Original: C ONCENTRIC S PHERICAL GNN FOR 3D R EPRESENTATION L EARNING
Revised: C ONCENTRIC S PHERICAL GNN FOR 3D R EPRESENTATION L EARNING

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: 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
Revised: 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

Original: 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.
Revised: 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.

Original: Our hierarchical architecture is based on alternatively learning to incorporate both intra-sphere and inter-sphere information.
Revised: Our hierarchical architecture is based on alternatively learning to incorporate both intra-sphere and inter-sphere information.

Original: 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.
Revised: 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.

Original: 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.
Revised: 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.

Original: We demonstrate the effectiveness of our approach in achieving state-of-the-art performance on 3D classiﬁcation tasks with rotated data
Revised: We demonstrate the effectiveness of our approach in achieving state-of-the-art performance on 3D classiﬁcation tasks with rotated data.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: While convolutional neural networks have been applied to great success to 2D images, extending the same success to geometries in 3D has proven more challenging.
Revised: While convolutional neural networks have been applied to great success to 2D images, extending the same success to geometries in 3D has proven more challenging.

Original: A desirable property and challenge in this setting is to learn descriptive representations that are also equivariant to any 3D rotation.
Revised: A desirable property and challenge in this setting is to learn descriptive representations that are also equivariant to any 3D rotation.

Original: Cohen et al.
Revised: Cohen et al.

Original: (2018) and Esteves et al.
Revised: (2018) and Esteves et al.

Original: (2018) showed that the spherical domain permits learning such rotationally equivariant representations, by deﬁning convolutions with respect to spherical harmonics.
Revised: (2018) showed that the spherical domain permits learning such rotationally equivariant representations, by deﬁning convolutions with respect to spherical harmonics.

Original: In practice, 3D convolutions are implemented via discretization of the sphere.
Revised: In practice, 3D convolutions are implemented via discretization of the sphere.

Original: Earlier spherical
Revised: Earlier spherical

Original: CNNs used spherical coordinate grids, but these discretizations result in non-uniform samplings of the sphere, which is non-ideal.
Revised: Convolutional Neural Networks (CNNs) used spherical coordinate grids, but these discretizations result in non-uniform samplings of the sphere, which is non-ideal.

Original: Furthermore, spherical convolutions deﬁned on these grids scale with O ( N 1 . 5 ) complexity ( N as the number of grid points).
Revised: Furthermore, spherical convolutions deﬁned on these grids scale with O ( N 1 . 5 ) complexity ( N as the number of grid points).

Original: Subequent works, Jiang et al.
Revised: Subequent works, Jiang et al.

Original:  Cohen et al.
Revised: (2019), Cohen et al.

Original: Defferrard et al.
Revised: Defferrard et al.

Original: (2020), designed more scalable O ( N ) convolutions focusing on more uniform spherical discretizations.
Revised: (2020), designed more scalable O ( N ) convolutions focusing on more uniform spherical discretizations.

Original: Existing spherical CNNs operate over a spherical image, resulting from projection of data to a bounding sphere.
Revised: Existing spherical CNNs operate over a spherical image, resulting from projection of data to a bounding sphere.

Original: We show that it is more expressive and general to instead operate over a concentric, multi-spherical discretization for representing 3D data.
Revised: We show that it is more expressive and general to instead operate over a concentric, multi-spherical discretization for representing 3D data.

Original: Our main innovation is introducing a new two-phase convolutional scheme for learning over a concentric spheres representation, by alternating between inter-sphere and intra-sphere convolutional blocks.
Revised: Our main innovation is introducing a new two-phase convolutional scheme for learning over a concentric spheres representation, by alternating between inter-sphere and intra-sphere convolutional blocks.

Original: We use graph convolutions to incorporate inter-sphere information, and 1D convolutions to incorporate radial information.
Revised: We use graph convolutions to incorporate inter-sphere information, and 1D convolutions to incorporate radial information.

Original: Similar to Jiang et al.
Revised: Similar to Jiang et al.

Original: (2019),Cohen et al.
Revised: and Cohen et al.

Original: (2019), we focus on the icosahedral spherical discretization, which produces a mostly regular sampling over the sphere.
Revised: (2019), we focus on the icosahedral spherical discretization, which produces a mostly regular sampling over the sphere.

Original: Our proposed architecture is hierarchical, following the recursive coarsening hierarchy of the icosahedron.
Revised: Our proposed architecture is hierarchical, following the recursive coarsening hierarchy of the icosahedron.

Original: Combining intra-sphere and inter-sphere convolutions has a conceptual analogy to gradually incorporating information over volumetric sectors.
Revised: Combining intra-sphere and inter-sphere convolutions has a conceptual analogy to gradually incorporating information over volumetric sectors.

Original: At the same time, the choice of convolutions allows our model to retain a high degree of rotational equivariance.
Revised: At the same time, the choice of convolutions allows our model to retain a high degree of rotational equivariance.

Original: We demonstrate the effectiveness and generality of our approach through two 3D classiﬁcation experiments with different types of input data: mesh objects and general point clouds.
Revised: We demonstrate the effectiveness and generality of our approach through two 3D classiﬁcation experiments with different types of input data: mesh objects and general point clouds.

Original: The latter poses an additional challenge for discretization-based methods, as native point clouds are non-uniformly distributed in 3D space.
Revised: The latter poses an additional challenge for discretization-based methods, as native point clouds are non-uniformly distributed in 3D space.

Original: To summarize our contributions:
Revised: To summarize our contributions:

Original: We propose a 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.
Revised: We propose a 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.

Original: We also introduce a novel convolutional architecture for multi-sphere discretization by introducing two different types of convolutions, conceptually separated as intra-sphere and inter-sphere.
Revised: We also introduce a novel convolutional architecture for multi-sphere discretization by introducing two different types of convolutions, conceptually separated as intra-sphere and inter-sphere.

Original: Combining graph convolutions (intra-sphere) with 1D radial convolutions (inter-sphere) leads to an expressive architecture that is also rotationally equivariant.
Revised: Combining graph convolutions (intra-sphere) with 1D radial convolutions (inter-sphere) leads to an expressive architecture that is also rotationally equivariant.

Original: Our proposed convolutions are also scalable, being linear with respect to total grid size.
Revised: Our proposed convolutions are also scalable, being linear with respect to total grid size.

Original: We design mappings of both 3D mesh objects and general point clouds to the proposed representation.
Revised: We design mappings of both 3D mesh objects and general point clouds to the proposed representation.

Original: We achieve state-of-art performance on ModelNet40 point cloud classiﬁcation, using the proposed model and a data mapping using radial basis functions.
Revised: We achieve state-of-art performance on ModelNet40 point cloud classiﬁcation, using the proposed model and a data mapping using radial basis functions.

Original: We also improve on existing Spherical CNN performance in SHREC17 3D mesh classiﬁcation by utilizing multi-radius information.
Revised: We also improve on existing Spherical CNN performance in SHREC17 3D mesh classiﬁcation by utilizing multi-radius information.

Original:  R ELATED W ORK
Revised:  R ELATED W ORK

Original: Spherical CNNs.
Revised: Spherical CNNs.

Original: The goal of learning rotationally invariant representations of 3D geometries has led to several ideas for rotationally equivariant convolutions in the spherical domain.
Revised: The goal of learning rotationally invariant representations of 3D geometries has led to several ideas for rotationally equivariant convolutions in the spherical domain.

Original: Cohen et al.
Revised: Cohen et al.

Original: (2018), Esteves et al.
Revised: (2018) and Esteves et al.

