Generalised Implicit Neural Representations
Authors: Daniele Grattarola
Pierre Vandergheynst
Venue: NeurIPS 2022 Accept (2022)
Keywords: implicit neural representations
neural fields

Abstract: We consider the problem of learning implicit neural representations (INRs) for signals on non-Euclidean domains. In the Euclidean case, INRs are trained on a discrete sampling of a signal over a regular lattice. Here, we assume that the continuous signal exists on some unknown topological space from which we sample a discrete graph.
In the absence of a coordinate system to identify the sampled nodes, we propose approximating their location with a spectral embedding of the graph. This allows us to train INRs without knowing the underlying continuous domain, which is the case for most graph signals in nature, while also making the INRs independent of any choice of coordinate system. We show experiments with our method on various real-world signals on non-Euclidean domains.

Generalised Implicit Neural Representations
Daniele Grattarola EPFL
Pierre Vandergheynst EPFL
Lausanne, Switzerland daniele daniele.grattann www.tarrllleeepl...c..
Lausanne, Switzerland Seirrrevvadderreenss@@eel..... ch
Abstract
We consider the problem of learning implicit neural representations (INRs) for signals on non-Euclidean domains. In the Euclidean case, INRs are trained on a discrete sampling of a signal over a regular lattice. Here, we assume that the continuous signal exists a discrete graph. the absence of a coordinate system to identify the sampled nodes, we propose approximating their location with spectral embedding of the graph. This allows us to train INRs without knowing the underlying continuous domain, which is the case for most graph signals in nature, while also making the INRs independent of any choice of coordinate system. We show experiments with our method on various real-world signals on non-Euclidean domains.
unknown topological space from which sample
fo
INR
Signal on non-Euclidean domain
Observed graph
Predicted signal
Spectral embedding
Figure Given continuous signal on con-Euclidean domain, we observe discrete graph realisation of it. A generalised implicit neural representation is neural network fo trained to map spectral embedding of each node to the corresponding signal value.
Introduction
Implicit neural representations (INRs) are a class of techniques to parametrise signals using neural networks [48] 41 45 344. INRs are trained to map each point in a given domain the corresponding value of a signal at that point. For example, INRs for images learn to map the 2D coordinates of pixels to their corresponding RGB values. INRs can also conditioned on latent vector, typically learned end-to-end as part of the model, that allows them to represent different signals on the same
domain. INRs have been successfully applied model complex signals like images 48 45 signed
distance functions 41 and radiance fields 40
INR, obtain continuous approximation of the signal the whole domain. This allows us to compute a higher-resolution approximation of the signal by sampling more points the domain (e.g., finer grid of pixels). Additionally, since INRs are fully differentiable.
By expressing signal
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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the latent vector of a conditional INR can be optimised with backpropagation and gradient descent to obtain a signal with some desired characteristics, For example, a conditional INR for signed distance functions can be used to design surfaces with a target aerodynamic profile 42
So far, the literature on INRs has only focused on signals on Euclidean domains. However, signals defined on non- Non-Euclidean domains are ubiquitous in nature and are especially relevant in artificial intelligence, as demonstrated by the recent rise of geometric deep learning [1].. In this paper, we propose an extension of the INR setting to signals on arbitrary non-Euclidean domains.
Contributions We formulate the generalised INR problem as the task of learning an implicit representation for signal on arbitrary topological space T where, instead of observing the signal sampled on a regular lattice, we observe a graph (i.e., a discretisation of T) and the corresponding graph signal. In most practical cases, T is unknown and we cannot represent the sampled vertices in a coordinate system. Even when T is known (as in the Euclidean case), training an INR on fixed coordinate system means that the model will depend on this choice. We solve both these issues by identifying the sampled nodes with an intrinsic spectral embedding obtained from the eigenvectors of the graph Laplacian. We then train neural network to map the spectral embeddings to the corresponding signal values. Figure [
shows a schematic view of the method.
Since the eigenvectors of the graph Laplacian are discrete approximation of the continuous eigen- functions of the Laplace-Belin Beltrami operator on T (when appropriately rescaled) [5] 77. at inference time we can map arbitrary points T to the corresponding approximation of the signal, as long as the discrete graph signal is sampled consistently. This allows us to compute higher-resolution signals to estimate the signal on different graph realisations of the same phenomenon (e.g., different social networks with a similar structure).
