts to increase again.
Conversely, if $h$ is set too high, the resulting surface is too smooth, ultimately leading to increased errors.
In fact, what we actually observe is that Matérn kernels under-fit the true surface when $h$ is set too big, and, conversely, over-fit if $h$ is chosen too small, see also Figure 4.
As we have shown, the optimal trade-off is reached if $h$ equals $h^{*}$.
Moreover, whilst the bandwidth $h$ does play a critical role, our results show it is not very sensitive in practice, especially for Matérn 1/2 and 3/2 (see inset figure).

###### Theorem 1 ( Seeger et al. ( 2008 ) , Theorem 3) .

###### Proposition 2 .

### 4.2 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 first-order arc-cosine kernels mimic the computation of a two-layer, infinite-width ReLU network $f:\mathbb{R}^{d}\longrightarrow\mathbb{R}$, $f(x)=m^{-1/2}\sum_{i=1}^{m}v_{i}[w^{\top}_{i}x+b_{i}]_{+}$ when the bottom layer weights $W=(w_{1},w_{2},\dots,w_{m})\in\mathbb{R}^{d\times m}$ and biases $b=(b_{1},b_{2},\dots,b_{m})\in\mathbb{R}^{m}$ are fixed from initialization and drawn from a standard normal distribution, see, e.g., Cho & Saul (); Williams et al. ().
Here, $[x]_{+}:=\max\{0,x\}$ denotes the ReLU activation function.

Under similar assumptions, we now show that Matérn kernels mimic the computation of two-layer Sirens Sitzmann et al. () (fully-connected networks with sine activation function) if the layer width approaches infinity.
We claim the following:

###### Theorem 3 .

Consider a two-layer fully-connected network $f:\mathbb{R}^{d}\longrightarrow\mathbb{R}$ with sine activation function, $m$ hidden neurons, and fixed bottom layer weights $W=(w_{1},w_{2},\dots,w_{m})\in\mathbb{R}^{d\times m}$ and biases $b=(b_{1},b_{2},\dots,b_{m})\in\mathbb{R}^{m}$. Let $h,\nu>0$ be given. If $w_{i}$ is randomly initialized from

$$ $p_{\nu}(\omega)=h^{-2\nu}C_{d,\nu}\left(\frac{2\nu}{h^{2}}+(2\pi\|\omega\|)^{2 }\right)^{-(\nu+d/2)}$ $$

and $b_{i}\sim\mathcal{U}(0,2\pi)$, the NTK of $f$ is a Matérn kernel with bandwidth $h$ and smoothness $\nu$ as $m\rightarrow\infty$.

A proof can be found in Appendix C.
This result establishes for the first time a connection between Matérn kernels and the computation in neural networks, in particular, Sirens.

Matérn kernels vs. arc-cosine kernels.
Lastly, we aim to compare Matérn kernels with the previously used first-order arc-cosine kernels.
Our analysis is based on:

###### Theorem 4 ( Chen & Xu ( 2021 ) , Theorem 1) .

When restricted to the hypersphere $\mathbb{S}^{d-1}$, the RKHS of the Matérn kernel with smoothness $\nu=1/2$ (the Laplace kernel) include the same set of functions as the RKHS induced by the NTK of a fully-connected ($L\geq 2$)-layer ReLU network.

In other words, Theorem 4 shows that the RKHS of the Laplace kernel (a Matérn kernel with smoothness $\nu=1/2$) is the same as the RKHS induced by the NTK of a fully-connected ReLU network when inputs are restricted to $\mathbb{S}^{d-1}$.
This immediately raises the question of whether the NTK for general input domains is significantly different from a standard Laplace kernel.
Indeed, empirical results (Belkin et al., ; Hui et al., ; Geifman et al., ) already suggest a negative answer; however, from a theoretical perspective, the claim is still unproven.
Based on Theorem 4, we obtain the following connection between Laplace and the first-order arc-cosine kernel:

###### Corollary 5 .

When restricted to the hypersphere $\mathbb{S}^{d-1}$, (1) the RKHS of the Matérn kernel with smoothness $\nu=1/2$ (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 D for a proof.
For general input domains, our empirical results presented next support the claim mentioned above (at least for ($L=2$)-layer networks), showing that the Laplace kernel even outperforms the arc-cosine kernel.

