Concentric Spherical GNN for 3D Representation Learning
Authors: James S Fox
Bo Zhao
Sivasankaran Rajamanickam
Rampi Ramprasad
Le Song
Venue:  (2021)
Keywords: spherical cnn
GNN
graph convolution
rotation equivariance
3D

Abstract: Learning 3D representations that generalize well to arbitrarily oriented inputs is a challenge of practical importance in applications varying from computer vision to physics and chemistry.
We propose a novel multi-resolution convolutional architecture for learning over concentric spherical feature maps, of which the single sphere representation is a special case.
Our hierarchical architecture is based on alternatively learning to incorporate both intra-sphere and inter-sphere information. 
We show the applicability of our method for two different types of 3D inputs, mesh objects, which can be regularly sampled, and point clouds, which are irregularly distributed. 
We also propose an efficient mapping of point clouds to concentric spherical images using radial basis functions, thereby bridging spherical convolutions on grids with general point clouds.
We demonstrate the effectiveness of our approach in achieving state-of-the-art performance on 3D classification tasks with rotated data.

Under review as a conference paper at ICLR 2021
CONCENTRIC SPHERICAL GNN FOR 3D REPRESENTATION LEARNING
Anonymous authors Paper under double-blind review
ABSTRACT
Learning 3D representations that generalize well to arbitrarily oriented inputs is a challenge of practical importance in applications varying from computer vision to physics and chemistry. We propose novel multi-resolution convolutional ar- chitecture for learning over concentric spherical feature maps, of which the single sphere representation is special case. Our hierarchical architecture is based on al- ternatively learning to incorporate both intra-sphere and inter-sphere information. We show the applicability of our method for two different types of 3D inputs, mesh objects, which can be regularly sampled, and point clouds, which are irregularly distributed. We also propose an efficient mapping of point clouds to concentric spherical images using radial basis functions, thereby bridging spherical convolu- tions on grids with general point clouds. We demonstrate the effectiveness of our approach in achieving state-of-the-art performance on 3D classification tasks with rotated data.
INTRODUCTION
While convolutional neural networks have been applied to great success to 2D images, extending the same success geometries in 3D has proven more challenging. desirable property and challenge in this setting is to learn descriptive representations that are also equivariant to any 3D rotation. Co- hen et al. (2018) and EEteves et al. (2018) showed that the spherical domain permits learning such rotationally equivariant representations, by defining convolutions with respect to spherical harmon- ics. In practice, 3D convolutions are implemented via discretization of the sphere. Earlier spherical Convolutional Neural Networks (CNNs) used spherical coordinate grids, but these discretizations result in non-uniform samplings of the sphere, which is non-ideal. Furthermore, spherical convo- lutions defined these grids scale with O(N1.5) complexity (N as the number of grid points). Subequent works, Jiang et al. (2019), Cohen et al. (2019), Defferrard et al. (2020), designed more scalable ONN) convolutions focusing on more uniform spherical discretizations.
Existing spherical CNNs operate over a spherical image, resulting from projection of data to a bounding sphere. We show that it is centric, multi-spherical discretization for representing 3D data. Our main innovation is introducing new two-phase convolutional scheme for learning over a concentric spheres representation, by al- ternating between inter-sphere and intra-sphere convolutional blocks. We use graph convolutions to incorporate inter-sphere information, and 1D convolutions to incorporate radial information. Similar to Jiang et al. (2019) and Cohen et al. (2019). we focus on the icosahedral spherical discretization, which produces mostly regular sampling over the sphere. Our proposed architecture is hierarchi- cal, following the recursive coarsening hierarchy of the icosahedron. Combining intra-sphere and inter-sphere convolutions has a conceptual analogy to gradually incorporating information over vol- umetric sectors. At the same time, the choice of convolutions allows model to retain high degree of rotational cquivariance.
expressive and general to instead operate over con-
We demonstrate the effectiveness and generality of approach through two 3D classification periments with different types of input data: mesh objects and general point clouds. The latter poses an additional challenge for discretization-base methods, as native point clouds are non -niforllyyly distributed in 3D space.
To summarize contributions:

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1. We propose new multi-sphere icosahedral discretization for representation of 3D data, and show that incorporating the radial dimension can greatly enhance representation ability over single-sphere representations.
2. We also introduce a novel convolutional architecture for multi-sphere discretization by in- troducing two different types of convolutions, conceptually separated as intra-sphere and inter-sphere. Combining graph convolutions (intra-sphere) with ID radial convolutions (inter-sphere) leads an expressive architecture that is also rotationally equivariant. Our proposed convolutions are also scalable, being linear with respect to total grid size.
3. We design mappings of both 3D mesh objects and general point clouds to the proposed representation. We achieve state-of-art performance on ModelNet40 point cloud classifica- tion, using the proposed model and a data mapping using radial basis functions. We also improve on existing Spherical CNN performance in SHREC17 3D mesh classification by utilizing multi- raltis information.
2 RELATED WORK
Spherical CNNs. The goal of learning rotationally invariant representations of 3D geometries has led to several ideas for rotationally equivariant convolutions in the spherical domain. [Cohen et al. (211)) and Esteves et al. (2018) defined spherical convolutions that are rotationally equivariant to rotations of the (3(3) group. However, these convolutions are restricted to non-uniform grid sam- plings and scale superlinearly with respect to grid resolution. Later works have explored more scalable convolutions on other spherical discretizations, achieving linear complexity with respect to grid resolution.
Jiang et al. (2019) proposed using parameterized differential operators to form convolutional ker- nels over the icosahedron, where equivariance is restricted to rotations about the z-axis. [ohen al.) (2019) proposed gauge equivariant convolutions on manifolds, operating on feature fields corre- sponding underlying geometric entities. This was applied achieve rotationally equivariant con- volutions over the icosahedral discretization. Defferrard et al. (2020) propose graph convolution- based spherical CNN using spectral filters, along with eestance-weightee nearest-neighbors graph construction scheme that allows balancing between rotational equivariance and efficiency, when applied to different types of grids.