Results In the experiments section, we show concrete examples of learning INRs for signals on graphs and manifolds, using real-world data from biology and social networks. We also show that INRs trained on one instance of a graph can be transferred to different graph realisations of the underlying domain. Then, we show applications of conditional generalised INRs to model different signals on the same irregular domain, and also different signals on different domains. Finally, we conclude the paper with experiment that consolidates all our results, modelling real-world meteorological signals on the spherical surface of the Earth.
Related works
INRs have recently attracted the attention of the machine learning community for their ability to represent complex signals, especially in computer vision, 3D rendering, and image synthesis applications [4] 38 22
40 46 32). Recent work has highlighted that INRs benefit from computing sinusoidal transformations of the coordinates, typically called Fourier features or positional encodings
(PEs) 52 50
53 [. Since the PEs typically used in INRs also the eigenfunctions of the
Laplace operator in Euclidean domains, our method is a generalisation of this approach.
However, these transformations not applicable to Euclidann domains and still depend choice of coordinate system. A proposal for making INRs equivariant under the SO(3) group has been presented by Deng et al. (15), although this technique does not apply in general to non-Euclinn nuc-eucli domains. The field of geometric deep learning has also explored the use of graph PEs similar to the we use in this paper, showing their usefulness node features for graph neural networks [47 Recent research has also investigated the idea of learnable graph PEs [8 13]
36
16
Related to our work, Belkin and Niyogi [5] Boguna et al. [1] and Levie et al. 30 have studied
the geometrical link between graphs and topological spaces. Keriven et al. [25] 26 investigated the convergence, in the limit, of graph neural networks trained on large graphs. An alternative view of graphs as discrete samples of continuous spaces is the concept of graphon, which can be seen both as generative model and as the limit of sequence of graphs 37
We also mention the work of Koestler et al. [28] who propose a similar idea to ours for learning fields on meshes. In this work, the authors train a neural network to map the eigenfunctions of the Laplacian to an RGB texture. By contrast, we focus on a wider range of tasks and experiments, including transferring to completely different graphs, conditioning the INRs, and modelling dynamical systems. Our scheme to compute generalised embeddings also different from Koestler et al.'s, not relying on knowing node coordinates.

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3 Method
Standard setting In the standard INR setting, we consider a signal f: f:X + y with X C Rd and Y C RP. We observe a discrete realisation of the signal f(x,, for i -,= 1,...,, n, where points x; are sampled on a regular lattice on X. Then, we train a neural network fo X D, with parameters 0, on input-output pairs (x, (x)).. Since we know that the signal domain is a subset of IR' at inference time we can sample points anywhere on X' to compute the approximated value of the signal at those points. For example, an INR for images maps equispaced points in the unit square (pixel coordinates) to points in the unit cube (RGB values normalised between 0 and 1). The specific image on which we train the INR is realisation of one such signal at given resolution, and at inference time we can sample finer lattice to super-resolve the image [48 45].
Generalised setting In the generalised setting, consider continuous signal f: + Y, with T an arbitrary topological space. We observe a discrete graph signal f(:)) on an undirected graph G = (V, E), with node set V {v,} for 1,..., and edge set E C V x V. Note that the graph can be weighted if metric is available on T. The meaning of sampling G from T is intuitive in the case of geometric meshes other physical structures like proteins (in which case we also know the coordinates of v;), but the same reasoning also applies to more complex domains with an abstract meaning (e.g., the space of all possible papers and their citations). We generally assume that the graph describes some measure of closeness between uniformly sampled points on T, be it function of the coordinates (e.g., a kernel) or some logical or functional relation that is given as part of the data. In general, we don't 't assume to know the true 'T from which G' is sampled or the coordinates of e/- We only observe G and the associated graph signal. For an in-depth discussion on the relation and (30
between graphs and topological spaces, see references [
Proposed method We approach the generalised INR problem by mapping es to f(;)) through a spectral embedding obtained from the graph Laplacian. Let A IR= be the weighted adjacency matrix of graph G, D the diagonal degree matrix, and L D - A the combinatorial Laplacian. The eigendecomposition of the Laplacian yields orthonormal basis of eigenvectors {uy E [R*) for
k = l...., n, with a canonical ordering given by their associated eigenvalues \ S >2 S ... S An
We define generalised spectral embedding of size k for node U, as
c; |u
u
E R
The generalised spectral embeddings are a discrete approximation of the eigenfunctions of the continu- ous Laplace- Beltraam operator on T and converge to them for n > oo. See the discussion ences 5 practice, rescaling the eigenvectors by /ni ensures that the embeddings for graphs of different sizes will have the same range component-wist and that similar nodes will have similar embeddings regardless of the graph size. As an example, we show a comparison between the eigenvectors and the generalised spectral embeddings of a path graph in Figure []
0.1
refer- [9, and (7] for formal analysis. In
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Figure 2: Laplacian eigenvectors (left) vs. rescaled spectral embeddings (right) of path graph, for k = 3 and n = 100, 1000, 2000. The curves on the right are shifted to improve visualisation.