Original: (2018) deﬁned spherical convolutions that are rotationally equivariant to rotations of the SO (3) group.
Revised: (2018) deﬁned spherical convolutions that are rotationally equivariant to rotations of the SO (3) group.

Original: However, these convolutions are restricted to non-uniform grid samplings and scale superlinearly with respect to grid resolution.
Revised: However, these convolutions are restricted to non-uniform grid samplings and scale superlinearly with respect to grid resolution.

Original: Later works have explored more scalable convolutions on other spherical discretizations, achieving linear complexity with respect to grid resolution.
Revised: Later works have explored more scalable convolutions on other spherical discretizations, achieving linear complexity with respect to grid resolution.

Original: Jiang et al.
Revised: Jiang et al.

Original: (2019) proposed using parameterized differential operators to form convolutional kernels over the icosahedron, where equivariance is restricted to rotations about the z -axis.
Revised: (2019) proposed using parameterized differential operators to form convolutional kernels over the icosahedron, where equivariance is restricted to rotations about the z -axis.

Original: Cohen et al.
Revised: Cohen et al.

Original: (2019) proposed gauge equivariant convolutions on manifolds, operating on feature ﬁelds corresponding to underlying geometric entities.
Revised: (2019) proposed gauge equivariant convolutions on manifolds, operating on feature ﬁelds corresponding to underlying geometric entities.

Original: This was applied to achieve rotationally equivariant convolutions over the icosahedral discretization.
Revised: This was applied to achieve rotationally equivariant convolutions over the icosahedral discretization.

Original: Defferrard et al.
Revised: Defferrard et al.

Original: (2020) propose a graph convolutionbased spherical CNN using spectral ﬁlters, along with a distance-weighted nearest-neighbors graph construction scheme that allows balancing between rotational equivariance and efﬁciency, when applied to different types of grids.
Revised: (2020) propose a graph convolutionbased spherical CNN using spectral ﬁlters, along with a distance-weighted nearest-neighbors graph construction scheme that allows balancing between rotational equivariance and efﬁciency, when applied to different types of grids.

Original: 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.
Revised: 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.

Original: Rao et al.
Revised: Rao et al.

Original: (2019) uses graphconvolution inspired message passing operators for learning over the icosahedral discretization.
Revised: (2019) uses graphconvolution inspired message passing operators for learning over the icosahedral discretization.

Original: Our work is similar to Rao et al.
Revised: Our work is similar to Rao et al.

Original: and Defferrard et al.
Revised: and Defferrard et al.

Original: in terms of using graph-based spherical convolutions, but we generalize to multi-sphere convolutions.
Revised: in terms of using graph-based spherical convolutions, but we generalize to multi-sphere convolutions.

Original: You et al.
Revised: You et al.

Original: (2020) is the most related work in terms of multi-sphere representation learning.
Revised: (2020) is the most related work in terms of multi-sphere representation learning.

Original: The authors propose a spherical voxel grid, and extending the SO3 convolutions of Cohen et al.
Revised: The authors propose a spherical voxel grid, and extending the SO3 convolutions of Cohen et al.

Original: (2018) to incorporate the radial dimension.
Revised: (2018) to incorporate the radial dimension.

Original: Our work treats spherical and radial convolutions as distinct, which observes much better results in practice.
Revised: Our work treats spherical and radial convolutions as distinct, which results in much better results in practice.

Original: We also use more scalable spherical convolutions deﬁned on the uniform icosahedral grid.
Revised: We also use more scalable spherical convolutions deﬁned on the uniform icosahedral grid.

Original: Pointwise Convolution Networks.
Revised: Pointwise Convolution Networks.

Original: There is a signiﬁcant body of work on learning point cloud representations using pointwise convolutions, beginning with with Qi et al.
Revised: There is a signiﬁcant body of work on learning point cloud representations using pointwise convolutions, beginning with with Qi et al.

Original: (2017) which proposed learning permutation invariant functions that directly operate on point coordinates.
Revised: (2017) which proposed learning permutation invariant functions that directly operate on point coordinates.

Original: Only more recently have such methods have been developed towards learning rotationally invariant representations.
Revised: Only more recently have such methods have been developed towards learning rotationally invariant representations.

Original: Thomas et al.
Revised: Thomas et al.

Original: (2018) and Poulenard et al.
Revised: (2018) and Poulenard et al.

Original: both propose pointwise convolutional ﬁlters based on spherical harmonic functions to achieve rotational equivariance (or invariance).
Revised: both propose pointwise convolutional ﬁlters based on spherical harmonic functions to achieve rotational equivariance (or invariance).

Original: Distance information is recorded through learned functions in the former, and radial sampling in the latter.
Revised: Distance information is recorded through learned functions in the former, and radial sampling in the latter.

Original: While these ﬁlters are deﬁned with respect to all-to-all convolution between points, in practice convolutions are limited to k -nearest neighbors (Poulenard et al.
Revised: While these ﬁlters are deﬁned with respect to all-to-all convolution between points, in practice convolutions are limited to k -nearest neighbors (Poulenard et al.

Original: for scalability.
Revised: for scalability.

Original: Chen et al.
Revised: Chen et al.

Original: (2019), Sun et al.
Revised: (2019), Sun et al.

Original: , Zhang et al.
Revised: , Zhang et al.

Original: (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.
Revised: (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.

Original: These features are hand-engineered based on carefully picking local frames of references, or global ones in the case of Sun et al.
Revised: These features are hand-engineered based on carefully picking local frames of references, or global ones in the case of Sun et al.

Original:  R EPRESENTATION BY C ONCENTRIC S PHERES
Revised:  R EPRESENTATION BY C ONCENTRIC S PHERES

Original: Existing work on spherical CNNs operate on spherical grids, where data is typically projected to and deﬁned on grid points.
Revised: Existing work on spherical CNNs operate on spherical grids, where data is typically projected to and deﬁned on grid points.

Original: However, projecting 3D data to a single sphere may not always be sufﬁcient or appropriate.
Revised: However, projecting 3D data to a single sphere may not always be sufﬁcient or appropriate.

Original: Simple projections may be lossy when describing highly non-convex shapes, for instance if the shape curves in on itself.
Revised: Simple projections may be lossy when describing highly non-convex shapes, for instance if the shape curves in on itself.

Original: To increase capacity to distinguish different data distributions, we introduce a new discretization based on concentric spheres, which additionally discretizes 3D space in the radial dimension.
Revised: To increase capacity to distinguish different data distributions, we introduce a new discretization based on concentric spheres which additionally discretizes 3D space in the radial dimension.

Original: The single sphere discretization is a special case in our proposed paradigm.
Revised: The single sphere discretization is a special case in our proposed paradigm.

Original: Spherical Discretization.
Revised: Spherical Discretization.

Original: We work with an icosahedral grid discretization of the sphere.
Revised: We work with an icosahedral grid discretization of the sphere.

Original: The base icosahedron I (0) has 12 vertices, forming 20 equilateral triangle faces (each face with 3 edges).
Revised: The base icosahedron I (0) has 12 vertices, forming 20 equilateral triangle faces (each face with 3 edges).

Original: Each vertex is incident with 5 triangles.
Revised: Each vertex is incident with 5 triangles.

Original: Each face can be subdivided, with the number of vertices scaling as | V | = 10 ∗ 4 l + 2 (where l is the discretization level).
Revised: Each face can be subdivided, with the number of vertices scaling as | V | = 10 ∗ 4 l + 2 (where l is the discretization level).

Original: See Fig.
Revised: See Fig.

Original: 2 for an illustration.
Revised: 3 for an illustration.

Original: Radial Discretization.
Revised: Radial Discretization.