Other spherical CNNs have been designed in the context of handling arbitrary point cloud data, which typically requires first mapping the data to a discretization. |Roo et al. (2019) uses graph- convolution inspired message passing operators for learning over the icosahedral discretization. Our work is similar to Rao et al. (2019) and Defferrard et al. (2020)) in terms of using graph-based spherical convolutions, but generalize to multi- sphere convolutions. You al. (2020) is the most related work in terms of multi selti--sphere representation learning. The authors propose a spherical voxel grid, and extending the SO3 convolutions Cohen et al. (2018) to incorporate the radial dimension. Our work treats spherical and radial convolutions as distinct, which results in much better results in practice. We also use more scalable spherical convolutions defined on the uniform icosahedral grid.
Pointwise Convolution Networks. There is significant body of work on learning point cloud representations using pointwise convolutions, beginning with with Qi al. (2017) which proposed learning permutation invariant functions that directly operate on point coordinates. Only more re- cently have such methods have been developed towards learning rotationally invariant representa- tions. Thomas et al. (2018) and Poulenard et al.] (2019) both propose pointwise convolutional filters based on spherical harmonic functions to achieve rotational equivariance (or invariance). Distance information is recorded through learned functions in the former, and radial sampling in the latter. While these filters are defined with respect to all-to-all convolution between points, in practice con- volutions are limited to k-nearest neighbors (Poulenard et al. (2019)) for scalability. Chen et al. 20119, Sun et al. (2019) Zhang et al. (2019) all extract rotationally invariant features (i.e. low-level geometric features such as angles and distances) from the point cloud as input to their respective convolutional architectures. These features are hand- hand-engineered based on carefully picking local frames of references, or global ones in the case of (un et al. (2019).
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Figure 1: (a) shows an example point cloud (black dots) contained within a bounding sphere. (b) shows the spherical partioning of 3D space in 2D cross section view, and zooms in on sector occupied by 3 data points. Vertices are white circles. Each point bounded by a neighborhood of 6 vertices, 3 from the sphere above and 3 from below. (c).(d) Each point is mapped to scalar values defined on the bounding vertex neighborhood using radial basis functions. Vertices affected by the mapping shaded gray. Dotted circles indicate vertices temporarily added in the radial dimension to increase resolution. (e) Vertex values are concatenated into feature channels of original vertices.
(e)
Figure 2: Two subsets of vertices from two concentric spheres, connected radially. 4 or Us are vertices on i-th sphere (a) Intra-sphere convolution and (b) inter-sphere convolution applied with respect to the target vertex u2 (bolded). Third sphere not shown for clarity. Vertices involved in convolution are connected by orange or green edges.
3 REPRESENTATION BY CONCENTRIC SPHERES
Existing work spherical CNNs operate on spherical grids, where data is typically projected to and defined on grid points. However, projecting 3D data to single sphere may not always be sufficient or appropriate. Simple projections may be lossy when describing highly non-convex shapes, for instance if the shape curves in on itself. To increase capacity to distinguish different data distribu- tions, we introduce new discretization based on concentric spheres, which additionally discretizes 3D space in the radial dimension. The single sphere discretization is special case in our proposed paradigm.
Spherical Discretization. We work with an icosahedral grid discretization of the sphere. The base icosahedron (0) has 12 vertices, forming 20 equilateral triangle faces (cach face with 3 edges). Each vertex is incident with triangles. Each face can be subdivided. with the number of vertices scaling as V = 10 x 4' + 2 (where / is the discretization level). See Fig. 3 for an illustration.
Radial Discretization. We construct the tical icosahedral grids. Assuming unit radius normalization, we use a uniform discretization that results in concentric spheres scaled radii 1 R ......1]
spherical discretization by stacking R iden-
Intra-sphere Convolutions. There is a growing body of work addressing design of rotation- equivariant filters over spherical feature maps. We focus on graph convolutional filters for intra- sphere convolutions, as graph convolutions are scalable and lead to equivariant representations, up to discretization effects Defferrard et al. (2020), Yang et al. (2020). This motivates our construction of the undirected graph ( (V(), EU) from level icosahedron 70) Vertices of the vertex set (f) correspond one-to-one with vertices of (t) projected to unit sphere. E()) is simply the set of all (bidirectional) face edges of the icosahedron (projected to unit sphere). Vertices are all degree 6, with exception of the the initial 12 vertices of the base icosahedron (U) that are degree 5, Since each edge is approximately equidistant between two points of the sphere Wang & Lee (2011). G()) is also treated an unweighted graph.
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While there is rich body of work graph convolutions and its variants, this work focuses on the graph convolution defined in Kipf & Welling (0077). We introduce notation to define this convolu- tion the context of our multi-spherical discretization in more detail. Let H E RXXIVXX denote a C' channel tensor of features. Also let Z E gCxF be shared parameters, where C and F are input and output channel sizes. N (u) denote neighbors of vertex u in graph G and deg(u) denotes |N (u)], the degree of vertex We assume that self-edges are added for every vertex in N()). Finally, introduce the subscript & to be the convolutional layer number, E [0, R 1] to index the radial dimension, and E [V] 1] to index the vertices. The layer L ++ 1 intra-sphere convolution output for vertex u of sphere is given by Eq. where a indicates a nonlinear activation function.:
((+1)) S,
= =o Σ
1
(1)
ddgg(u) deg(v)
vEN(u)
Inter-sphere Convolutions. We introduce radial convolutions incorporate inter-sphere infor- mation, implemented as ID convolutions where the radial dimension is treated as the sequence length. Importantly, radial convolutions are also rotationally invariant, as ID convolution operates over channels of the same vertex. See Fig. 2 for illustration of graph convolutions with respect to concentric spheres representation. We introduce some additional notation to describe radial convo- lutions. Let K be 1D convolution kernel size. We assume K' is odd valued, and pad inputs in the radial dimension such that a dimension of R is maintained across convolutions. Let W E IR XxCxF be a tensor of shared parameters, where C' and F are input and output channel sizes. The layer & ++ inter-sphere convolution output for vertex u of sphere is given by Eq. 2
H (++1) a
Σ
H
W
K+141
Vertex Pooling. Pooling is widely used alongside convolutional filters in CNN architectures to learn invariance to transformations of the input. The icosahedron, due to its recursive refinement by discretization level, defines natural hierarchy for pooling and downsampling (see Fig. 3 We
introduce overloaded notation (() E [x [xC to denote the feature tensor corresponding to
VOO the vertex set corresponding to level 1 icosahedron. We define pooling as H = f((H))
E N()))), where N(u) is the neighborhood of vertex u E V(O and f a permutation invariant function (e.g. max operator). Pooling is followed by downsampling, where only vertices of the smaller vertex set (((+1) are retained (see Sec. A.4 for details.) Only vertices within same sphere are involved; there no pooling or downsampling of vertices between spheres (except right before classification).