We train neural network fo e; + f((:) on the observed graph signal. Since the generalised spectral embeddings are an intrinsic property of the graph, the INR will not depend on any choice of coordinate system but only on the topology of the underlying continuous domain.
At inference time, we can compute the approximated value of the signal at an arbitrary location on T by computing the associated spectral embedding. If we know T, or if we estimate it, we can sample new vertices directly from the domain and apply the same procedure used to construct the training graph. If T is unknown (e.g., in the case of natural graphs that describe some abstract concept like citations, social interactions, or biological relations), then we just assume to observe a similar graph sampled from T. An important difference between generalised INRs and Euclidean INRs that generally we must observe the full graph in order to compute the inputs. This is necessary because, while in the Fuclidean case we completely know the domain of the signal a priori, in the generalised case we need to estimate the topology of the domain by sampling.
3

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Figure 3: Ground truth signals signals predicted by the INR. Best viewed in colour.
Complexity Although computing the full eigendecomposition of the Laplacian has complexity of O(n)), here we are only interested the first & eigenvectors. Also, the Laplacian of most graphs is very sparse, with nodes having average degree d < n. We therefore the implicitly restarted Lanczos method for eigendeconposition implemented by the ARPACK software, which has complexity of (ddn²) and can be easily parallelised (29]. In practice, all computations for this paper were easily managed on a commercial laptop with 10 CPU cores, scaling up to graphs in the order of 10a nodes.
We also note that at inference time, depending on how the edges are constructed, it could be possible to add new nodes and edges the training graph instead of sampling entirely new graph from T. In this case, the spectral embeddings for the new nodes can be estimated using the Nyström method without needing to compute the full eigendecomposition 3 9
Alternative approaches The generalised Laplacian embeddings used in this paper are only one of many possibilities to represent nodes sampled from TT. name a few, locally linear embeddings [43, Isomap [51. Laplacian eigenmaps [6] and diffusion maps [14 are all based on the idea of embedding data using the first few principal eigenvectors of similarity matrix. Here we focus on the Laplacian since it is well known, sparse, and easy to compute, and its cigendecomposition is stable to graph perturbations. One disadvantage of Laplacian eigenvectors that they are only unique up to sign, leading to 2 possible eigenbases for a given T and the consequent ambiguity when transferring an INR to a different graph realisation. Additionally, eigenvalues of multiplicity greater than also introduce ambiguity in the eigenvectors, since all rotations and reflections of the associated eigenspace are valid choices. However, simple heuristics can be used to eliminate sign ambiguity, and we did not encounter any other practical issue in this regard. Alternative ways to resolve the ambiguity is to use PEs based on random walks [18 39 31, heat kernels [49 19] or the more recent
sign-and-basis invariant neural networks [33. We leave the exploration of these alternatives to future work, since they do not significantly impact our main contributions.
4 Experiments
Setting We implement the generalised INR SIREN multi-layer perceptron 455 The model has 6 layers with 512 hidden neurons and a skip connection from the input to the middle layer. We use the same hyperparameters and initialisation scheme suggested by Sitzmann et al. [45]. We train the model using Adam (27) with learning rate of 10 and an annealing schedule that halves the learning rate if the loss does not improve for 1000 steps. At each step, we sample 5000 nodes randomly from the graph as a mini-batch. We use spectral embeddings of size k = 100, value that we found empirically as good trade-off between complexity and performance. We report an analysis on the effect of k in Section 4. We ran all experiments on an Nvidia Tesla V100 GPU. The code to reproduce our results and the high-resolution version of all figures are available at whhtpss: //gittubb com/ danieleggatteenn documented in the supplementary material, where we also report an additional experiment on solving differential equations with generalised INRs.