###### Theorem 3 .

###### Theorem 4 ( Chen & Xu ( 2021 ) , Theorem 1) .

###### Corollary 5 .

## 5 Experiments and Results

We systematically evaluated the effectiveness of Matérn kernels in the context of implicit surface reconstruction, presenting results on ShapeNet (Chang et al., ) and the Surface Reconstruction Benchmark (SRB; Berger et al. ()) in Section 5.1.
For ShapeNet, we use the train/val/test split provided by Mescheder et al. ().
Leveraging Neural Kernel Fields (Williams et al., ), we present and evaluate learnable Matérn kernels in Section 5.2.

### 5.1 Surface Reconstruction on ShapeNet and SRB

ShapeNet. We compare Matérn kernels in a sparse setting against the arc-cosine kernel as implemented in Neural Splines (NS, Williams et al. ()).
To do so, we randomly sample $m=1,000$ 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 a unified framework to ensure fair comparison.
Notably, we did not use Nyström sampling as in NS, and employ PyTorch’s direct Cholesky solver to solve the KRR problem in Eq. (1) instead of an iterative conjugate gradient solver.
Moreover, similar to NS, we approximate the gradient part in Eq. (1) with finite differences.
See Appendix E for details.

Table: Table 1: Results on ShapeNet (left) and SRB (right) for non-learnable kernels. Arc-cosine kernel and Matérn $\nu\in\{\text{1/2},\text{3/2},\text{5/2},\infty\}$ are implemented in a unified framework and based on the exact same parameters. ”NS*” denotes the best result we could achieve by running the official implementation of NS with default parameters, see Appendix F. Runtime is measured on a single NVIDIA V100. Bold marks best result, underline second best.

Quantitative results in terms of F-Score (with a threshold of 0.01) and Chamfer distance (CD; always reported $\times$ 10^3) can be found in Table 1.
Matérn 1/2 and 3/2 consistently outperform the arc-cosine kernel as well as popular SPSR (Kazhdan & Hoppe, ).
In addition, our experiments show that the bandwidth parameter, $h$, can be used to tune the surface reconstruction (lower the error), while not being too sensitive in practice.
Matérn 5/2 and the Gaussian kernel (for $\nu\rightarrow\infty$) perform significantly worse, being simply too smooth.
Figure 2 and Appendix E shows qualitative results.

SRB. Additionally, we evaluate Matérn kernels on the challenging Surface Reconstruction Benchmark (Berger et al., ) which consists of five complex shapes simulated from noisy range scans.
For this, we integrated Matérn kernels into the NS framework, using KeOps (Charlier et al., ) and the FALKON (Rudi et al., ) solver.
We used the same parameter settings as in the original NS paper.
In addition, for Matérn kernels, we did a modest parameter sweep over the bandwidth, $h\in\{0.5,1,2\}$, and took the reconstruction with the lowest Chamfer distance.

Figure: Figure 5: Qualitative results for non-learnable kernels on SRB (left) and learned kernels on ShapeNet (right). Please see Appendix F and G for more qualitative results.
Refer to caption: x8.png

Quantitative results obtained by employing Chamfer and Hausdorff distance (HD) can be found in Table 1, demonstrating that Matérn 3/2 performs on par with NS while being faster to compute (compared to what we measured for NS).
See Figure 5 (left) and Appendix F for qualitative results and per-object metrics.