Concentric Spherical GNN (CSGNN) Architecture. Fig. 3 gives an example illustration of an end-to-end architecture using both convolutions. Importantly, radial convolution blocks are intro- duced alongside graph convolutional blocks at every level of the spherical discretization hierarchy, to incorporate inter-sphere information gradually. From a icosahedron of level L refinement, we construct a sequences of graphs (G(L), ((L-1), .... GXO)). Each (i)) carries an additional R di- mension, corresponding to spheres at different radial levels. Each level 1 features two blocks of convolutions: graph convolutions, followed by radial convolution. These correspond to intra-sphere and inter-sphere convolutions respectively. Vertex neighborhood pooling downsamples the graph from GO to ((!-1), The size of the radial dimension remains constant, until final pooling.
Complexity Analysis. In the intra-sphere convolution of Eq. [ the neighborhood size (vertex degree) is bounded by a constant. Dimensions of parameters and feature vectors are also bounded by a constant. Therefore, the overall complexity of intra-sphere convolution is linear with respect to the multi-spherical grid size, or (R)VV)), where |V| is the number of vertices in the spherical grid. R is can be effectively kept very small relative to [V], as we show in our experiments. Similarly, each inter-sphere convolution (Eq. 2 with respect to a vertex is bounded a fixed kernel size for the neighborhood. The overall complexity of inter-sphere convolution is also O (R) and both convolutions introduced in this work are linear with respect to the multi-spherical grid size. We
further provide experimental parameter analysis in Tables and time analysis in Appendix A.2
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Vietex
Radius
Radius 2
Pool
Verdex
Radius
Figure 3: Example multi-radius architecture with = 3 concentric spheres. Graph convolu- tions, followed by radial convolutions, applied over sequence of discretization levels. Pooling coarsens the discretization to a lower level. Vertex-wise and radial-wise pooling is applied to obtain final representation for classifier. Icosahedron visualization from Satoh et al. (2014).
4 POINT CLOUD TO CONCENTRIC SPHERICAL SIGNAL
We consider the problem of mapping a point cloud P E NNx point cloud to an initial spherical feature map E IR. R where C' is number of input channels. While the concentric grid representation defined discretely at fixed positions, the space of data point locations is continuous. We also aim to capture the distribution of points in a continuous way. To do so, we summarize the contribution of points using the Gaussian radial basis function (RBF):
N
f(x)
a
P P)
N is the number of data points, and exp( --))) parameterized by the bandwidth In practice we limit computation to local neighborhood (instead of considering all points), and choose ? accordingly. See Fig. for visualization of the local neighborhood and mapping, and Sec. A.I for additional details.
One possible mapping is to compute Eq. every vertex position of the spherical discretiza- tion, resulting in a single channel feature map. However, it is possible to obtain better resolution in capturing distribution of surrounding points by further sub-diving the discretized space, taking inspiration from Meng et al.] (2019), along the radial dimension. Subdividing along radial dimen- sion by factor K2 results in new spherical discretization with increased radial dimension of R' = Rt * Ke. The RBF is evaluated at every vertex position of this new discretization, resulting in feature map of dimension of M' E R x We map back to the original discretization by assigning M... [M
(iKe, iK 1, .... 2 2 + 1.... 3iK, 1)}, resulting size [R |V| 2k.] spherical feature map. In summary, multiple RBF values assigned each vertex by further sub-dividing space in the radial dimension.
EXPERIMENTS
MODELNET40 POINT CLOUD CLASSIFICATION
We consider the ModelNet40 3D shape classification task, with 12308 shapes and 40 classes. Each point cloud has 1024 points. For all experiments, 9840 shapes are used for training and 2468 for testing.
Architecture and Hyperparameters
Fig. shows complete architecture overview. Point clouds are first mapped to 16 concentric
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spheres with level 4 icosahedral discretization (L 4, = 16), using RBF kernels with threshold 0.01. ID convolutions use kernel size of Each graph and 1D convolution is followed by batch normalization and ReLU nonlinear activation. Additionally, skip connections are added between every graph convolution layer, whenever input dimension matches output. The model trained with Adam optimizer for 30 epochs using initial learning rate of 1e-3, along with learning rate decay by 0.1 at 15 and 25 epochs. Batch size is 32.
Gronj
Vantax
Vertte
Fool
Verdez
Pool
Verso
Veree
Pool
Radial Pool
Goond)112d11
Conrled
Consteproddi
Figure 4: Architecture for ModelNet40 classification. Input dimension is 16, resulting from point cloud RBF data mapping- "Geonv" is graph convolution applied over graph connectivity of the sphere. "Convld is 1D convolution, applied over the radial dimension. I denotes discretization level, as the representation is coarsened following each vertex pooling step. A final pooling of radial dimension results in 1024 dimensional vector.
Results
Method
Input
Params
z/z
z/SO3
NR/SO3
SO3/SO3
Pointwise Convolution
PointNet Qi et al. (2017
xyz
3.5M
0.875
0.229
0.081
0.849
ClusterNet Chen et al. (201
xyz
0.871
0.871
0.871
RIConv Zhang ct al. (2019)
xyz
0.7M
0.870
0.870
0.872
0.872
SPHNet Poulenard et (2019)
Xyz
2.9M
0.865
0.856
0,854
0,870
SRINet Sun et al.
2019
xyz+
0.9M
0.844
0.829
0.834
0.837
normal
Spherical CNN
SFCNN Rao 2019
xyz
9.2M
0.888
0.831
0.350
0.874
PRIN You et al. 2020
xyz
1.7M
0.819
0.765
0.753
0.810
Ours (CSGNN)
xyz
2.8M
0.884
0.874
0.833
0.884
ModelNet40 classification results, across four train/test data orientation settings. NR de- notes original data (no rotations), is arbitrary rotation about axis, and SO3 is arbitrary rotation. For example, SO3/SO3 means training and testing with arbitrary rotations the data. Params is number of parameters in millions.