Original: We construct the multi-radius spherical discretization by stacking R identical icosahedral grids.
Revised: We construct the multi-radius spherical discretization by stacking R identical icosahedral grids.

Original: Assuming unit radius normalization, we use a uniform discretization that results in concentric spheres scaled to radii [ 1 R , 2 R , ..., 1] .
Revised: Assuming unit radius normalization, we use a uniform discretization that results in concentric spheres scaled to radii [ 1 R , 2 R , ..., 1] .

Original: Intra-sphere Convolutions.
Revised: Intra-sphere Convolutions.

Original: There is a growing body of work addressing design of rotationequivariant ﬁlters over spherical feature maps.
Revised: There is a growing body of work addressing design of rotationequivariant ﬁlters over spherical feature maps.

Original: We focus on graph convolutional ﬁlters for intrasphere convolutions, as graph convolutions are scalable and lead to equivariant representations, up to discretization effects Defferrard et al.
Revised: We focus on graph convolutional ﬁlters for intrasphere convolutions, as graph convolutions are scalable and lead to equivariant representations, up to discretization effects Defferrard et al.

Original: (2020), Yang et al.
Revised: (2020), Yang et al.

Original: This motivates our construction of the undirected graph G ( l ) =
Revised: This motivates our construction of the undirected graph G ( l ) =

Original: ( V ( l ) , E ( l ) ) from a level l icosahedron I ( l ) .
Revised: ( V ( l ) , E ( l ) ) from a level l icosahedron I ( l ) .

Original: Vertices of the vertex set V ( l ) correspond one-to-one with vertices of I ( l ) projected to unit sphere.
Revised: Vertices of the vertex set V ( l ) correspond one-to-one with vertices of I ( l ) projected to unit sphere.

Original: E ( l ) is simply the set of all (bidirectional) face edges of the icosahedron (projected to unit sphere).
Revised: E ( l ) is simply the set of all (bidirectional) face edges of the icosahedron (projected to unit sphere).

Original: Vertices have degree 6, with exception of the the initial 12 vertices of the base icosahedron I (0) that have degree 5.
Revised: Vertices are all degree 6, with exception of the the initial 12 vertices of the base icosahedron I (0) that are degree 5.

Original: Since each edge is approximately equidistant between two points of the sphere Wang & Lee (2011), G ( l ) is also treated as an unweighted graph.
Revised: Since each edge is approximately equidistant between two points of the sphere Wang & Lee (2011), G ( l ) is also treated as an unweighted graph.

Original: While there is a rich body of work on graph convolutions and its variants, this work focuses on the graph convolution deﬁned in Kipf & Welling (2017).
Revised: While there is a rich body of work on graph convolutions and its variants, this work focuses on the graph convolution deﬁned in Kipf & Welling (2017).

Original: We introduce notation to deﬁne this convolution in the context of our multi-spherical discretization in more detail.
Revised: We introduce notation to deﬁne this convolution in the context of our multi-spherical discretization in more detail.

Original: Let H ∈ R R ×| V |× C denote a C channel tensor of features.
Revised: Let H ∈ R R ×| V |× C denote a C channel tensor of features.

Original: Also let Z ∈ R C × F be  parameters, where C and F are input and output channel sizes.
Revised: Also let Z ∈ R C × F be shared parameters, where C and F are input and output channel sizes.

Original: N ( u ) denote neighbors of vertex u in graph G and deg( u ) denotes | N ( u ) | , the degree of vertex u .
Revised: N ( u ) denote neighbors of vertex u in graph G and deg( u ) denotes | N ( u ) | , the degree of vertex u .

Original: We assume that self-edges are added for every vertex in N ( u ) .
Revised: We assume that self-edges are added for every vertex in N ( u ) .

Original: Finally, we introduce the subscript t to be the convolutional layer number, i ∈ [0 , R − 1] to index the radial dimension, and u ∈ [0 , | V | − 1] to index the vertices.
Revised: Finally, we introduce the subscript t to be the convolutional layer number, i ∈ [0 , R − 1] to index the radial dimension, and u ∈ [0 , | V | − 1] to index the vertices.

Original: The layer t + 1 intra-sphere convolution output for vertex u of sphere i is given by Eq.
Revised: The layer t + 1 intra-sphere convolution output for vertex u of sphere i is given by Eq.

Original: [Equation] where σ indicates a nonlinear activation function.
Revised: 1, where σ indicates a nonlinear activation function.: [Equation]

Original: Inter-sphere Convolutions.
Revised: Inter-sphere Convolutions.

Original: We introduce radial convolutions to incorporate inter-sphere information, implemented as 1D convolutions where the radial dimension is treated as the sequence length.
Revised: We introduce radial convolutions to incorporate inter-sphere information, implemented as 1D convolutions where the radial dimension is treated as the sequence length.

Original: Importantly, radial convolutions are also rotationally invariant, as 1D convolution operates over channels of the same vertex.
Revised: Importantly, radial convolutions are also rotationally invariant, as 1D convolution operates over channels of the same vertex.

Original: See Fig.
Revised: See Fig.

Original: 1 for illustration ofradial and graph convolutions with respect to concentric spheres representation.
Revised: 2 for illustration of graph convolutions with respect to concentric spheres representation.

Original: We introduce some additional notation to describe radial convolutions.
Revised: We introduce some additional notation to describe radial convolutions.

Original: Let K be 1D convolution kernel size.
Revised: Let K be 1D convolution kernel size.

Original: We assume K is odd valued, and pad inputs in the radial dimension such that a dimension of R is maintained across convolutions.
Revised: We assume K is odd valued, and pad inputs in the radial dimension such that a dimension of R is maintained across convolutions.

Original: Let W ∈ R K × C × F be a tensor of  parameters, where C and F are input and output channel sizes.
Revised: Let W ∈ R K × C × F be a tensor of shared parameters, where C and F are input and output channel sizes.

Original: The layer t +inter-sphere convolution output for vertex u of sphere i is given by Eq. 2:
Revised: The layer t +inter-sphere convolution output for vertex u of sphere i is given by Eq. 2:

Original: [Equation]
Revised: [Equation]

Original: Complexity Analysis.
Revised: Vertex Pooling.

Original: In the intra-sphere convolution ofEq. 1, the neighborhood size (vertex degree) is bounded by  a constant. Dimensions of parameters and feature vectors are also bounded by a constant.
Revised: 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 a natural hierarchy for pooling and downsampling (see Fig. 3).

Original: Therefore, the overall complexity of intra-sphere convolution is linear with respect to the multi-spherical grid size, or O (  R | V | ) ,where | V | is the number of vertices in the spherical grid.
Revised: We introduce overloaded notation H ( l ) ∈ R R ×| V ( l ) |× C to denote the feature tensor corresponding to V ( l ) , the vertex set corresponding to level l icosahedron. We deﬁne pooling as H ( l − 1) i,u = f ( { H ( l ) i

Original: 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  isbounded a ﬁxed kernel size for the neighborhood.
Revised: [Equation] : v ∈ N ( u ) } ) , where N ( u ) is the neighborhood of vertex u ∈ V ( l ) and f is a permutation invariant function (e.g. max operator). Pooling is followed by downsampling, where only vertices of the smaller vertex set V ( l − 1) are retained (see Sec.

Original: The overall complexity of inter-sphere convolution is also O ( R | V | ) , and so both convolutions introduced in this work are linear with respect to the multi-spherical grid size. We further provide experimental parameter analysis in Tables 1, 2, and time analysis in Appendix A.2.
Revised: A.4 for details.) Only vertices within same sphere are involved; there is no pooling or downsampling of vertices between spheres (except right before classiﬁcation).