### 5.2 Data-dependent Matérn Kernels

We leverage the Neural Kernel Field (NKF) framework introduced by Williams et al. () to make Matérn kernels learnable.
Specifically, NKF proposed to feed points $x,y\in\mathbb{R}^{d}$ through an input-conditioned neural network $\varphi:\mathbb{R}^{d}\longrightarrow\mathbb{R}^{q}$ before evaluating a kernel function, $k^{\varphi}(x,y;\mathcal{D})=k\left([x,\varphi(x;\mathcal{D})],[y,\varphi(y;
\mathcal{D})]\right)$.
We re-implemented the NKF framework since the authors did not provide code.
We set $q=32$ for our experiments.
Please refer to (Peng et al., ; Williams et al., ) for more details.

Sparse surface reconstruction and extreme generalization.
We compare learned Matérn kernels against the arc-cosine kernel originally used within the NKF framework and NKSR (Huang et al., ) in a sparse setting on ShapeNet.
Again, we sample $m=1,000$ surface points and corresponding normals for each shape and set $h=1$ for all Matérn kernels.
Additionally, we evaluate Matérn kernel’s 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 2 reports the results, quantified using intersection-over-union (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—almost twice as long as for Matérn kernels, which require about three days (measured on a single NVIDIA A100).
Additionally, although Matérn 1/2 is not able to surpass NKSR, it comes very close (95.3 vs. 95.6 in IoU) while having a shorter training time.
Regarding out-of-category generalization, we observe that Matérn 1/2 performs best, followed by Matérn 3/2, NKF, and NKSR.
Please see Figure 5 (right) and Appendix G for qualitative results.

Table: Table 2: Results on ShapeNet for learnt kernels. NKF* denotes a re-implemented variant of NKF. Importantly, NKF* and Matérn $\nu\in\{\text{1/2},\text{3/2},\text{5/2},\infty\}$ are based on the exact same framework and parameters; they differ only in the employed kernel. Runtime is measured on a single NVIDIA A100 with a batch size of one to ensure fair comparison. Bold marks best result, underline second best.

Table: Table 3: Robustness against noise. We compare Matérn kernels’ robustness against noise on a subset of the ShapeNet dataset (including about 1,700 shapes) using different noise levels, $\sigma\in\{0,0.0025,0.005\}$ and bandwidths, $h\in\{0.5,1,2\}$. Again, NKF* denotes a re-implementation of NKF. Bold marks best result, underline second best.

Robustness against noise.
We evaluate Matérn kernel’s robustness against different noise levels, $\sigma\in\{0,0.0025,0.005\}$, 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 3 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.

[Uncaptioned image]: x9.png

Convergence speed.
Finally, we analyze Matérn kernel’s convergence 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 4.1.
Matérn 1/2 converges the fastest, followed by Matérn 3/2, 5/2, and the Gaussian kernel.

## 6 Conclusion

In this work, we have proposed to use the family of Matérn kernels for tunable implicit surface reconstruction, acting as a drop-in replacement of the recently employed first-order arc-cosine kernel within frameworks such as Neural Splines (Williams et al., ) and Neural Kernel Fields (Williams et al., ).
We presented an in-depth theoretical analysis of Matérn kernels, analyzing its relation to Fourier feature mappings, Siren networks, as well as arc-cosine kernels.
Our systematic evaluation confirmed our theoretical results and showed that Matérn kernels are significantly faster to compute and train, and, especially Matérn 1/2 and 3/2, lead to better surface reconstructions compared to the arc-cosine kernel—both, in a non-learnable as well as learnable regime.
In the future, we plan to incorporate Matérn kernels into the NKSR framework to enhance robustness against noise.

#### 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 German Federal Ministry of Education and Research (BMBF), FKZ: 01IS22082 (IRRW).
The authors are responsible for the content of this publication.