//IN All deviations from the default setting are
4.1 Learning generalised INRs
We begin by verifying that typical INR architecture can successfully learn arbitrary graph signals in the generalised setting. We consider three datasets test cases.

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Bunny We generate texture on the Stanford bunny mesh using the Gray-Scott model given by:
Aa - D.La a b b F(1, a); Ab DDLL + a (F+ K)b
a
(2)
for state vectors a, b E Ik" and parameters Da: Da, F.K E R, where @ indicates element- wise product [21]. The mesh has 34834 nodes and 104288 edges. We configure the system to yield "coral" patterns] and evolve the simulation for 10 steps starting from random pos- itive values for a, b. At the end of the evolution, use vector
target graph signal.
Table I: R and mean squared error for the bunny, protein, and US election signals.
Protein As real-world domain, we consider the solvent excluded surface of protein struc- ture Fl The continuous signal is the value of the electrostatic field generated by the amino acid residues at the surface. Protein function can be largely understood by studying their surfaces and the corresponding chemical features, like
Bunny
Protein
US Election
R² MSE
1.000
1,000
0,999
9.14 10-8
1.17 10
1.45 10-3
the electrostatic charge, so this represents potentially interesting application of INRs to biology. We use a combination of the MSMS [4] and APBS [24] software packages to estimate the surface, sample the nodes, and get the corresponding signal, following the same workflow of Gainza et al. 200 The final graph has 1 1966 nodes and 35892 edges.
US election Finally, we consider a social net-
Table 2: Performance comparison (RP2) of typical INRs and generalised INRs (GINRs), using ReLU
work dataset introduced by Jia and Benson [23]. in which nodes represent United States coun- ties and edges are estimated using the Facebook Social Connectedness Index. The target signal represents the county-wiss outcome of the 2012 US presidential election, as values in the range [- (indicating the proportion of votes in favour of one candidate vs. the other). The graph has 3106 nodes and 22574 edges.
(r) or SIREN (s) activation.
INR (r)
INR (S)
GINR (r)
GINR (s)
Bunny Protein
0.000
0.919
0.932
0.885
0.799
0.275
0.921
0.916
Results In all cases, the generalised INR achieves an R2 close to indicating that the model can indeed learn non-trivial signals on non-Euclidean domains. We report the R82 and mean squared error for all datasets Tab. [ We compare the true graph signals and those learned by the INR, for the bunny and protein, in Fig. 3 As second test, we compare typical INRs trained on node coordinates with generalised INRs trained on the spectral embeddings. We consider the bunny and protein since they have input coordinates for the typical INRs. We also test different activation functions- ReLU or SIREN. We report in Tab. the R² for a held-out set of nodes (to evaluate whether the INRs are overfitting instead of learning meaningful representation) All results are averaged over 5 runs and have negligible standard deviations. We see that the generalised INRs perform almost always better than the typical ones. Note that these experiments only verify that the model can indeed learn the target signals. We will test the transferability of the model in-depth in Sec. 4.2 In the following sections, we mostly focus on known spaces T or synthetic graphs that we can better answer other research questions by having full control of the data.
Size of the spectral embeddings Typical INRs take as input the d-dimensional coordinates of points on the sampled lattice. In our general setting, the dimension k of the spectral embeddings is a hyperparameter of the method. To evaluate the impact of & on the performance of the INR, we train the same model for different values of & on the Stanford bunny.
We report the results in Figure 4 showing that the model fails to learn a meaningful rep- resentation for & < 3. This is reasonable because at least 3 non-trivial eigenvectors are needed correctly distinguish the main structural features of the bunny (ears, tail, etc.), as shown in Figure 5 We also see that, while the INR has no issues in learning the signal for = 100, for lower values of k the model struggles to represent the signal on the ears.
Available at
stanford. edu//aata//Dscaarepp/
2 DD 0.64, Do = 0.32, F = 0.06, K = 0.062
3 Prootein Data Bank identifier: IAA7
We report training details in the supplementary material.
5

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2
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(b) =
(c) 4
Figure 4: Left: R2 vs. k; Right: signals learned by the INR for k 2,
Figure 5 lgain provides possible explanation for this result, since the first few cigenvectors tend to collapse on the ears and higher frequency eigenvectors are needed to distinguish points towards the narrow tips. We will show another consequence of this in Section 4.2 We also mention that the drop in perfor- mance for lower values & on the other two datasets less pronounced than on the bunny, likely because of their lack of any high-frequency features.