Table 1:
We present results and compare against other related works in Table in four different train/test data orientation settings. When training with rotations, a new rotation is sampled per instance in each epoch. Rotation the only augmentation used in comparisons. We report accuracy average validation score across last 5 epochs of training, due to lack of standard validation/test split and to account for variation.
Our method achieves state-of-the-art results in z/SO3 and SO3/SO3 settings, i.c. testing on arbitrar- ily rotated data. For more detailed comparison, we loosely categorize compared works by method into two categories: pointwise convolution networks and spherical CNNs. Methods in the former category operate directly on data points in 3D space, while methods in the latter operate on a spher- ical discretization. Our work is most closely related to methods in the spherical CNN category.
Similar to our work, PRIN also explored learning concentric spherical representation based on extending SO3 convolutions from Cohen et al. (2018). Our method is based on separate graph and radial convolutions, which achieves much better performance in all settings. SFCNN has similar- ity to work in using graph convolution-inspire message passing filters and learning over the icosahedral discretization of the sphere. However, SFCNN is restricted to a single-sphere repre- sentation, and also relies on PointNet-like learned module project points to spherical features.
Using available numbers reported in authors' paper, as the authors' code was not publically available at the time of this writing.

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While this learned projection should be able capture some degree of multi-radius information in the point distribution, best results seem to be achieved by learning from both intra-sphere and inter- sphere convolutions. Compared to approach, SFCNN also has relatively higher performance gap between z/z. and any SO3 test setting (significantly so for z/SO3 and NR/SO3), which suggests greater difficulty achieving rotational invariance. CSGNN, similar to SFCNN and PRIN, exhibits some drop in performance in the z/SO3 and NR/SO3 settings, This could partly be due to effects of discretization and data mapping. However, this gap is relatively smallest in CSGNN, and there is essentially no gap in z/z and SO3/SO3 performance.
ClusterNet, RIConv, and SRINet use hand-crafted rotationally invariant geometric features as inputs, and so there is negligble to no performance gap in testing with or without rotations. By contrast, our method largely learns to extract features directly from the input (outside of an initial step map- ping points to vertices). PointNet, unlike the other baselines, was not designed to be rotationally equivariant. This reflects in the relatively large difference in performance when comparing testing with without rotations. Even when training with rotations and using a learned alignment module that attempts to learn a canonical transformation, PointNet SO3/SO3 performance is not competitive with that of most other baselines.
5.2 SHREC17 3D SHAPES
The SCHREC17 task has 51300 3D models and 55 categories. We use the version where all models have been randomly perturbed by rotations. Here the inputs are not point clouds, but mesh objects. Cohen et al. (2018) Esteves al. (018)) presented a ray-casting scheme regularly sample in- formation incident to outermost mesh surfaces and obtain features maps defined over the spherical discretization. For sufficiently non-convex mesh objects, a single sphere projection may result in information loss. For example, when a ray is incident to multiple surfaces occurring at different radii, this information is discarded by existing methods. We propose a new data mapping that gener- alizes single sphere representation to a concentric spherical representation, thereby preserving more information. Fig. g in the appendix shows visual examples of where the proposed representation may be helping.
Representation
In the case of single-sphere representations, a single ray is projected from source point (vertex) on the enclosing sphere towards the center of the object. The first hit incident with the mesh recorded. To extend ray-casting to multiple concentric spheres, we rescale the source point to the radii of each respective sphere. This results in multiple co-linear source points, one per sphere. The 1st hit incident with the mesh is recorded for each ray cast from those source points, resulting in a multi- radius projection. While this new scheme is not sufficient to capture all incident surface information (e.g. if there are multiple hits sandwiched between two radial levels). it provides more samples that scales with the number of spheres. We use a uniform 1 / .... 1) radii division assuming inputs are normalized to unit radius. From each point of intersection with the mesh, the distance (with respect to outermost sphere) to the point of incidence as well as sin and cos features are recorded, resulting in features per vertex. These are similar features to those collected in related work, except we do not use the object's convex hull information.
Method
Params
F
Cohen et al. (018)
(equiangular, b 64)
0.4M²
0.7892
Esteves et al.
(2018)
(equiangular, b = 64)
0.5M²
0.7942
DeepSphere Defferrard et al. 2020) (equiangular, b 64)
0.2M²
0.7942
DeepSphere Defferrard et al. (020) (HEALPix, Nuide 32)
0.2M²
0.8072
CSGNN (icosahedral, 4, R=
1.3M
0.805
CSGNN (icosahedral, -= R- 16)
2.9M
0.823
Table 2: SHREC17 classification performance in terms of Fi metric (micro-average). CSGNN is implementation. Equiangular, HEALPix, and icosahedral different discretizations of the sphere. CSGNN (this work) uses level 4 icosahedral discretization, R is number of concentric spheres (specific to this work). Params is number of parameters in millions.
Numberrs as reported in Defferrard 2020
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Architecture and Hyperparameters
The architecture for SHREC17 is identical to the one used for ModelNet4O in Fig. with the exception that the imput dimension is 3 (corresponding to features obtained from ray-casting). We consider two model variations, single-sphere (R = 1) and multi-sphere (R = 16). For R = 16, we 1D convolution kernel of size 1. Note this
1D convolution kernel of size 3. For R 1, is equivalent to applying fully connected layers; we found adding additional fully connected layers after graph convolutions helped improve performance in the single-sphere case. Training settings are also same as for ModelNet40, except learning rate decays at epochs 10 and 20.
Results
See 2 for classification results. The reported metric is Fi micro-average classification score. Results from three other Spherical CNN works designed for general rotational equivariance are cluded for reference. DeepSphere is most similiar to our work in terms of graph-based spherical convolutions, where the authors explored design of paationally-equivariini ters with respect the type of grid and neighborhood size. This work focuses graph construction to the icosahedral discretization, using a minimal set of roughly equidistant neighbors.
graph convolutional fil-
We additionally introduce inter-sphere convolutions with concentric spheres, which is largely or- thogonal to the design of intra-sphere representation and convolutions. Single-sphere (R 1) CSGNN achieves competitive performance with other single-sphere baselines. However, it is diffi- cult to draw comparative conclusions in this particular case, due to differences in feature extraction, spherical discretization type and size, and model size. More significantly, using multiple spheres (R = 16) outperforms single-sphere baselines, including a 2.2% relative performance improvement over our R version. It also seems likely that the concentric spheres approach can improve other spherical convolutional designs as well, but this is beyond the scope of this work.