## Appendix A Proof of Proposition 2

See 2

###### Proof.

Based on a simple bound

$$ $\displaystyle I$ $\displaystyle=\int_{\mathbb{R}^{d}}\left(\frac{2\nu}{h^{2}}+\left(2\pi\|\omega \|\right)^{2}\right)^{\nu+d/2}|\mathcal{F}[f](\omega)|^{2}\mathrm{d}\omega$ $\displaystyle\leq\int_{\mathbb{R}^{d}}\left(\frac{2\nu}{h^{2}}\right)^{\nu+d/2 }|\mathcal{F}[f](\omega)|^{2}\mathrm{d}\omega+\int_{\mathbb{R}^{d}}\left(\left (2\pi\|\omega\|\right)^{2}\right)^{\nu+d/2}|\mathcal{F}[f](\omega)|^{2}\mathrm {d}\omega$ $\displaystyle=\left(\frac{2\nu}{h^{2}}\right)^{\nu+d/2}\underbrace{\int_{ \mathbb{R}^{d}}|\mathcal{F}[f](\omega)|^{2}\mathrm{d}\omega}_{=:C^{\prime}( \mathcal{F}[f])}+\underbrace{\int_{\mathbb{R}^{d}}\left(\left(2\pi\|\omega\| \right)^{2}\right)^{\nu+d/2}|\mathcal{F}[f](\omega)|^{2}\mathrm{d}\omega}_{=:C ^{\prime}_{d,\nu}(\mathcal{F}[f])}$ $\displaystyle=\left(\frac{2\nu}{h^{2}}\right)^{\nu+d/2}C^{\prime}(\mathcal{F}[ f])+C^{\prime}_{d,\nu}(\mathcal{F}[f])$ $$

we get

$$ $\displaystyle\|f\|_{H_{\nu}}^{2}$ $\displaystyle=h^{2\nu}\left((2\pi)^{d/2}C_{d,\nu}\right)^{-1}I$ $\displaystyle\leq h^{2\nu}\left((2\pi)^{d/2}C_{d,\nu}\right)^{-1}\left(\left( \frac{2\nu}{h^{2}}\right)^{\nu+d/2}C^{\prime}(\mathcal{F}[f])+C^{\prime}_{d, \nu}(\mathcal{F}[f])\right)$ $\displaystyle=h^{2\nu}\left(\frac{1}{h^{2\nu+d}}\underbrace{(2\nu)^{\nu+d/2} \left((2\pi)^{d/2}C_{d,\nu}\right)^{-1}C^{\prime}(\mathcal{F}[f])}_{=:C^{1}_{d ,\nu}(\mathcal{F}[f])}+\underbrace{\left((2\pi)^{d/2}C_{d,\nu}\right)^{-1}C^{ \prime}_{d,\nu}(\mathcal{F}[f])}_{=:C^{2}_{d,\nu}(\mathcal{F}[f])}\right)$ $\displaystyle=h^{2\nu}\left(\frac{1}{h^{2\nu+d}}C^{1}_{d,\nu}(\mathcal{F}[f])+ C^{2}_{d,\nu}(\mathcal{F}[f])\right)$ $$

which concludes the proof.
∎

###### Proof.

## Appendix B Derivation of h ∗ superscript ℎ h^{*} italic_h start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT in Eq. ( 13 )

Taking derivative of the bound presented in Proposition 2 w.r.t. $h$ yields

$$ $\frac{\mathrm{d}}{\mathrm{d}h}\left(h^{2\nu}\left(\frac{1}{h^{2\nu+d}}C^{1}_{d ,\nu}(\mathcal{F}[f])+C^{2}_{d,\nu}(\mathcal{F}[f])\right)\right)=\frac{1}{h^{ d+1}}\left(2\nu C^{2}_{d,\nu}(\mathcal{F}[f])h^{2\nu+d}-dC^{1}_{d,\nu}( \mathcal{F}[f])\right).$ $$

Since $d,h>0$, the leading factor $1/h^{d+1}$ never becomes zero, so we must have

$$ $2\nu C^{2}_{d,\nu}(\mathcal{F}[f])h^{2\nu+d}-dC^{1}_{d,\nu}(\mathcal{F}[f])=0 \iff h^{*}=\left(\frac{d}{2\nu}\frac{C^{1}_{d,\nu}(\mathcal{F}[f])}{C^{2}_{d, \nu}(\mathcal{F}[f])}\right)^{1/(2\nu+d)}$ $$

which shows what we have stated in Eq. (13) of the main paper.