4.2 Transferability of generalised INRs
Figure 5: Spectral drawing (blue) of the bunny using the first three non-trivial eigenvectors. In grey, 2d projections of the 3d drawing.
Because the generalised spectral embeddings approximation of the continuous eigenfunctions of the Laplace- Beltrami operator T, we apply a trained INR differ- ent graph realisation of T as long as its spectral structure is consistent with the training data.
discrete
In practical terms, we need know 1) how to sample nodes and 2) how the nodes connected. we know T, then the problem simply boils down to choosing a sampling strategy for the points and a suitable measure of closeness (e.g. kernel or edge-generating function like the k-nearest neighbours algorithm). T unknown, then we assume to observe a new graph through the same process that generated the training data (e.g., observe new social interactions among the same similar set of people).
Note that, when sampling new graph from T, the spectral embeddings will be slightly different (possibly better) approximation the continuous eigenfunctions. As this effect more evident on high-frequency eigenvectors, empirically observed that training the INR with smaller k and ReLU activations improved its transferability (cf. Table 2). See the supplementary material for more details. We investigate the transferability of generalised INRs in two settings.
Transferring to similar graphs As first example, we study the case in which T is unknown and we only observe the discrete graphs. We consider stochastic block model (SBM) with two communities of equal size (in total, n = 1000), parametrised by an inter-connection probability and intra-connection probability
We train an INR on graph with a strong community structure (p= 0.1, = 0.5) and a graph signal indicating the community of each node (effectively node classification task). Then, test the INR on graphs sampled in the ranges E [0.1, 1] and r E [0.1, 1] to test the robustness of the model to changes in the community structure.
We report our results in Figure First, we note that perfectly recovering the correct clusters is only theoretically guaranteed in certain regions of the parameter space M1, delimited with orange line in the figure 5 We see that the trained INR transfers perfectly to all graphs sampled this region and that its performance sharply drops only for p r, where graphs have community structure (see Figure 66 a. b). Additionally, we see that the INR maintains a good performance also in the region i.e., region in which graphs still exhibit some
between the theoretical boundary and the line p community structure although perfect recovery is not guaranteed. Overall, this indicates that training the INR on an instance of a graph allows to transfer it to graphs with similar structure.
$ Thhe region is above curve r In(n)/n n. (V2+ v np/ In(m)

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(inter)
Figure 6: Left: normalised mutual information at test time, evaluating on graphs with different inter- and intra-connection probabilities. We trained the INR on p 0.1 and 0.5, marked with x the plot. The orange line indicates the theoretical boundary above which it is possible to perfectly recover the signal. Best viewed colour. Right: the first two non-trivial eigenvectors of two SBMs, respectively exhibiting strong and no community structure.
10a
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Figure 7: Zoomed-in comparison between the nodes of the train- ing graph (left) and its super-resolved version (right), plotted on a black surface to aid visualisation. An interactive version in vector graphics of this figure available in the supplementary material.
Figure 8: Distribution of the error in the super-resolution experiment.
Super-resolution As a second test case, we consider super-resolution of a signal. We consider again the Stanford bunny mesh with the same texture graph signal as Section 4.1 we apply Loop's method for mesh subdivision to obtain higher-resolution mesh with 139122 nodes and 41 416929 edges (35) Then, compute the spectral embeddings for the new graph and predict the signal with the INR trained on the low-resolution graph.
Because the unit-norm cigenvectors are only unique up to sign, the INR will transfer to different graph only its eigenvectors are aligned the training ones (i.e., similar nodes must have similar spectral embeddings
in the previous experiment, we did not have to deal with this problem due to
symmetry in the SB.M). However, many cases, simple comparison the eigenvectors' histograms provides useful heuristic for automatically aligning them. For simplicity, here we also verify the alignment the two eigenbases manually.