Original: Concentric Spherical GNN  Architecture.
Revised: Concentric Spherical GNN (CSGNN) Architecture.

Original: Fig.
Revised: Fig.

Original: 2 gives an example illustration of an end-to-end architecture using both convolutions.
Revised: 3 gives an example illustration of an end-to-end architecture using both convolutions.

Original: Importantly, radial convolution blocks are introduced alongside graph convolutional blocks at every level of the spherical discretization hierarchy, to incorporate inter-sphere information gradually.
Revised: Importantly, radial convolution blocks are introduced alongside graph convolutional blocks at every level of the spherical discretization hierarchy, to incorporate inter-sphere information gradually.

Original: From a icosahedron of level L reﬁnement, we construct a sequences of graphs [ G ( L ) , G ( L − 1) , ..., G (0) ].
Revised: From a icosahedron of level L reﬁnement, we construct a sequences of graphs [ G ( L ) , G ( L − 1) , ..., G (0) ].

Original: Each G ( l ) carries an additional R dimension, corresponding to spheres at different radial levels.
Revised: Each G ( l ) carries an additional R dimension, corresponding to spheres at different radial levels.

Original: Each level l features two blocks of convolutions: graph convolutions, followed by radial convolution.
Revised: Each level l features two blocks of convolutions: graph convolutions, followed by radial convolution.

Original: These correspond to intra-sphere and intersphere convolutions respectively.
Revised: These correspond to intra-sphere and inter-sphere convolutions respectively.

Original: Vertex neighborhood pooling downsamples the graph from G ( l ) to G ( l − 1) .
Revised: Vertex neighborhood pooling downsamples the graph from G ( l ) to G ( l − 1) .

Original: The size of the radial dimension remains constant, until ﬁnal pooling.
Revised: The size of the radial dimension remains constant, until ﬁnal pooling.

Revised: Complexity Analysis.

Original: P OINT C LOUD TO C ONCENTRIC S PHERICAL S IGNAL
Revised: In the intra-sphere convolution of Eq. 1, 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 O ( R | V | ) , 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.

Revised: Similarly, each inter-sphere convolution (Eq. 2) with respect to a vertex is bounded a ﬁxed kernel size for the neighborhood.

Revised: The overall complexity of inter-sphere convolution is also O

Revised: ( R | V | ) , and so both convolutions introduced in this work are linear with respect to the multi-spherical grid size.

Revised: We further provide experimental parameter analysis in Tables 1, 2, and time analysis in Appendix A.2.

Original: We consider the problem of mapping a point cloud P ∈ R N × 3 point cloud to an initial spherical feature map M ∈ R R ×| V |× C , where C is number of input channels.
Revised: We consider the problem of mapping a point cloud P ∈ R N × 3 point cloud to an initial spherical feature map M ∈ R R ×| V |× C , where C is number of input channels.

Original: While the concentric grid representation is deﬁned discretely at ﬁxed positions, the space of data point locations is continuous.
Revised: While the concentric grid representation is deﬁned discretely at ﬁxed positions, the space of data point locations is continuous.

Original: We also aim to capture the distribution of points in a continuous way.
Revised: We also aim to capture the distribution of points in a continuous way.

Original: To do so, we summarize the contribution of points using the Gaussian radial basis function (3).
Revised: To do so, we summarize the contribution of points using the Gaussian radial basis function (RBF):

Original: [Equation]
Revised: [Equation]

Original: N is the number of data points, and φ = exp( − γr 2 ) , parameterized by the bandwidth γ .
Revised: N is the number of data points, and φ = exp( − γr 2 ) , parameterized by the bandwidth γ .

Original: In practice we limit computation to a local neighborhood (instead of considering all points), and choose γ accordingly.
Revised: In practice we limit computation to a local neighborhood (instead of considering all points), and choose γ accordingly.

Original: See Fig.
Revised: See Fig.

Original: 1 for visualization of the local neighborhood and mapping, and Sec.
Revised: 1 for visualization of the local neighborhood and mapping, and Sec.

Original: A.1 for additional details.
Revised: A.1 for additional details.

Original: One possible mapping is to compute Eq.
Revised: One possible mapping is to compute Eq.

Original: 3 at every vertex position of the spherical discretization, resulting in a single channel feature map.
Revised: 3 at every vertex position of the spherical discretization, resulting in a single channel feature map.

Original: 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.
Revised: 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.

Original: (2019), along the radial dimension.
Revised: (2019), along the radial dimension.

Original: Subdividing along radial dimension by a factor K e results in a new spherical discretization with increased radial dimension of
Revised: Subdividing along radial dimension by a factor K e results in a new spherical discretization with increased radial dimension of

Original: R (cid:48)
Revised: R (cid:48)

Original: =
Revised: =

Original: R ∗ K e .
Revised: R ∗ K e .

Original: The RBF is evaluated at every vertex position of this new discretization, resulting in a feature map of dimension of M (cid:48)
Revised: The RBF is evaluated at every vertex position of this new discretization, resulting in a feature map of dimension of M (cid:48)

Original: ∈ R R (cid:48)
Revised: ∈ R R (cid:48)

Original: × V × 1 .
Revised: × V × 1 .

Original: We map back to the original discretization by assigning M i,u =
Revised: We map back to the original discretization by assigning M i,u =

Original: [ M (cid:48)
Revised: [ M (cid:48)

Original: j,u : j ∈ ( iK e , iK e + 1 , ..., 2 iK e , 2 iK e + 1 , ..., 3 iK e − 1)] , resulting in a size [ R × | V | × 2 K e ] spherical feature map.
Revised: j,u : j ∈ ( iK e , iK e + 1 , ..., 2 iK e , 2 iK e + 1 , ..., 3 iK e − 1)] , resulting in a size [ R × | V | × 2 K e ] spherical feature map.

Original: In summary, multiple RBF values are assigned to each vertex by further sub-dividing space in the radial dimension.
Revised: In summary, multiple RBF values are assigned to each vertex by further sub-dividing space in the radial dimension.

Original:  E XPERIMENTS 5. M ODEL N ET 40 P OINT C LOUD C LASSIFICATION
Revised:  E XPERIMENTS 5. M ODEL N ET 40 P OINT C LOUD C LASSIFICATION

Original: We consider the ModelNet40 3D shape classiﬁcation task, with 12308 shapes and 40 classes.
Revised: We consider the ModelNet40 3D shape classiﬁcation task, with 12308 shapes and 40 classes.

Original: Each point cloud has 1024 points.
Revised: Each point cloud has 1024 points.

Original: For all experiments, 9840 shapes are used for training and 2468 for testing.
Revised: For all experiments, 9840 shapes are used for training and 2468 for testing.

Original: Architecture and Hyperparameters
Revised: Architecture and Hyperparameters

Original: Fig.
Revised: Fig.

Original: 3 shows a complete architecture overview.
Revised: 4 shows a complete architecture overview.

Original: Point clouds are ﬁrst mapped to 16 concentric spheres with level 4 icosahedral discretization ( L = 4 , R = 16 ), using RBF kernels with threshold
Revised: Point clouds are ﬁrst mapped to 16 concentric spheres with level 4 icosahedral discretization ( L = 4 , R = 16 ), using RBF kernels with threshold

Original: T = 0 .
Revised: T = 0 .

Original: 1D convolutions use a kernel size of 3.
Revised: 1D convolutions use a kernel size of 3.

Original: Each graph and 1D convolution is followed by batch normalization and ReLU as nonlinear activation.
Revised: Each graph and 1D convolution is followed by batch normalization and ReLU as nonlinear activation.