## Appendix C Proof of Theorem 3

The proof of Theorem 3 is based on Bochner’s theorem (in harmonic analysis) which reads:

###### Theorem 6 (Bochner) .

A continuous function $k:\mathbb{R}^{d}\longrightarrow\mathbb{R}$ with $k(0)=1$ is positive definite if and only if there exists a finite positive Borel measure $\mu$ on $\mathbb{R}^{d}$ such that

$$ $k(\tau)=\int_{\mathbb{R}^{d}}e^{i\omega^{\top}\tau}\mathrm{d}\mu(\omega).$ $$

In essence, Bochner’s theorem states that $k$ and $\mu$ are Fourier duals; given a stationary kernel $k$, one can obtain $\mu$ (better known as spectral density if normalized) by applying the inverse Fourier transform to $k$.
On the other hand, given a spectral density $\mu$, one can obtain the corresponding kernel function $k$ by applying the Fourier transform to $\mu$.
Based on Bochner’s theorem,

$$ $k(\tau)=k(x-y)=\int_{\mathbb{R}^{d}}e^{i\omega^{\top}(x-y)}\mathrm{d}\mu( \omega)=\mathbb{E}_{\omega\sim\mu}\left[\cos(\omega^{\top}(x-y))\right]$ $$

because $k$ is real and for even spectral densities.
On the other hand, Rahimi & Recht () showed

$$ $\displaystyle k(\tau)=k(x-y)$ $\displaystyle=\mathbb{E}_{\omega\sim\mu}[\cos(\omega^{\top}(x-y))]$ $\displaystyle=\mathbb{E}_{\omega\sim\mu}[\cos(\omega^{\top}(x-y))]+\underbrace {\mathbb{E}_{\omega\sim\mu,b\sim\mathcal{U}(0,2\pi)}[\cos(\omega^{\top}(x+y)+2 b)]}_{=0\,(*)}$ $\displaystyle=\mathbb{E}_{\omega\sim\mu,b\sim\mathcal{U}(0,2\pi)}[\cos(\omega^ {\top}(x-y))+\cos(\omega^{\top}(x+y)+2b)]$ $\displaystyle=\mathbb{E}_{\omega\sim\mu,b\sim\mathcal{U}(0,2\pi)}[2\cos(\omega ^{\top}x+b)\cos(\omega^{\top}y+b)],$ $$

where $(*)$ follows because $b$ is uniformly distributed in $[0,2\pi]$ (hence, the inner expectation w.r.t. $b$ becomes zero).

We will now prove Theorem 3, copied below for convenience:

See 3

###### Proof.

Let $f:\mathbb{R}^{d}\longrightarrow\mathbb{R}$ be a two-layer fully-connected network with sine activation function, i.e.,

$$ $f(x)=\sqrt{\frac{2}{m}}\sum_{i=1}^{m}v_{i}\sin(w_{i}^{\top}x+b_{i})$ $$

with bottom-layer weights $W=(w_{1},w_{2},\dots,w_{m})\in\mathbb{R}^{d\times m}$, biases $b=(b_{1},b_{2},\dots,b_{m})\in\mathbb{R}^{m}$, and top-layer weights $v=(v_{1},v_{2},\dots,v_{m})\in\mathbb{R}^{m}$.
Assume that $W$ and $b$ are fixed (that is, we only allow the second layer to be trained), and initialized according to some distribution.
Choose

$$ $w_{i}\sim p_{\nu}(\omega)=h^{-2\nu}C_{d,\nu}\left(\frac{2\nu}{h^{2}}+(2\pi\| \omega\|)^{2}\right)^{-(\nu+d/2)}$ $$

for $h,\nu>0$ and $b_{i}\sim\mathcal{U}(0,2\pi)$ for all $i\in\{1,2,\dots,n\}$.
Then, based on the fact that $\partial_{v_{i}}f(x)=\sqrt{2}\sin(\omega_{i}^{\top}x+b_{i})/\sqrt{m}$, it is easy to see that the NTK of $f$ is given by