We show in Figure 7 qualitative comparison between the training and super-resolved signals, from which see that the INR can predict the graph signal correctly. Quantitatively, since mesh subdivision does not remove the original nodes from the mesh, we can compute the H² between the training signal and part of the super-resolved signal. The model achieves an Rt² of 0.39 compared to training R of 0.99. However, this apparent drop in performance is mostly due to the high- frequecyy ears of the bunny, for which the model evidently overfits and cannot tolerate even small changes in the cigenvectors. Figure Sluure&Sshone shows the distribution of the squared error between the ground truth and predicted signal, from which we see that the error close to zero for most of the nodes (note that the y-axis is logarithmic). By ignoring the nodes above the 90th percentile of the squared ervor distribution, the R² score 0.94. Increasing resulted better prediction of the ears region (cf. Figure at the cost of worse overall transfcrability.

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(a) t = 625
(b) t. = 875
(c) = 1125
(d) 1375
(c) f 1625
Figure 9: Signals predicted by the conditional INR fo t) equispaced time steps [625, 1625]. All time steps shown were not part of the training set. Also, each test time step is as far as possible from its two closest training samples (i.e., training samples are at t = 0 (mod 10) while test samples are at / = 5 (mod 10). An animated version of this figure with 600 predicted samples is available in the supplementary material.
4.3 Conditional generalised INRs
We also consider the setting in which a generalised INR conditioned on a latent vector z E R allow the representation of different signals by the same neural network fo(e; z). Unlike the Euclidean setting, here we can investigate two cases: I) learning a conditional INR for different signals the same domain (the typical setting), and 2) learning conditional INR for different signals on different domains.
Reaction- -iffuionni process For the first case, we train a generalised INR to approximate the Gray-Scott reaction-diffusion process that we used to generate the graph signal in Section 4.1 given by Equation (2). We consider a signal fw, t). where E N is the number of updates since the start of the simulation. We evolve the system for 3000 steps starting from the same random initialisation used in Section |.1 and we train the INR fe (e,, t) using 300 time steps sampled at t = 0 (mod 10). The graph is fixed throughout the evolution, the INR must learn to exploit the conditioning input t to correctly predict the signal at different times.
At test time, we predict the signal for all the remaining time steps and compare the prediction with the ground truth signals. The INR achieves test Rt of 0.999 + 0.006 indicating that it successfully learned to approximate the true dynamics. We report Figure a few samples predicted by the INR at test time. By comparison, simple baseline using linear interpolation between training samples achieves similar R², while requiring to keep all 300 training steps in memory. The INR, on the other hand, represents the signal and its evolution with single set of weights and scalar control parameter.
0.98
0.96
0.92
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200
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1000
Multi-protein INR For the second case, we consider a dataset D {(Tm, fm)] of protein surfaces Tm with associated electro- static signals f'm T. - R for = 1,..., [D), obtained following the same protocol as Section 4.1 We configure the neural network
Figure 10: R² size of the
dataset [D].
fo(e;, )m) an autodecoder [1 where Z.m E R is a vector of free parameters learned end-to-end alongside the weights of the neural network. We assign a separate vector Zm to each training sample, allowing the model to find the best latent representation for the dataset. Since the continuous surface and the associated signals are unique to each protein in the dataset, the latent vector Zm must capture both aspects. We set the latent vector size 8, which we empirically found to give consistently good performance across datasets. Larger values of 2 did not significantly improve the performance.
We test the ability of the model to represent datasets of different sizes, by sampling random proteins from the dataset used in reference [20]. Figure 10 shows the R² of the model for different values of (D). Understandably, the performance decreases the number of samples grows, although the model achieves R² 2 0.9 for datasets of up to 500 different proteins. Overall, these results indicate that the model can represent many domains and signals with single set of weights.

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(a) Temperature
(b) Clouds
(c) Wind
prediction of the INR for = 4.5 (randomly chosen, not in the training data) at the original resolution. Bottom:
Figure 11: Top:
The super-resolved versions of the top row. Note that the apparently lighter colours are simply artifacts of the plotting software. Best viewed in colour and high resolution. An animated and uncompressed version of this figure is available in the supplementary material.
4.4
Weather modelling
As final experiment, we consolidate results from previous sections to showcase an application of generalised INRs in a real-world setting. We consider signals [(; t) representing hourly me- teorological measurements on the spherical domain of the Earth's 's surface. Our goal is to train a conditional INR at given spatial and temporal resolution and then use it to super-resolve the signal over space and time.