5.3 MODEL ARCHITECTURE ABLATION STUDY
To study the impact of multi-radius spherical discretization, we vary the number of radial levels and present results in Table 3 ModelNet40 is used for all ablation experiments. We use a base model with R 16 and L model. For this particular ablation, for simplicity we use single channel, indicator feature map a special case of the RBF mapping where 7 and F 1. Adding radial convolutions in the case of 1 is equivalent to adding additional dense layers after graph convolutions. The architecture the same as the one in 4 except input dimension is 1. Performance consistently improves with higher radial dimension, peaking R = 16 with 4.8% relative accuracy improvement over the R = 1 version. Performance declines from R = 16 to = 32, which suggests diminishing returns for fixed parameter budget.
and keep the number of parameters identical across all versions of the
Setting S03/S03
R=1
R 4
R = 8
R R= 16
R = 32
0.839
0.857
0.869
0.879
0.872
Table 3:
ModelNet40 ablation with number radial levels (R).. Number of model parameters fixed across all settings.
More ablation studies are presented in Table We study the impact of varying the radial kernel size KRC [1,3, 3.... KRC = 1 is same as learning representations independently learned at each radial level. While this still improves over single-sphere representation, using spatial filters nc = [3. 5)) over the radial dimension is important for best performance. Varying the number of graph and radial convolutional layers shows that between I and 2 layers per block leads to comparable performance. Finally, we compare using only graph convolutions or only radial convolutions. Results suggest that it is essential to combine both types convolutions for best performance. Interestingly, restricting to radial convolutions achieves slightly better performance than restricting to graph convolutions over the single sphere. This provides further empirical support for the expressiveness of our proposed representation and radial convolutions.
5.4 POINT CLOUD DATA MAPPING ABLATION STUDY
In this section we compare alternative choices of neighborhood the mapping of points to vertices, well as alternative functions the Gaussian radial basis function (RBF). The bounding vertices
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Setting Radial kernel size
S03/SO3
Setting Convolution layers
S03/SO3
KBC = 1, Mcc = 1, MC = 1
0.853
KRC = 3, Mcc 1, MRC = 1 0.880
KRC 3, Mec 1, RR
0.880
KRC 3, Mac 2, MRG
0.876
K KRC- = 5, Mac = 1, MRC =
0.882
Graph convolution only
Radial convolution only Knc = 3, MRC =1
R = Mac =1
0.837
0.845
Table 4: Ablation study on ModelNet40. KRC is size of radial convolutional kernel, Mac and MR are number of graph and radial convolutional layers per block. R and L 4, unless stated otherwise.
Name
Function
SO3/SO3
Neighborhood
S03/SO3
Constant
((r) =1
0.879
k-NN, k 4
0.875
Linear
((r) = max 0)
0.877
k-NN, A: 8
0.870
Inverse Quadratic
<(r: Ityn
0.882
k-NN, K 16
0.875
Gaussian
(r: = ( 0.884
Bounding vertices
0,884
Table 5: Ablation study on mapping point clouds to concentric spherical signal with ModelNet40. Left table compares different decay functions. Right table compares k-nearest neighbor approach with bounding vertices approach. The former is vertex-centric, considering points are neighbors, whereas the latter is point-centric, influencing its neighboring vertices.
approach (detailed in Sec. A.1 is point-centered, in that each data point influences its immediate surrounding vertices. These vertices form two triangles which "bound" the point in vertices in space (see Fig. corresonding
neighhorhood of 6 vertices. An alternative is to define vertex-centered neighborhood consisting of the k-nearest points to each vertex. We use the Gaussian RBF to map the contribution of each point based on its distance to the vertex. The Gaussian parameter 7 is chosen such that the contribution of points to a maximum distance of πf decay to small threshold value (I' 0.01 in this case). Results from Table 5 show the bounding vertices approach outperforming the vertex-centered k-NN approach. One possible reason is the difficulty in controlling for scale, especially when combined with fixed- sxre-parreeter decay function, as vertices closer to the center are more densely situated in space compared to vertices further from the center.
We also compare four different decay functions applied to bounding vertices neighborhood. The constant function is trivial in that it is independent to distance, and essentially serves an indicator. The linear function decays to zero with distance, but is non-smooth. c is a specified parameter for distance cutoff. Inverse quadratic and Gaussian are two smoothly decaying functions commonly used as radial basis functions. The latter two perform slightly better than the constant and linear functions, suggesting some advantage to using RBFs to summarize neighborhood contributions.
6 DISCUSSION AND CONCLUSIONS
multi-sphere convolutional architecture, CSGNN, for learning tationally invariant representations of 3D data. We introduced distinct intra-sphere and inter-sphere convolutions, which can be combined to learn more expressive representations compared to being re- stricted to single-sphere representation. Our use of graph and ID convolutions preserves rotational cquivariance, while achieving linear scalability with respect to size of discretization. We achieve state-of-the-art performance in ModelNet classification for testing on arbitrary rotations among both spherical CNN and pointwise convolutional models. We also show that our approach generalizes to classification of 3D mesh objects by improving on single-sphere representation and performance for the SHREC17 task. One avenue of future work is to explore more descriptive mappings of point cloud data to the discretization. A learned assignment may better learn vertex features for de- scribing nearby points. There is also room to explore other kinds of convolutions for incorporating inter-sphere information,
In this work we proposed
well as other radial division schemes. Finally, existing implementa- tions in this work can be more efficient implementations based on using regular properties of the icosahedral grid, as opposed to using general graph construction.

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REFERENCES
Chao Chen, Guanbin Li, Ruijia Xu, Tianshui Chen, Meng Wang, and Liang Lin. Clusternet: Deep hierarchical cluster network with rigorously iaation-nnaariant representation for point cloud anal- ysis. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2019.
Taco Cohen, Maurice Weiler, Berkay Kicanaoglu, and Max Welling. Gauge equivariant convolu- tional networks and the icosahedral CNN. In International Conference on Machine Learning, pp- 1321-1330, 2019.
Taco S Cohen, Mario Geiger, Jonas Köhler, and Max Welling. Spherical CNNs. Conference on Learning Representations, 2018.