$$ $\displaystyle k_{\text{NTK}}(x,y)$ $\displaystyle=\sum_{i=1}^{m}\partial_{v_{i}}f(x)\partial_{v_{i}}f(y)$ $\displaystyle=\frac{1}{m}\sum_{i=1}^{m}2\sin(w_{i}^{\top}x+b_{i})\sin(w_{i}^{ \top}y+b_{i})$ $\displaystyle=\frac{1}{m}\sum_{i=1}^{m}2\cos(w_{i}^{\top}x+b_{i})\cos(w_{i}^{ \top}y+b_{i})-2\cos(w_{i}^{\top}(x+y)+2b_{i})$ $\displaystyle=\frac{1}{m}\sum_{i=1}^{m}2\cos(w_{i}^{\top}x+b_{i})\cos(w_{i}^{ \top}y+b_{i})-\frac{1}{m}\sum_{i=1}^{m}2\cos(w_{i}^{\top}(x+y)+2b_{i}).$ $$

As $m\rightarrow\infty$, we finally obtain

$$ $\displaystyle k_{\text{NTK}}(x,y)$ $\displaystyle=\mathbb{E}_{(w,b)}\left[2\cos(w^{\top}x+b)\cos(w^{\top}y+b) \right]-2\underbrace{\mathbb{E}_{(w,b)}\left[\cos(w^{\top}(x+y)+2b)\right]}_{= 0\,(*)}$ $\displaystyle{=}k_{\nu}(x,y)$ $$

where $(*)$ follows because $b$ is uniformly distributed, and $(**)$ since $w$ is distributed according to $p_{\nu}$ which is, in fact, the spectral density of $k_{\nu}$, see Eq. (10) of the main paper.
∎

###### Theorem 6 (Bochner) .

###### Proof.

## Appendix D Proof of Corollary 5

We first re-state the following result, needed to prove the second part of Corollary 5:

###### Theorem 7 ( Saitoh & Sawano ( 2016 ) , Theorem 2.17) .

Let $k_{1}$ and $k_{2}$ be two positive definite kernels. Denote by $H_{1},H_{2}$ the RKHSs induced by $k_{1}$ and $k_{2}$. Then,

$$ $H_{1}\subset H_{2}\iff\gamma^{2}k_{2}-k_{1}\text{ is positive definite for } \gamma>0.$ $$

We are now ready to prove Corollary 5 which is copied below for convenience:

See 5

###### Proof.

(1) Follows from Theorem 4 by noting that the NTK of a fully-connected $(L=2$)-layer ReLU network is the first-order arc-cosine kernel, see, e.g., Cho & Saul (); Williams et al. (). For (2), we know that $H_{\nu}\subset H_{\text{AC}}$ and $H_{\text{AC}}\subset H_{\nu}$, so, based on Theorem 7, $k:=\gamma^{2}k_{\text{AC}}-k_{\nu}$ and $\bar{k}:=\bar{\gamma}^{2}k_{\nu}-k_{\text{AC}}$ are both valid kernels for $\gamma,\bar{\gamma}>0$. Re-arranging both equations, we have

$$ $k_{\text{AC}}=ak_{\nu}+b,\quad\text{where}\quad a=1/\gamma^{2},b=k/\gamma^{2}$ $$

and, equivalently,

$$ $k_{\nu}=\bar{a}k_{\text{AC}}+\bar{b},\quad\text{where}\quad\bar{a}=1/\bar{ \gamma}^{2},\bar{b}=\bar{k}/\bar{\gamma}^{2}.$ $$

This shows that the Laplace and arc-cosine kernel are indeed related by an affine transformation when restricted to $\mathbb{S}^{d-1}$, concluding the proof.
∎

###### Theorem 7 ( Saitoh & Sawano ( 2016 ) , Theorem 2.17) .

###### Proof.

## Appendix E Additional Details for ShapeNet Experiment

In this section, we provide additional details on the ShapeNet evaluation as presented in Section 5.1 of the main paper.