Data We collected data from the National Oceanic and Aumospheric Administration (NOAA) Operational Model Archive and Distribution System, specifically from the Global Forecast System (GFS). The data consists of three different signals representing: the dew point temperature 2m Above-ground measured in K (dpr2m), the atmosphere's total cloud cover percentage (tedcclm), and surface wind speed m/s (gustsfc). We consider a 24-hour period sampled at 1-hour increments over an equiangular spherical grid sampled at 1 increments We generate spherical mesh from the 65160 locations of the grid by computing their convex hull with the Qhuil software for a total of 193320 edges.
Results We train a conditional INR fa(e; t) at the given spatial and temporal resolution, for t 0, ..... 23. Then, evaluate its predictions a high-resolution spherical mesh with 258480 nodes and 773280 edges (obtained with the same procedure of Section 4.2). and at time increments of 30 minutes. We report in Figure II the prediction of the INR for a random test time, on both the original mesh and the high-resolution one. We refer the reader to the animated version of the figure available in the supplementary material for a better appreciation of the results. Overall, our results confirm that generalised INRs can model complex real-world signals on non-Euclidean domains, learning realistic continuous approximation of the signals from low-resolution samples.
Conclusion
We have presented the problem of learning generalised implicit neural representations for signals on non- non-Euclidean domains. Our method learns to map a spectral embedding of the domain the value of the signal, without relying on a choice of coordinate system. We have shown applications of our method on biological, social, and meteorological data, highlighting the potential usefulness of such INRs in wide range of scientific fields. We hope that future work will explore more real-world applications of our method.
"Specifically the period from 00:00 to 23:59 of May 9, 2022. All times f given in the text are in hours relative the start of the period.

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Checklist
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13


Reviews:
[{"review_id": "onKdQ7iuvPr", "reviewer": "Reviewer_8jka", "summary": "The authors propose to parameterize implicit neural representations (INRs) as neural networks that take as input the Laplacian eigenvectors of a non-Euclidean domain. This allows for learning INRs over spaces without an external coordinate system. These INRs are tested over various application areas, each of which defines a suitable discrete graph for computing the Laplacian eigenvectors from.\n\n--------\n\nAfter reading the authors replies, I have increased my score to a 7. The comparison with other choices / baselines, and the improved exposition / comparison to literature have greatly improved the paper in my eyes.", "questions": "1. For the Bunny mesh, do you use the cotangent Laplacian scheme, or do you still use the combinatorial graph Laplacian of the mesh? I'm curious as the cotangent Laplacian scheme is widely used for meshes, but there appears to be no mention of specific Laplacian normalization for this mesh.\n2. You treat the Earth as a sphere, correct? If so, it would be possible to evaluate the exact Laplacian eigenfunctions, how does this compare?", "limitations": "Are the authors aware of [Koestler et al. 2022]? These two papers do the same thing. Given that the idea is so simple, this would be an issue for novelty. I see that it is from about 2 months before the NeurIPS submission deadline, so we may ignore the effect on novelty it if the policy requires.\n\nReferences  \n[Koestler et al. 2022] Intrinsic Neural Fields: Learning Functions on Manifolds. https://arxiv.org/abs/2203.07967", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. The paper is simple and straightforward, in a good way.\n2. This is a natural generalization of the Euclidean case, in which signals are usually defined on a grid. INRs have found much success in the Euclidean case, and they can presumably be very useful for non-Euclidean domains.\n\nWeaknesses:\n1. Please be more precise with the claim that the INRs are ``equivariant under the symmetry group of the domain.'' In particular, it would be great to include a reference, or to include a precise statement and proof. I think what you mean is that the eigenvectors are equivariant under the symmetry group, and / or the entire INR algorithm (i.e. the mapping from input to final trained model) is equivariant.\n2. Would be nice to compare different choices of Laplacian normalizations, or to compare against using the ground truth eigenfunctions of a continuous surface like the sphere (see Questions section for more). E.g. various normalized graph Laplacians are often used for many applications.\n3. The ad-hoc manual choice of sign for the eigenvectors seems a bit hacky for the super-resolution task. Past work has given ways to get rid of sign ambiguity (there is also a basis ambiguity for higher dimensional eigenspaces), see [Lim et al. 2022], which also tests on a generalized INR task on non-Euclidean data.\n\nMinor notes:\n1. For graph positional encodings, in addition to random walk PEs there are various other PEs that are sign invariant, see [Lim et al. 2022] Proposition 3 for examples. These include the well known Heat Kernel Signature [Sun et al. 2009].