International
Michal Defferrard, Martino Milani, Frdrick Gusset, and Nathanal Perraudin. Deepsphere: graph- based spherical cnn. In International Conference on Learning Representations, 2020. URL
htttp:://openrewe
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Carlos Esteves, Christine Allen-Blanchette, Ameesh Makadia, and Kostas Daniilidis. Learning SO(3) equivariant representations with spherical CNNs. In Proceedings of the European Con- ference on Computer Vision (ECCV), pp. 52-68, 2018.
Chiyu Max Jiang, Jingwei Huang, Karthik Kashinath, Prabhat, Philip Marcus, and Matthias Niess- ner. Spherical CNNs on unstructured grids. In International Conference on Learning Represen- tations, 2019, URL
net/ forum Battrummid-Bkkk-4400 43C9FQ
Thomas N Kipf and Max Welling. Semi-supervised classification with graph convolutional net- works. International Conference Learning Representations, 2017.
Hsien- Yu Meng, Lin Gao, Yu-Kun Lai, and Dinesh Manocha. Vv-net: Voxel vae net with group convolutions for point cloud segmentation. In Proceedings of the IEEE International Conference on Computer Vision, pp. 8500-8508, 2019.
Adrien Poulenard, Marie-Julie Rakotosaona, Yann Ponty, and Maks Ovsjanikov. Effective rotation- invariant point cnn with spherical harmonics kernels. In 2019 International Conference on 3D Vision (3DV), pp. 47-56. IEEE, 2019.
Charles R Qi. Hao Su, Kaichun Mo, and Leonidas Guibas. Pointnet: Deep learning on point sets for 3d classification and segmentation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 6602 2017.
Yongming Rao, Jiwen Lu, and Jie Zhou Spherical fractal convolutional neural networks for point cloud recognition. In Proceedings of the IEEE Conference Computer Vision and Pattern Recognition, pp. 452-460, 2019.
Masaki Satoh, Hirofumi Tomita, Hisashi Yashiro, Hiroaki Miura, Chihiro Kodama, Tatsuya Seiki, Akira T Noda, Yohei Yamada, Daisuke Goto, Masahiro Sawada, et al. The non-hydrostatic icosa- hedral atmospheric model: Description and development. Progress Earth and Planetary Sci- ence, 1(1):88, 2014.
Xiao Sun, Zhouhui Lian, and Jianguo Xiao. Srinet: Learning strictly rotation-invariant representa- tions for point cloud classification and segmentation. In Proceedings of the 27th ACM Interna- tional Conference on Multimedia, pp. 980-988, 2019.
Nathaniel Thomas, Tess Smidt, Steven Kearnes, Lusann Yang, Li Li, Kai Kohlhoff, and Patrick Riley. Tensor field networks: Rotation-and translation-equiirrian neural networks for 3d point clouds. arXiv preprint arXiv: 2018
Ning Wang and Jin-Luen Lec. Geometric properties of the coosahedral-hexaagoee grid on the two- sphere. SIAM Journal Scientific Computing, 33(5):2536-255 2559, 2011.
Qin Yang, Chenglin Li, Wenrui Dai, Junni Zou, Guo-Jun Qi, and Hongkai Xiong. Rotation equiv- ariant graph convolutional network for spherical image classification. In Proceedings of the IEFF/CVF Conference Computer Vision and Pattern Recognition, pp. 4303-4312, 2020.
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Yang You, Yujing Lou, Qi Liu, Yu-Wing Tai, Weiming Wang, Lizhuang Ma, and Cewu Lu. Point- wise rotation rotation-invariane invariant network with adaptive sampling and 3D spherical voxel convolution. The AAAI Conference Artificial Intelligence, 2020.
Zhiyuan Zhang, Binh-Son Hua, David W Rosen, and Sai-Kit Yeung. Rotation invariant convolutions for 3d point clouds deep learning. In 2019 International Conference on 3D Vision (3DV), pp. 204- 213. IEEE, 2019.
A APPENDIX
A.1 POINT CLOUD TO SPHERICAL. SIGNAL
Instead of computing the summation in Eq. 3 with respect to all points, for each data point we update the features of vertices in local neighborhood. The radial basis function decays exponentially, and so points beyond a local neighborhood have little influence (depending on choice of bandwidth 7). Restricting a constant size local neighborhood improves computation from ((NV) O(N).
To define the local neighborhood of data point in this work: any given point is contained within two bounding "triangles" of the discretization (ignoring boundary conditions and degenerate cases). These correspond to the vertices S()) {u(f) v(9), vo(0) and S(++1) (u((+1), u((+1), w((+1)),, where indexes radial level. However, using a single value for the RBF results in scaling incon- sistency: distances between vertices progressively shrink moving to inner spheres. Based the assumption that RBF values should be invariant to scale, a different y and corresponding RBF is defined with respect radial level. To define 7/s tween vertices in ( (S(+1)) Specifically, set 7/ d(t)
the maximum pairwise distance ( be- log T where T is lower bound target
RBF value. For example, T = 1 would correspond to % = O, or RBF value of any distance. E (0, 1] is a tuning parameter that enables toggling the overall sensitivity of the RBF to distances.
Based on the approximation that (i) is similar for any data point, d is precomputed once.
A.2 NET TIME ANALYSIS
Baseline CSGNN
PointNet
RIConv
SPHNet
SRINet
SFCNN
PRIN
Training (hrs)
5.1
3.3
1.9
2.4
4.8
6.1
Inference (S)
0.16
0.016
0.062
0.099
0.097
0.069
0.004
Time comparisons of baselines for ModelNet40 dataset. Total training times reported in hours. Inference time, in seconds, is for batch size of 32. CSGNN our model.
Table 6:
We compare total training time and batch inference time of baselines from Sec. 5. for all instances where code was available. Results are reported in Table 6 All baselines were run on the NVIDIA Tesla P100 GPU. Total training time includes data loading and transformation time. Inference time is based on batch size of 32, and computed from averaging 32 different batches. The inference times reported do not include data loading or transformation time.
A.3 CHOICE CENTER
Center
Centroid
Component 0.878
Component 2 Component 3
S03/SO3
0.884
0.873
0.866
Table 7: Comparison of different choices center. Centroid is the positional average of the point cloud. Components refer to the right singular vectors from singular value decomposition of the point cloud, decreasing order of singular value. The components serve as alternative centers. Performance accuracy from training and testing with rotations.