### E.1 Implementation Details

Following common practice (see, e.g., Williams et al. ()), we use finite differences to approximate the gradient part of the KRR problem in Eq. (1) of the main paper.
Specifically, denote $\mathcal{X}^{\prime}:=\mathcal{X}^{+}\cup\mathcal{X}^{-}$, where $\mathcal{X}^{+}:=\{x_{1}+{\epsilon}n_{1},x_{2}+{\epsilon}n_{2},\dots,x_{m}+{
\epsilon}n_{m}\}$ and $\mathcal{X}^{-}:=\{x_{1}-{\epsilon}n_{1},x_{2}-{\epsilon}n_{2},\dots,x_{m}-{
\epsilon}n_{m}\}$ for a fixed ${\epsilon}>0$.
The Representer Theorem (Kimeldorf & Wahba, ; Schölkopf et al., ) tells us that the solution to Eq. (1) is of the form

$$ $f(x)=\sum_{x^{\prime}\in\mathcal{X}^{\prime}}\alpha_{i}k(x,x^{\prime})$ $$

which is linear in the coefficients $\alpha\in\mathbb{R}^{2m}$, given by

$$ $\alpha=(K+\lambda I)^{-1}y,\quad\text{where}\quad K=(K)_{ij}=k(x_{i},x_{j})\in \mathbb{R}^{2m\times 2m},$ $$

and

$$ $y_{i}=\begin{cases}+{\epsilon}&\text{if }x^{\prime}_{i}\in\mathcal{X}^{+},\\ -{\epsilon}&\text{if }x^{\prime}_{i}\in\mathcal{X}^{-}.\end{cases}$ $$

$I\in\mathbb{R}^{2m\times 2m}$ denotes the identity matrix.
We employ PyTorch’s built-in Cholesky solver to numerically stable solve for $\alpha$ and set ${\epsilon}=0.005$ for all experiments.

### E.2 Further Qualitative Results

We show additional qualitative results on ShapeNet in Figure 6.

Figure: Figure 6: More qualitative results on ShapeNet, comparing Matérn 1/2 and 3/2 for different values of $h$ to the arc-cosine kernel as implemented in Neural Splines (Williams et al., ).
Refer to caption: x10.png

## Appendix F Further Details on the Surface Reconstruction Benchmark

### F.1 Experimental Setting

For Neural Splines (NS; Williams et al. ()), we used the official implementation provided here: [https://github.com/fwilliams/neural-splines](https://github.com/fwilliams/neural-splines).
Similar to the original paper, we used 15,000 Nyström samples and did the same parameter sweep over the regularization parameter, $\lambda\in\{0,10^{-13},10^{-12},10^{-11},10^{-10}\}$.
For the rest of the parameters (not mentioned in the paper), default 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 parameter sweep over the bandwidth parameter, $h\in\{0.5,1,2\}$.
We utilized the SRB data from here: [https://github.com/fwilliams/deep-geometric-prior](https://github.com/fwilliams/deep-geometric-prior).

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_size, eps, cg-max-iters, cg-stop-thresh, outer-layer-variance.
In Table 1 of the main paper, we report the best result we could achieve on a V100, which is 0.19 for Chamfer and 3.19 for the Hausdorff distance; the authors of (Williams et al., ) reported 0.17 and 2.85 for Chamfer and Hausdorff distance, respectively.
In order to achieve this, we had to lower outer-layer-variance form $10^{-3}$ 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 $\lambda=10^{-11}$.
Due to the absence of further information about the experimental setting in the original paper, we 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., ).

### F.2 Additional Results

Per-object metrics are shown in Table 4, and more qualitative result in Figure 7.