Original: Additionally, skip connections are added between every graph convolution layer, whenever input dimension matches output.
Revised: Additionally, skip connections are added between every graph convolution layer, whenever input dimension matches output.

Original: The model is 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.
Revised: The model is 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.

Original: Batch size is 32.
Revised: Batch size is 32.

Original: Each training epoch took 5 minutes and inference (of entire validation set) took 7-8 seconds, data loading and transformation times included.

Original: Results
Revised: Results

Revised: Method

Revised: Input

Revised: Params z/

Revised: z z/SO

Original: We present our results and compare against other related works in Table 1, in four different train/test data orientation settings.
Revised: NR/SO SO3/SO Pointwise Convolution PointNet Qi et al.

Revised: (2017) xyz 3.5M 0.

Revised: ClusterNet Chen et al.

Original: {+xyz * 0.8710.871* 0.
+}
Revised: {+xyz * 0.8710.871* 0.
+}

Revised: RIConv Zhang et al.

Revised: (2019) xyz 0.7M 0.

Revised: SPHNet Poulenard et al.

Revised: (2019) xyz 2.9M 0.

Revised: SRINet Sun et al.

Revised: xyz+ 0.9M 0.

Revised: normal

Revised: Spherical CNN

Revised: SFCNN Rao et al.

Revised: (2019) xyz 9.2M

Revised: [Table]

Revised: We present our results and compare against other related works in Table 1, in four different train/test data orientation settings.

Original: When training with rotations, a new rotation is sampled per instance in each epoch.
Revised: When training with rotations, a new rotation is sampled per instance in each epoch.

Original: Rotation is the only augmentation used in comparisons.
Revised: Rotation is the only augmentation used in comparisons.

Original: We report accuracy as average validation score across last 5 epochs of training, due to lack of standard validation/test split and to account for variation.
Revised: We report accuracy as average validation score across last 5 epochs of training, due to lack of standard validation/test split and to account for variation.

Original: Our method achieves state-of-the-art results in z/SO3 and SO3/SO3 settings, i.e. testing on arbitrarily rotated data.
Revised: Our method achieves state-of-the-art results in z/SO3 and SO3/SO3 settings, i.e. testing on arbitrarily rotated data.

Original: For more detailed comparison, we loosely categorize compared works by method[Table] into two categories: pointwise convolution networks and spherical CNNs.
Revised: For more detailed comparison, we loosely categorize compared works by method into two categories: pointwise convolution networks and spherical CNNs.

Original: Methods in the former category operate directly on data points in 3D space, while methods in the latter operate on a spherical discretization.
Revised: Methods in the former category operate directly on data points in 3D space, while methods in the latter operate on a spherical discretization.

Original: Our work is most closely related to methods in the spherical CNN category.
Revised: Our work is most closely related to methods in the spherical CNN category.

Original: Similar to our work, PRIN also explored learning a concentric spherical representation based on extending SO3 convolutions from Cohen et al.
Revised: Similar to our work, PRIN also explored learning a concentric spherical representation based on extending SO3 convolutions from Cohen et al.

Original: Our method is based on separate graph and radial convolutions, which achieves much better performance in all settings.
Revised: Our method is based on separate graph and radial convolutions, which achieves much better performance in all settings.

Original: SFCNN has similarity to our work in using graph convolution-inspired message passing ﬁlters and learning over the icosahedral discretization of the sphere.
Revised: SFCNN has similarity to our work in using graph convolution-inspired message passing ﬁlters and learning over the icosahedral discretization of the sphere.

Original: However, SFCNN is restricted to a single-sphere representation, and also relies on a PointNet-like learned module to project points to spherical features.
Revised: However, SFCNN is restricted to a single-sphere representation, and also relies on a PointNet-like learned module to project points to spherical features.

Original: 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 intersphere convolutions.
Revised: 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 intersphere convolutions.

Original: Compared to our approach, SFCNN also has a 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.
Revised: Compared to our approach, SFCNN also has a 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.

Original: CSGNN, similar to SFCNN and PRIN, exhibits some drop in performance in the z/SO3 and NR/SO3 settings.
Revised: CSGNN, similar to SFCNN and PRIN, exhibits some drop in performance in the z/SO3 and NR/SO3 settings.

Original: This could partly be due to effects of discretization and data mapping.
Revised: This could partly be due to effects of discretization and data mapping.

Original: However, this gap is relatively smallest in CSGNN, and there is essentially no gap in z/z and SO3/SO3 performance.
Revised: However, this gap is relatively smallest in CSGNN, and there is essentially no gap in z/z and SO3/SO3 performance.

Original: 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.
Revised: 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.

Original: By contrast, our method largely learns to extract features directly from the input (outside of an initial step mapping points to vertices).
Revised: By contrast, our method largely learns to extract features directly from the input (outside of an initial step mapping points to vertices).

Original: PointNet, unlike the other baselines, was not designed to be rotationally equivariant.
Revised: PointNet, unlike the other baselines, was not designed to be rotationally equivariant.

Original: This reﬂects in the relatively large difference in performance when comparing testing with or without rotations.
Revised: This reﬂects in the relatively large difference in performance when comparing testing with or without rotations.

Original: 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.
Revised: 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.

Original: 5. SHREC17 3D S HAPES
Revised: 5. SHREC17 3D S HAPES

Original: The SCHREC17 task has 51300 3D models and 55 categories.
Revised: The SCHREC17 task has 51300 3D models and 55 categories.

Original: We use the version where all models have been randomly perturbed by rotations.
Revised: We use the version where all models have been randomly perturbed by rotations.

Original: Here the inputs are not point clouds, but mesh objects.
Revised: Here the inputs are not point clouds, but mesh objects.

Original: Cohen et al.
Revised: Cohen et al.

Original: (2018),Esteves et al.
Revised: (2018),Esteves et al.

Original: (2018) presented a ray-casting scheme to regularly sample information incident to outermost mesh surfaces and obtain features maps deﬁned over the spherical discretization.
Revised: (2018) presented a ray-casting scheme to regularly sample information incident to outermost mesh surfaces and obtain features maps deﬁned over the spherical discretization.

Original: For sufﬁciently non-convex mesh objects, a single sphere projection may result in information loss.
Revised: For sufﬁciently non-convex mesh objects, a single sphere projection may result in information loss.

Original: For example, when a ray is incident to multiple surfaces occurring at different radii, this information is discarded by existing methods.
Revised: For example, when a ray is incident to multiple surfaces occurring at different radii, this information is discarded by existing methods.

Original: We propose a new data mapping that generalizes single sphere representation to a concentric spherical representation, thereby preserving more information.
Revised: We propose a new data mapping that generalizes single sphere representation to a concentric spherical representation, thereby preserving more information.

Original: Fig.
Revised: Fig.

Original: 4 in the appendix shows visual examples of where the proposed representation may be helping.
Revised: 8 in the appendix shows visual examples of where the proposed representation may be helping.

Original: Representation
Revised: Representation

Original: In the case of single-sphere representations, a single ray is projected from a source point (vertex) on the enclosing sphere towards the center of the object.
Revised: In the case of single-sphere representations, a single ray is projected from a source point (vertex) on the enclosing sphere towards the center of the object.

Original: The ﬁrst hit incident with the mesh is recorded.
Revised: The ﬁrst hit incident with the mesh is recorded.

Original: To extend ray-casting to multiple concentric spheres, we rescale the source point to the radii of each respective sphere.
Revised: To extend ray-casting to multiple concentric spheres, we rescale the source point to the radii of each respective sphere.

Original: This results in multiple co-linear source points, one per sphere.
Revised: This results in multiple co-linear source points, one per sphere.

Original: The 1st hit incident with the mesh is recorded for each ray cast from those source points, resulting in a multiradius projection.
Revised: The 1st hit incident with the mesh is recorded for each ray cast from those source points, resulting in a multiradius projection.

Original: 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.
Revised: 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.

Original: We use a uniform [ 1 R , 2 R , ..., 1] radii division assuming inputs are normalized to unit radius.
Revised: We use a uniform [ 1 R , 2 R , ..., 1] radii division assuming inputs are normalized to unit radius.

Original: 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 3 features per vertex.
Revised: 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 3 features per vertex.

Original: These are similar features to those collected in related work, except we do not use the object’s convex hull information. 
Revised: These are similar features to those collected in related work, except we do not use the object’s convex hull information. [Table] et al. [Table]

Revised: [Table]

Revised: , b = 64 )

Original: [Table]
Revised: [Table]

Revised: Esteves et al.

Original: Architecture and Hyperparameters
Revised: (2018) (equiangular, b = 64 ) [Table] Defferrard et al. (2020) (equiangular, b = 64 )

Revised: [Table]

Revised: DeepSphere Defferrard et al.

Revised: (HEALPix, N side = 32)

Revised: [Table]

Revised: 2M

Revised: [Table]

Revised: CSGNN (icosahedral, L = 4 , R = 1 )

Revised: [Table]

Revised: 3M

Revised: [Table]

Revised: CSGNN (icosahedral, L = 4 , R = 16 )

Original:  The architecture for SHREC17 is identical to the one used for ModelNet40 in Fig.
Revised: [Table] [Table] Architecture and Hyperparameters The architecture for SHREC17 is identical to the one used for ModelNet40 in Fig.

Original: 3, with the exception that the input dimension is 3 (corresponding to features obtained from ray-casting).
Revised: 4, with the exception that the input dimension is 3 (corresponding to features obtained from ray-casting).

Original: We consider two model variations, single-sphere ( R = 1 ) and multi-sphere
Revised: We consider two model variations, single-sphere ( R = 1 ) and multi-sphere

Original: ( R = 16 ).
Revised: ( R = 16 ).

Original: For R = 16 , we use a 1D convolution kernel of size 3.
Revised: For R = 16 , we use a 1D convolution kernel of size 3.

Original: For R = 1 , we use a 1D convolution kernel of size 1.
Revised: For R = 1 , we use a 1D convolution kernel of size 1.

Original: Note this 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.
Revised: Note this 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.

Original: Training settings are also same as for ModelNet40, except learning rate decays at epochs 10 and 20.
Revised: Training settings are also same as for ModelNet40, except learning rate decays at epochs 10 and 20.

Original: For our R =model, each training epoch took 19 minutes, while inference (of validation set) took approximately 2 minutes, data loading and transformation time included.

Original: Results
Revised: Results

Original: See Table 2 for classiﬁcation results.
Revised: See Table 2 for classiﬁcation results.

Original: The reported metric is F 1 micro-average classiﬁcation score.
Revised: The reported metric is F 1 micro-average classiﬁcation score.

Original: Results from three other Spherical CNN works designed for general rotational equivariance are included for reference.
Revised: Results from three other Spherical CNN works designed for general rotational equivariance are included for reference.

Original: DeepSphere is most similiar to our work in terms of graph-based spherical convolutions, where the authors explored design of rotationally-equivariant graph convolutional ﬁlters with respect to the type of grid and neighborhood size.
Revised: DeepSphere is most similiar to our work in terms of graph-based spherical convolutions, where the authors explored design of rotationally-equivariant graph convolutional ﬁlters with respect to the type of grid and neighborhood size.

Original: This work focuses graph construction to the icosahedral discretization, using a minimal set of roughly equidistant neighbors.
Revised: This work focuses graph construction to the icosahedral discretization, using a minimal set of roughly equidistant neighbors.

Original: We additionally introduce inter-sphere convolutions with concentric spheres, which is largely orthogonal to the design of intra-sphere representation and convolutions.
Revised: We additionally introduce inter-sphere convolutions with concentric spheres, which is largely orthogonal to the design of intra-sphere representation and convolutions.

Original: Single-sphere ( R = 1 )
Revised: Single-sphere ( R = 1 )

Original: CSGNN achieves competitive performance with other single-sphere baselines.
Revised: CSGNN achieves competitive performance with other single-sphere baselines.

Original: However, it is difﬁcult to draw comparative conclusions in this particular case, due to differences in feature extraction, spherical discretization type and size, and model size.
Revised: However, it is difﬁcult to draw comparative conclusions in this particular case, due to differences in feature extraction, spherical discretization type and size, and model size.

Original: More signiﬁcantly, using multiple spheres ( R = 16 ) outperforms single-sphere baselines, including a 2.2% relative performance improvement over our R = 1 version.
Revised: More signiﬁcantly, using multiple spheres ( R = 16 ) outperforms single-sphere baselines, including a 2.2% relative performance improvement over our R = 1 version.

Original: 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. 
Revised: 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. [Figure]

Original: 5. M ODEL A RCHITECTURE A BLATION S TUDY
Revised: A RCHITECTURE A BLATION S TUDY

Original: To study the impact of multi-radius spherical discretization, we vary the number of radial levels and present results in Table 3.
Revised: To study the impact of multi-radius spherical discretization, we vary the number of radial levels and present results in Table 3.

Original: ModelNet40 is used for all ablation experiments.
Revised: ModelNet40 is used for all ablation experiments.

Original: We use a base model with R = 16 and L = 4 , and keep the number of parameters identical across all versions of the model.
Revised: We use a base model with R = 16 and L = 4 , and keep the number of parameters identical across all versions of the model.

Original: For this particular ablation, for simplicity we use a single channel, indicator feature map–a special case of the RBF mapping where γ = 0 and F = 1 .
Revised: For this particular ablation, for simplicity we use a single channel, indicator feature map–a special case of the RBF mapping where γ = 0 and F = 1 .

Original: Adding radial convolutions in the case of R = 1 is equivalent to adding additional dense layers after graph convolutions.
Revised: Adding radial convolutions in the case of R = 1 is equivalent to adding additional dense layers after graph convolutions.

Original: The architecture is the same as the one in 3, except input dimension is 1.
Revised: The architecture is the same as the one in 4, except input dimension is 1.

Original: Performance consistently improves with higher radial dimension, peaking at R = 16 with 4.8% relative accuracy improvement over the
Revised: Performance consistently improves with higher radial dimension, peaking at R = 16 with 4.8% relative accuracy improvement over the

Original: R = 1 version.
Revised: R = 1 version.

Original: Performance declines from R = 16 to R = 32 , which suggests diminishing returns for a ﬁxed parameter budget. 
Revised: Performance declines from R = 16 to R = 32 , which suggests diminishing returns for a ﬁxed parameter budget. Setting R = [Table]

Revised: =

Original: [Table]
Revised: [Table]

Revised: =

Revised: [Table]

Revised: =

Revised: [Table]

Revised: =

Original:  More ablation studies are presented in Table 4.
Revised: SO3/ [Table] [Table] More ablation studies are presented in Table 4.

Original: We study the impact of varying the radial kernel size
Revised: We study the impact of varying the radial kernel size

Original: K RC =
Revised: K RC =

Original: K RC = 1 is same as learning representations independently learned at each radial level.
Revised: K RC = 1 is same as learning representations independently learned at each radial level.

Original: While this still improves over single-sphere representation, using spatial ﬁlters ( K RC =
Revised: While this still improves over single-sphere representation, using spatial ﬁlters ( K RC =

Original: [3 , 5] ) over the radial dimension is important for best performance.
Revised: [3 , 5] ) over the radial dimension is important for best performance.