A choice of center is needed to represent the 3D data using concentric spheres, and also is necessary to define rotations of the data. The ModelNet40 dataset from MinQQii et al. (2077) centers the data based
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(GGC22 + BN + ReLU x 2) + [RC32 + BN + ReLU RC + BN + ReLU
MarPool
+ ([GC61 + BN + ReLU L3 2) + [RCss + BN + ReLU + 12C + BN + ReLU L3
MaxPool
+ ([GC128 BN + ReT.U 1. x 2) [RC128 + BN + ReLUJ + RRC256 + BN + ReLU
MarPool
+ [GC256 + BN + ReLU L1 + [RCs12 + BN + ReLU + MaxPool
[GCs12 BN ReIUU Lo G. Perticcs [RC1024 BN + ReLU Lo GG
+ FC + Soflmax
Figure 5: Architecture used for Sec. 5.1 and 5.2 experiments. Layers proceed from left to right. GC and RC indicate graph and radial convolution layers, subscripted by output channel dimension. BN is Batch Normalizatinn, FC' is fully connected layer. GP indicates global pooling with max operation. Subscript "L" indicates icosahedron level, corresponding discretization level of the sphere.
Figure 6: A vertex (red) and its neighbors (yellow/orange) shown in local patch of the icosa- hedral spherical discretization. These are the basic units for intra-sphere convolution and pooling. Downsampling results in coarser grid and vertex set, where neighborhood is defined (orange vertices)
on the centroid of a point cloud, while the SHREC17 dataset uses the centroid of mesh shapes. Our experiments in Sec. [.1 and 5.2 follow this convention. In this section also consider other choices of center besides centroid, and show that our representation relatively robust to other centers of rotation, in application to the ModelNet40 dataset. We apply singular value decomposition to an input point cloud to obtain principal component vectors. These dimensional vectors each serve new reference point by which the point cloud is re-centered. Results from Table / show that centroid-basee center produces performs best with our model. Performance of principal component- based center declines according to singular value. A possible reason for worse performance in the non-centroid center examples is that for the same number of points, a relatively smaller fraction of the concentric spherical discretizaton is utilized (following re-centering), effectively lowering the resolution of the input represenation. Overall the results suggest that CSGNN can be effective even when using con-centroid centers of representation and data rotation.
A.4 ARCHITECTURE DETAILS
Fig. c provides visualization of the vertex pooling and downsampling operations. Fig. provides more detail to the base architecture used in our experiments. Radial convolutions are implemented using 1D convolution layers with kernel size of 3 in the ModelNer40 experiment, and either 1 or the SHREC17 experiment. Radial convolutions are followed by max pooling and downsampling to lower level of spherical discretization, reducing the number of vertices. We also add residual connections between every graph convolution layer, whenever the number of input channels matches output channels.
A.5 VISUALIZING CONCENTRIC SPHERICAL FEATURES
To better understand the impact of the multi-sphere aspect of the concentric spherical representation, visualize the learned features from 3 different point cloud examples in Fig. 7 We the 1st
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Figure 7: Visualization of point clouds from ModelNet40 and learned features. Example instances from left to right, shown in 3 different orientations: airplane, sofa, and toilet. In the table of spherical visualizations, each sphere corresponds to a single feature channel. Rows correspond to radial level (16 total), with bottom rows corresponding to outer spheres. Columns correspond to discretization level of the sphere, from level 4 to 3 to 2 (left to right). Colors are interpolated between blue and red, corresponding to low or high normalized feature values.
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(a) Remote
(b) Watereraft
(c) Cabinet
(d) Table
(e) Lamp
(1) Tower
(g) Dish washer
(h) File cabinet
Figure 8: SHREC17 mis-predicted class pairs from single-sphere model where the multi-sphere (R = 16) model showed biggest relative improvement. Each image is a representative sample from the class. Note that watercraft, table, and tower all have more non-convex features that distinguish them from their mis-predicted counterparts. The concentric spherical model seems to better capture these differences.
feature channel in each visualization instance, and the features are obtained after radial convolution layers but before pooling and downsampling. The results visually demonstrate that different radial spheres within the same representation are capturing different features. At the same time, there is also a high degree of continuity between consecutive spheres in most cases, suggesting that there information sharing between spheres resulting from radial convolutions.
A.6 SHREC17 VISUALIZATION
See Fig. g for SHRECI7 visualization.
14


Reviews:
[{"review_id": "pSyLMymRuV2", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Summary:\nThis paper presents a novel multi-sphere icosahedral discretization for representation of 3D data. Given meshes or point cloud, the authors map them to multiple layers spheres and apply graph conv on the spheres to learn rotation-invariant features. In the final stage, all the layers are merged via a radial pool operator. The authors claim that such a structure could better preserve information of 3D objects and showcase on mesh and point cloud classification tasks.\n\nComments:\nThe authors extend the spherical CNN from a single layer to multiple layers and propose corresponding graph conv and pool operations for it. However, I have several concerns, as seen below:\n\nHierarchy v.s. partition: In my understanding, the authors use RBF to map each point to its nearest sphere vertex, as shown in Fig.1. Does it indicate the multilayer spherical CNN is not hierarchical but more about partition?  Also, RBF needs some explanation. I only see the abbreviation.\n\nMore explanation should be added in Sec.5.2, especially the way to do ray marching in multi-layer spherical CNN. It is a valid concern that the regular ray marching in Esteves 2018 would lose information for non-convex objects. But after reading the section, I am not exactly clear about how you preserve information for the multiple surfaces. Also, such new ray marching is more about hierarchy or partition?\n\nIn the experiments, I find the authors directly use the numbers in others' papers. This is not super fair since the experiments may be done in different settings. Why the number in Table 1(16 layers, SO3/SO3 is 0.884) and Table 3(16 layers, SO3/SO3 is 0.879?) are different?  Also, some time analysis is worth adding.\n\nConclusion: Overall, I think extending the spherical CNN from 1 layer to multiple layers is an interesting idea and one can foresee that such an extension would bring better performance, as shown in the paper. I think validating such an idea is a solid contribution and would bring insights to others. Thus I would like to accept this paper.\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "khRsLZyDkNV", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper proposes a Spherical GNN approach for effective representation of 3D data (point clouds and surface meshes). \nThe main ideas are interesting and reasonably well presented. The results are convincing, being comparable to or better than SOTAs on benchmarks. \nHowever, I am not an expert in this area: It seems to be an interesting new idea proposed in this work; meanwhile, I feel a lack of sufficient technical details for reproduction.\nIt would be helpful if the authors could explain some of the key ingredients, such as the radial convolutions, inter/intro sphere convolutions in greater details. The proposed NN architecture, as illustrated in Fig. 3, still lacks in many details, and possibly is not sufficient for a faithful reimplementation/reproduction by other researchers.\nThe authors should make their code and processed data and results publicly available.\nThe empirical evaluation is also somewhat lacking in its breadth and depth. The shapenet, for example, also contains lots of 3D shapes of various types of objects, and could be a good dataset for evaluaiton. In terms of applications, it would be helpful to showcase a few application studies and to demonstrate the advantages of adopting the proposed scheme.\nThe authors should also present more visual results of their 3D shape representation in a supplementary video.\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "FvTuX_SAT4", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper addresses an important problem in 3D representation learning, which is how to design a robust convolution neural network for arbitrarily oriented inputs. They propose to use multi-sphere icosahedral discretization to representation 3D data at first. And then, several alternative convolution operations are used to further process features. The novel convolution operation is consisting of intra-sphere and inter-sphere convolution. They also design a mapping function to convert point cloud data to mesh data. The experimental results indicate the model can work well.\n\nStrengths:\n1.\tA novel method to discretize space, which can be used for represent 3D data.\n2.\tThe convolution combines intra-sphere information with inter-sphere information, it can keep rotation invariant.\n3.\tNew mapping is proposed to convert point cloud data to mesh data, it can make the model more general to deal with 3D data.\n4.\tResults show the model is effective and achieves SOTA.\n\nWeakness:\n1.\tMath symbols need to be unified.\n2.\tSome repeated words. E.g. “stacking” in “Radial Discretization”, Page 3, “point cloud” in “POINT CLOUD TO CONCENTRIC SPHERICAL SIGNAL”, Page 4.\n3.\tSome math symbols have no explanation, which may make readers cannot get your idea. E.g. d_u、d_v in Eq(1)。\n4.\tI don’t know what is Z_g in Eq(1), and the shape of this parameter. The oversimplified description of Z_g makes me can’t understand why Eq(1) can keep rotation invariant.\n5.\tWhat is the shape of W_{k+K/2} in Eq(2), is it a matrix ? Please add more description of your math symbels.\n6.\tHow to determine the neighbor range of point x in Eq(3) ? Are you use KNN or ball query methods ? Do you have some contrast experiments?\n7.\tOnly use the norm-2 to calculate Eq(3) may loss some direction information, how do you think of this question?\n\nAlthough the paper seems to propose an effective convolutional operation for point cloud representation, it is unclear how to maintain its rotation invariance. Possibly this is because of inconsisent and unclear symbols. Therefore, if the authors could show this factor satisfactory in the rebuttal and address the above concerns, I would like to move to a positive rating. ", "extra_scores": {}, "extra_text": {}}, {"review_id": "-1HEgbz6Su4", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "In this paper, a multi-resolution convolutional architecture is proposed to learn from concentric feature maps. Different from single sphere representation, both graph convolutions and radial convolutions are employed to extract the intra-sphere and inter-sphere information. Benefit form the radial discretization, the proposed CSGNN achieves state-of-the-art results in testing on arbitrarily rotated data under ModelNet40 dataset. However, there are several drawbacks in the  draft, such as ambiguous figure and  insufficient ablation study.\n\n## Summary\n\nIn CSGNN,  a  two-phase convolutional scheme for learning over a concentric spheres representation, by alternating between inter-sphere and intra-sphere convolutional blocks. Specifically, the graph convolutional network is applied to incorporate intra-sphere information, and 1D convolution to incorporate radial information. Different from previous works,  a multiple sphere representations are integrated in CSGNN, which is robust to rotation operations. In general, the proposed method is innovative on the basis of spherical discretization, but still has some limitations.\n\n## Strengths\n\n#### 1. A multi-sphere icosahedral discretization for representation of 3D data is proposed, which can enhance  representation ability and keep more details over single-sphere representations.\n\n#### 2. Both graph convolutional networks and 1D radial convolutions are employed to capture the intra-sphere and inter-sphere information of 3D objects.\n\n#### 3. The proposed CSGNN achieves state-of-the-art results in testing on arbitrarily rotated data.\n\n## Weaknesses\n\n#### 1. The complexity analysis is insufficient. In the draft, the author only provide the rough overall complexity. A better way is to show the comparison between the proposed method and some other methods, including the number of model parameter and network forwarding time.\n\n#### 2. In the converting of point cloud to concentric spherical signal,  the Gaussian radial basis function is adopted to   summarize the contribution of points. Is there any other function that can accomplish this job? The reviewer would like to the discussion about this.\n\n#### 3. The Figure 2 is a little ambiguous, where some symbols are not explained clearly. And the reviewer is curious about that  whether there is information redundancy and interference in the multi-sphere icosahedral discretization process.\n\n#### 4. There are some typos in the draft. The first is the wrong use of \"intra-sphere\" and \"inter-sphere\". The second is the use of two consecutive \"stacking\" in the Spherical Discretization subsection. Please check the full text carefully.\n\n#### 5. The center choice of the concentric spheres should be discussed both theoretically and experimentally. In the opinion, the center of spheres play a important role in the representation capturing of 3D point clouds in a sphere convolution manner.\n", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper proposes  to effectively learn representation of 3D data (point clouds/meshes) using a spherical GNN architecture over concentric spherical maps. A method for converting point clouds to concentric spherical images is also proposed. Evaluation is done via 3D classification tasks on rotated data.

Strengths:
- Interesting novel method for learning 3D representations
- Technically sound
- Performs similarly to spherical CNNs and other STOA on the Modelnet40 dataset

Weaknesses:
- Presentation of the work needs to be further improved such that it is easier for others to reproduce
- More in-depth experiments are needed to justify how much Spherical GNN improves over other STOA, particular given how classification accuracy is very similar to STOA.
Decision: Reject