Figure: Figure 7: Qualitative results on the Surface Reconstruction Benchmark, comparing Matérn 1/2 and 3/2 to Neural Splines (NS; Williams et al. ()).
Refer to caption: x11.png

**Table 4: Per-object quantitative results on the Surface Reconstruction Benchmark.**
|  | Anchor | Daratech | DC | Gargoyle | Lord Quas |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  | CD $\downarrow$ | HD $\downarrow$ | CD $\downarrow$ | HD $\downarrow$ | CD $\downarrow$ | HD $\downarrow$ | CD $\downarrow$ | HD $\downarrow$ | CD $\downarrow$ | HD $\downarrow$ |
| 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 $\infty$ | 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 |

## Appendix G Additional Results for Data-dependent Kernels

We show additional qualitative results for learned kernels on ShapeNet in Figure 8.

Figure: Figure 8: Additional qualitative results for learned kernels on ShapeNet. We compare against Neural Kernel Fields (NKFs; Williams et al. ()) and Neural Kernel Surface Reconstruction (NKSR; Huang et al. ()).
Refer to caption: x12.png

## Peer Reviews

confidence: 3
contribution: 2
initial_rating: 5
presentation: 3
questions: figure on page 6 needs labels and a quick caption
soundness: 3
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.
- I particularly liked the connection to SIRENS
- I found the experiments to be well made. Reconstructions and evaluations are indeed favorable in comparison to their competing methods
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. 
This 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. 

On 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.
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)
- I did not fully understand and appreciate the excessive focus on bettering the arc-cosine kernel

---

confidence: 3
contribution: 2
initial_rating: 6
presentation: 4
questions: I would appreciate any response to the questions stated above.
soundness: 3
strengths: The paper is well-organized and clearly written, with a sufficient amount of background material to make it self-contained.

The incorporation of Matérn kernels appears to be a valuable contribution, supported by theoretical analysis on their ability to capture high-frequency details effectively.

Both quantitative and qualitative comparisons are provided, highlighting key properties of the proposed method.
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.
weaknesses: Unclear Benefits of Matérn Kernel Properties

Section 3.1 describes Matérn kernel properties, yet it’s unclear how differentiability and stationarity specifically benefit surface reconstruction. More specifically,
  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?
Stationarity: 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?

The Role of Local Support

Since 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.

Comparison Quality

Comparison 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?

Comparison 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?

Unclear Connection to SIREN

Theorem 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?

---

confidence: 3
contribution: 3
initial_rating: 6
presentation: 3
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.
(2) In the sparse surface reconstruction, in addition to quantitative results, including visualizations of the sampled points would make the effects more intuitive.
soundness: 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.
(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.
(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.
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.

Key contributions include:
1.Highlighting the Matérn kernels’ stationary properties, which allow tunable spectral properties similar to Fourier feature mappings, addressing spectral bias issues in coordinate-based MLPs.
2.Providing theoretical insights into the Matérn kernel’s relationship with SIREN networks and its distinctions from arc-cosine kernels previously used in surface reconstruction.
3.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.
weaknesses: (1) Lacks comparison with the latest implicit surface reconstruction algorithms.
(2) Some figures are missing ground truth.

---

confidence: 4
contribution: 2
initial_rating: 3
presentation: 2
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? 

Can 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 $\nu$?
soundness: 3
strengths: Overall, the authors investigation into the suitability of Matern kernels for surface reconstruction in section 4 is the standout. 

While 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.

The 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).

To the best of my knowledge, the mathematical formulations are sound, though I did not check them closely.
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.
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. 

Additionally, 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.

The 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.

2). 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. 

Another 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.

In 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?

Together, 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.

Meta-review: This paper suggests Matern Kernels for applications of implicit surface representation and reconstruction from oriented point clouds (i.e., point clouds with normals). The Matern Kernels are used, similar to other kernel functions, to build (via linear combinations) implicit functions representing the signed distance function, the zero level of which represents the surface of interest. The significance in this particular choice of kernels is in the tunable parameters that provide a family including known kernels, such as the Gaussian kernel, and provide refined control over smoothness of the reconstructed surface and the reconstruction error. Experiments with these kernels for various surface reconstruction benchmarks demonstrate the efficacy. On the down side, some reviewers feel this paper presents a rather straightforward application of a known family of kernels in a standard way which diminishes from its contribution a bit.

Decision: Accept (Poster)