Original: Varying the number of graph and radial convolutional layers shows that between 1 and 2 layers per block leads to comparable performance.
Revised: Varying the number of graph and radial convolutional layers shows that between 1 and 2 layers per block leads to comparable performance.

Original: Finally, we compare using only graph convolutions or only radial convolutions.
Revised: Finally, we compare using only graph convolutions or only radial convolutions.

Original: Results suggest that it is essential to combine both types of convolutions for best performance.
Revised: Results suggest that it is essential to combine both types of convolutions for best performance.

Original: Interestingly, restricting to radial convolutions achieves slightly better performance than restricting to graph convolutions over the single sphere.
Revised: Interestingly, restricting to radial convolutions achieves slightly better performance than restricting to graph convolutions over the single sphere.

Original: This provides further empirical support for the expressiveness of our proposed representation and radial convolutions.[Table]
Revised: This provides further empirical support for the expressiveness of our proposed representation and radial convolutions.

Original: 5. P OINT C LOUD D ATA M APPING A BLATION S TUDY

Original: [Table]
Revised: [Equation]

Original: In this section we compare alternative choices of neighborhood in the mapping of points to vertices, as well as alternative functions to the Gaussian radial basis function (RBF).
Revised: In this section we compare alternative choices of neighborhood in the mapping of points to vertices, as well as alternative functions to the Gaussian radial basis function (RBF).

Original: The bounding vertices approach (detailed in Sec.
Revised: The bounding vertices [Table]

Original: 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.
Revised: [Table]

Original: 1 (b)) corresonding to a neighborhood of 6 vertices.
Revised: in vertices in space (see Fig. 1) corresonding to a neighborhood of 6 vertices.

Original: An alternative is to deﬁne vertexcentered neighborhood consisting of the k -nearest points to each vertex.
Revised: An alternative is to deﬁne vertex-centered neighborhood consisting of the k -nearest points to each vertex.

Original: We use the Gaussian RBF to map the contribution of each point based on its distance to the vertex.
Revised: We use the Gaussian RBF to map the contribution of each point based on its distance to the vertex.

Original: The Gaussian parameter γ is chosen such that the contribution of points up to a maximum distance of 2 R decay to a small threshold value ( T = 0 .
Revised: The Gaussian parameter γ is chosen such that the contribution of points up to a maximum distance of 2 R decay to a small threshold value ( T = 0 .

Original: 01 in this case).
Revised: 01 in this case).

Original: Results from Table 5 show the bounding vertices approach outperforming the vertex-centered KNN approach.
Revised: Results from Table 5 show the bounding vertices approach outperforming the vertex-centered k-NN approach.

Original: One possible reason is the difﬁculty in controlling for scale, especially when combined with a ﬁxed-parameter decay function, as vertices closer to the center are more densely situated in space compared to vertices further from the center.
Revised: One possible reason is the difﬁculty in controlling for scale, especially when combined with a ﬁxed-parameter decay function, as vertices closer to the center are more densely situated in space compared to vertices further from the center.

Original: We also compare four different decay functions applied to bounding vertices neighborhood.
Revised: We also compare four different decay functions applied to bounding vertices neighborhood.

Original: The constant function is trivial in that it is independent to distance, and essentially serves as an indicator.
Revised: The constant function is trivial in that it is independent to distance, and essentially serves as an indicator.

Original: The linear function decays to zero with distance, but is non-smooth.
Revised: The linear function decays to zero with distance, but is non-smooth.

Original: c is a speciﬁed parameter for a distance cutoff.
Revised: c is a speciﬁed parameter for a distance cutoff.

Original: Inverse quadratic and Gaussian are two smoothly decaying functions commonly used as radial basis functions.
Revised: Inverse quadratic and Gaussian are two smoothly decaying functions commonly used as radial basis functions.

Original: The latter two perform slightly better than the constant and linear functions, suggesting some advantage to using RBFs to summarize neighborhood contributions.
Revised: The latter two perform slightly better than the constant and linear functions, suggesting some advantage to using RBFs to summarize neighborhood contributions.

Original:  D ISCUSSION AND C ONCLUSIONS
Revised:  D ISCUSSION AND C ONCLUSIONS

Original: In this work we proposed a new multi-sphere convolutional architecture, CSGNN, for learning rotationally invariant representations of 3D data.
Revised: In this work we proposed a new multi-sphere convolutional architecture, CSGNN, for learning rotationally invariant representations of 3D data.

Original: We introduced distinct intra-sphere and inter-sphere convolutions, which can be combined to learn more expressive representations compared to being restricted to single-sphere representation.
Revised: We introduced distinct intra-sphere and inter-sphere convolutions, which can be combined to learn more expressive representations compared to being restricted to single-sphere representation.

Original: Our use of graph and 1D convolutions preserves rotational equivariance, while achieving linear scalability with respect to size of discretization.
Revised: Our use of graph and 1D convolutions preserves rotational equivariance, while achieving linear scalability with respect to size of discretization.

Original: We achieve state-of-the-art performance in ModelNet classiﬁcation for testing on arbitrary rotations among both spherical CNN and pointwise convolutional models.
Revised: We achieve state-of-the-art performance in ModelNet classiﬁcation for testing on arbitrary rotations among both spherical CNN and pointwise convolutional models.

Original: 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.
Revised: 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.

Original: One avenue of future work is to explore more descriptive mappings of point cloud data to the discretization.
Revised: One avenue of future work is to explore more descriptive mappings of point cloud data to the discretization.

Original: A learned assignment may better learn vertex features for describing nearby points.
Revised: A learned assignment may better learn vertex features for describing nearby points.

Original: There is also room to explore other kinds of convolutions for incorporating inter-sphere information, as well as other radial division schemes.
Revised: There is also room to explore other kinds of convolutions for incorporating inter-sphere information, as well as other radial division schemes.

Original: Finally, existing implementations in this work can be more efﬁcient implementations based on using regular properties of the icosahedral grid, as opposed to using a general graph construction.
Revised: Finally, existing implementations in this work can be more efﬁcient implementations based on using regular properties of the icosahedral grid, as opposed to using a general graph construction.

Original: A A PPENDIX A. P OINT C LOUD TO S PHERICAL S IGNAL
Revised: A A PPENDIX A. P OINT C LOUD TO S PHERICAL S IGNAL

Original: Instead of computing the summation in Eq.
Revised: Instead of computing the summation in Eq.

Original: 3 with respect to all points, for each data point we update the features of vertices in a local neighborhood.
Revised: 3 with respect to all points for each data point we update the features of vertices in a local neighborhood.

Original: The radial basis function φ decays exponentially, and so points beyond a local neighborhood have little inﬂuence (depending on choice of bandwidth γ ).
Revised: The radial basis function φ decays exponentially, and so points beyond a local neighborhood have little inﬂuence (depending on choice of bandwidth γ ).

Original: Restricting to a constant size local neighborhood improves computation from O ( NV ) to O ( N ) .
Revised: Restricting to a constant size local neighborhood improves computation from O ( NV ) to O ( N ) .

Original: To deﬁne the local neighborhood of data point p in this work: any given point p is contained within two bounding “triangles” of the discretization (ignoring boundary conditions and degenerate cases).
Revised: To deﬁne the local neighborhood of data point p in this work: any given point p is contained within two bounding “triangles” of the discretization (ignoring boundary conditions and degenerate cases).

Original: These correspond to the vertices S ( i ) =
Revised: These correspond to the vertices S ( i ) =

Original: { u ( i ) , v ( i ) , w ( i ) } and S ( i +1)
Revised: { u ( i ) , v ( i ) , w ( i ) } and S ( i +1)

Original: = { u ( i +1) , v ( i +1) , w ( i +1) } , where i indexes radial level.
Revised: = { u ( i +1) , v ( i +1) , w ( i +1) } , where i indexes radial level.

Original: However, using a single γ value for the RBF results in scaling inconsistency: distances between vertices progressively shrink moving to inner spheres.
Revised: However, using a single γ value for the RBF results in scaling inconsistency: distances between vertices progressively shrink moving to inner spheres.

Original: Based on the assumption that RBF values should be invariant to scale, a different γ and corresponding RBF is deﬁned with respect to radial level.
Revised: Based on the assumption that RBF values should be invariant to scale a different γ and corresponding RBF is deﬁned with respect to radial level.

Original: To deﬁne γ i, we use the maximum pairwise distance d ( i ) max between vertices in { S ( i ) , S ( i +1) } .
Revised: To deﬁne γ i we use the maximum pairwise distance d ( i ) max between vertices in { S ( i ) , S ( i +1) } .

Original: Speciﬁcally, we set γ i = − log T d ( i ) max 2 , where T is a lower bound target
Revised: Speciﬁcally, we set γ i = − log T d ( i ) max 2 , where T is a lower bound target

Original: RBF value.
Revised: RBF value.

Original: For example, T = 1 would correspond to γ i = 0 , or a RBF value of 1 at any distance.
Revised: For example, T = 1 would correspond to γ i = 0 , or a RBF value of 1 at any distance.

Original: T ∈ (0 , 1] is a tuning parameter that enables toggling the overall sensitivity of the RBF to distances.
Revised: T ∈ (0 , 1] is a tuning parameter that enables toggling the overall sensitivity of the RBF to distances.

Original: Based on the approximation that d ( i ) max is similar for any data point, d ( i ) max is precomputed once.
Revised: Based on the approximation that d ( i ) max is similar for any data point, d ( i ) max is precomputed once.

Original: A. M ODEL N ET 40 T IME A NALYSIS
Revised: A. M ODEL N ET 40 T IME A NALYSIS

Original: [Table]

Original: [Table]
Revised: [Table]

Original: We compare total training time and batch inference time of baselines from Sec. 5.1, for all instances where code was available.
Revised: We compare total training time and batch inference time of baselines from Sec. 5.1, for all instances where code was available.

Original: Results are reported in Table 6.
Revised: Results are reported in Table 6.

Original: All baselines were run on the NVIDIA
Revised: All baselines were run on the NVIDIA

Original: Tesla P100 GPU.
Revised: Tesla P100 GPU.

Original: Total training time includes data loading and transformation time.
Revised: Total training time includes data loading and transformation time.

Original: Inference time is based on batch size of 32, and computed from averaging 32 differentbatches.
Revised: Inference time is based on batch size of 32, and computed from averaging 32 different

Original: The inferencetimes reported do not includedata loading ortransformation time.
Revised: The inference reported do not include or

Original: A. C HOICE OF C ENTER
Revised: A. C HOICE OF C ENTER

Original: [Table]
Revised: [Table]

Original: A choice of center is needed to represent the 3D data using concentric spheres, and also is necessary to deﬁne rotations of the data.
Revised: A choice of center is needed to represent the 3D data using concentric spheres, and also is necessary to deﬁne rotations of the data.

Original: The ModelNet40 dataset from Qi et al.
Revised: The ModelNet40 dataset from Qi et al.

Original: (2017) centers the data based on the centroid of a point cloud, while the SHREC17 dataset uses the centroid of mesh shapes.
Revised: (2017) centers the data based ([ GC 32 + BN + ReLU ] L 4 × 2) + [ RC 32 + BN + ReLU ] L 4 + [ RC 64 + BN + ReLU ]

Revised: L+ MaxPool + ([ GC 64 + BN + ReLU ]

Revised: L 3 × 2) + [ RC 64 + BN + ReLU ]

Revised: L 3 + [ RC 128 + BN + ReLU ]

Revised: L+ MaxPool + ([ GC 128 + BN + ReLU ]

Revised: L 2 × 2) + [ RC 128 + BN + ReLU ]

Revised: L 2 + [ RC 256 + BN + ReLU ]

Revised: L+ MaxPool + [ GC 256 + BN + ReLU ]

Revised: L 1 + [ RC 512 + BN + ReLU ]

Revised: L 1 + MaxPool + [ GC 512 + BN + ReLU ]

Revised: L 0 + GP vertices + [ RC 1024 + BN + ReLU ]

Revised: L 0 + GP radial

Revised: [Equation]

Revised: Downsampling results in a coarser grid and vertex set, where a new neighborhood is deﬁned (orange vertices) on the centroid of a point cloud, while the SHREC17 dataset uses the centroid of mesh shapes.

Original: Our experiments in Sec.
Revised: Our experiments in Sec.

Original: 5.1 and 5.2 follow this convention.
Revised: 5.1 and 5.2 follow this convention.

Original: In this section we also consider other choices of center besides centroid, and show that our representation is relatively robust to other centers of rotation, in application to the ModelNet40 dataset.
Revised: In this section we also consider other choices of center besides centroid, and show that our representation is relatively robust to other centers of rotation, in application to the ModelNet40 dataset.

Original: We apply singular value decomposition to an input point cloud to obtain 3 principal component vectors.
Revised: We apply singular value decomposition to an input point cloud to obtain 3 principal component vectors.

Original: These 3 dimensional vectors each serve as a new reference point by which the point cloud is re-centered.
Revised: These 3 dimensional vectors each serve as a new reference point by which the point cloud is re-centered.

Original: Results from Table 7 show that centroid-based center produces performs best with our model.
Revised: Results from Table 7 show that centroid-based center produces performs best with our model.

Original: Performance of principal componentbased center declines according to singular value.
Revised: Performance of principal componentbased center declines according to singular value.

Original: 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.
Revised: 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.

Original: Overall the results suggest that CSGNN can be effective even when using non-centroid centers of representation and data rotation.
Revised: Overall the results suggest that CSGNN can be effective even when using non-centroid centers of representation and data rotation.

Original: A. SHREC17 V ISUALIZATION
Revised: A. A RCHITECTURE D ETAILS

Original: [Figure]
Revised: Fig. 6 provides visualization of the vertex pooling and downsampling operations. Fig. 5 provides more detail to the base architecture used in our experiments.

Revised: Radial convolutions are implemented using 1D convolution layers with a kernel size of 3 in the ModelNet40 experiment, and either 1 orin the SHREC17 experiment.

Revised: Radial convolutions are followed by max pooling and downsampling to a lower level of spherical discretization, reducing the number of vertices.

Revised: We also add residual connections between every graph convolution layer, whenever the number of input channels matches output channels.

Revised: A. V ISUALIZING C ONCENTRIC S PHERICAL F EATURES

Revised: To better understand the impact of the multi-sphere aspect of the concentric spherical representation, we visualize the learned features from 3 different point cloud examples in Fig.

Revised: We use the 1st

Revised: [Figure]

Revised: feature channel in each visualization instance, and the features are obtained after radial convolution layers but before pooling and downsampling.

Revised: The results visually demonstrate that different radial spheres within the same representation are capturing different features.

Revised: At the same time, there is also a high degree of continuity between consecutive spheres in most cases, suggesting that there is information sharing between spheres resulting from radial convolutions.

Revised: A. SHREC17 V ISUALIZATION

Revised: See Fig.

Revised: 8 for SHREC17 visualization.