\n2. May be worth pointing out that the sinusoidal positional encodings / Fourier features used in Euclidean INRs are eigenfunctions of the Laplace operator on Euclidean space. Thus, your method is a direct generalization.\n\nReferences:  \n[Lim et al. 2022] Sign and Basis Invariant Networks for Spectral Graph Representation Learning. https://arxiv.org/abs/2202.13013  \n[Sun et al. 2009] A Concise and Provably Informative Multi-Scale Signature Based on Heat Diffusion. Eurographics Symposium on Geometry Processing.\n", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "Uzd70WSUtag", "reviewer": "Reviewer_NW5a", "summary": "The paper proposes to extend implicit neural representation (INR) to non-euclidean settings, in particular graphs. This is achieved by using the top K eigenvectors of the graph laplacian to replace euclidean coordinates in INR. Experiments on several datasets, including protein and social networks, demonstrate the ability of the generalized INR to represent the signal and generalize to similar graphs faithfully.\n", "questions": "- The paper claims that the proposed model is “equivariant” to domain symmetries. I understand that the generation process is “invariant” to the symmetry transformations that leave the graph invariant. What am I missing? Also, these transformations are not necessarily the symmetries of the original manifold but those of the graph.\n\n- For experiments on conditional generation with the Stanford bunny, the figure (figure 9) does not convey the error in the generation process. The same is true for weather modelling (figure 10). Since you have the ground truth, please report the error in your approach. You can also compare to a simple baseline that, for example, uses averaging for interpolation? \n", "limitations": "Limitations are not currently discussed.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n\n- The paper has a polished and easy-to-read presentation, with many figures that help present the main idea and experimental results. \n- The paper does a good job of reviewing related works both on INR and related developments in geometric deep learning. \n- A wide range of experiments considering different tasks and families of graphs are used for evaluation.\n\n**Weaknesses**\n\n- Some experiments lack proper baselines and quantitative evaluation. Also, although quite pleasing to the eye, figures do not convey the quality of results since comparison to ground truth is missing.   \n- Discussions and experiments put more emphasis on reconstruction rather than generalization in using INR, especially generalization to new graphs where this approach can be most interesting could receive more attention. \n", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "PWJM9m8vSFK", "reviewer": "Reviewer_ofDT", "summary": "This paper generalizes implicit neural representation (INR) to arbitrary topological spaces beyond the regular Euclidean space. Technically, the authors utilize a graph as an approximation of topological space, and compute its spectral embedding as the point-wise encoding to regress an INR. In their experiments, they show the learned INR can be transferable to other graph approximations of the topological space and their techniques are also applicable to conditional INRs.", "questions": "- Solving spectral embedding suffers from the problem of scalability, stability, multiplicity and ambiguity of sign. How did authors solve these practical challenges in this paper?\n\n- Typical INR maps spatial coordinates to the corresponding value. How to bind generalized INR with positional information?\n\n- Can generalized INR solve differential/integral equations like SIREN?\n", "limitations": "Generalizing INR to arbitrary topology seems a promising direction. However, the strategy of  approximation via graphs seems to have some limitations. In vanilla INR, arbitrary coordinates in Euclidean space can be directly fed into the INR and query for the value. However, graph discretizes the signal and thus only vertices in the graph can be fed into the INR. That being said, the represented signals are no more continuous. Authors may discuss this limitation and provide preliminary solutions to it.", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n+ The paper is clearly written and easy to follow.\n\n+ The authors put forward an interesting question, i.e., how to learn an INR over the domain of arbitrary topology. They further propose a simple yet effective and well motivated method. It also has lots of practical usage in geometric, molecular, and climate data representation.\n\n+ The experimental discussion is comprehensive. It demonstrates different applications and verifies the transferability of the learned INR across graphs sampled from the same topological space.\n\nWeaknesses:\n\n- The reviewer cannot agree that the represented signals are still continuous. Approximating a topological space via a graph discretizes the signal space, which makes the input points restricted to the vertex set of the graph. To show the interpolation ability, the authors may first construct an over-dense graph, and then only train the INR on a subset of vertices.\n\n- In regards to the experiment, there is no comparison with other methods. To demonstrate the benefit of involving topological information, authors may compare their model with Euclidean domain INR. For example, also show the texture map regression results on SIREN.\n", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept