Original: G AUGE E QUIVARIANT M ESH CNN S A NISOTROPIC CONVOLUTIONS ON GEOMETRIC GRAPHS
Revised: G AUGE E QUIVARIANT M ESH CNN S A NISOTROPIC CONVOLUTIONS ON GEOMETRIC GRAPHS

Original: Anonymous authors

Original: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: A common approach to deﬁne convolutions on meshes is to interpret them as a graph and apply graph convolutional networks (GCNs).
Revised: A common approach to deﬁne convolutions on meshes is to interpret them as a graph and apply graph convolutional networks (GCNs).

Original: Such GCNs utilize isotropic kernels and are therefore insensitive to the relative orientation of vertices and thus to the geometry of the mesh as a whole.
Revised: Such GCNs utilize isotropic kernels and are therefore insensitive to the relative orientation of vertices and thus to the geometry of the mesh as a whole.

Original: We propose Gauge Equivariant
Revised: We propose Gauge Equivariant

Original: Mesh CNNs which generalize GCNs to apply anisotropic gauge equivariant kernels.
Revised: Mesh CNNs which generalize GCNs to apply anisotropic gauge equivariant kernels.

Original: Since the resulting features carry orientation information, we introduce a geometric message passing scheme deﬁned by parallel transporting features over mesh edges.
Revised: Since the resulting features carry orientation information, we introduce a geometric message passing scheme deﬁned by parallel transporting features over mesh edges.

Original: Our experiments validate the signiﬁcantly improved expressivity of the proposed model over conventional GCNs and other methods.
Revised: Our experiments validate the signiﬁcantly improved expressivity of the proposed model over conventional GCNs and other methods.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Convolutional neural networks (CNNs) have been established as the default method for many machine learning tasks like speech recognition or planar and volumetric image classiﬁcation and segmentation.
Revised: Convolutional neural networks (CNNs) have been established as the default method for many machine learning tasks like speech recognition or planar and volumetric image classiﬁcation and segmentation.

Original: Most CNNs are restricted to ﬂat or spherical geometries, where convolutions are easily deﬁned and optimized implementations are available.
Revised: Most CNNs are restricted to ﬂat or spherical geometries, where convolutions are easily deﬁned and optimized implementations are available.

Original: The empirical success of CNNs on such spaces has generated interest to generalize convolutions to more general spaces like graphs or Riemannian manifolds, creating a ﬁeld now known as geometric deep learning (Bronstein et al., 2017).
Revised: The empirical success of CNNs on such spaces has generated interest to generalize convolutions to more general spaces like graphs or Riemannian manifolds, creating a ﬁeld now known as geometric deep learning (Bronstein et al., 2017).

Original: A case of speciﬁc interest is convolution on meshes , the discrete analog of 2-dimensional embedded
Revised: A case of speciﬁc interest is convolution on meshes , the discrete analog of 2-dimensional embedded

Original: Riemannian manifolds.
Revised: Riemannian manifolds.

Original: Mesh CNNs can be applied to tasks such as detecting shapes, registering different poses of the same shape and shape segmentation.
Revised: Mesh CNNs can be applied to tasks such as detecting shapes, registering different poses of the same shape and shape segmentation.

Original: If we forget the positions of vertices, and which vertices form faces, a mesh M can be represented by a graph G .
Revised: If we forget the positions of vertices, and which vertices form faces, a mesh M can be represented by a graph G .

Original: This allows for the application of graph convolutional networks (GCNs) to processing signals on meshes.
Revised: This allows for the application of graph convolutional networks (GCNs) to processing signals on meshes.

Original: However, when representing a mesh by a graph, we lose important geometrical information.
Revised: However, when representing a mesh by a graph, we lose important geometrical information.

Original: In particular, in a graph there is no notion of angle between or ordering of two of a node’s incident edges (see ﬁgure 1).
Revised: In particular, in a graph there is no notion of angle between or ordering of two of a node’s incident edges (see ﬁgure 1).

Original: Hence, a GCNs output at a node p is designed to be independent of relative angles and invariant to any permutation of its neighbours q i ∈ N ( p ) .
Revised: Hence, a GCNs output at a node p is designed to be independent of relative angles and invariant to any permutation of its neighbours q i ∈ N ( p ) .

Original: A graph convolution on a mesh graph therefore corresponds to applying an isotropic convolution kernel.
Revised: A graph convolution on a mesh graph therefore corresponds to applying an isotropic convolution kernel.

Original: Isotropic ﬁlters are insensitive to the orientation of input patterns, so their features are strictly less expressive than those of orientation aware anisotropic ﬁlters.
Revised: Isotropic ﬁlters are insensitive to the orientation of input patterns, so their features are strictly less expressive than those of orientation aware anisotropic ﬁlters.

Original: To address this limitation of graph networks we propose Gauge Equivariant Mesh CNNs (GEM-CNN), which minimally modify GCNs such that they are able to use anisotropic ﬁlters while sharing weights[Table] across different positions and respecting the local geometry.
Revised: To address this limitation of graph networks we propose Gauge Equivariant Mesh CNNs (GEM-CNN), which minimally modify GCNs such that they are able to use anisotropic ﬁlters while sharing weights across different positions and respecting the local geometry.

Original: One obstacle in sharing anisotropic kernels, which are functions of the angle θ pq of neighbour q with respect to vertex p , over multiple vertices of a mesh is that there is no unique way of selecting a reference neighbour q 0 , which has the direction θ pq 0 = 0 .
Revised: One obstacle in sharing anisotropic kernels, which are functions of the angle θ pq of neighbour q with respect to vertex p , over multiple vertices of a mesh is that there is no unique way of selecting a reference neighbour q 0 , which has the direction θ pq 0 = 0 .

Original: The reference neighbour, and hence the orientation of the neighbours, needs to be chosen arbitrarily.
Revised: The reference neighbour, and hence the orientation of the neighbours, needs to be chosen arbitrarily.

Original: In order to guarantee the equivalence of the features resulting from different choices of orientations, we adapt Gauge Equivariant CNNs (Cohen et al., 2019b) to general meshes.
Revised: In order to guarantee the equivalence of the features resulting from different choices of orientations, we adapt Gauge Equivariant CNNs (Cohen et al., 2019b) to general meshes.

Original: The kernels of our model are thus designed to be equivariant under gauge transformations , that is, to guarantee that the responses for different kernel orientations are related by a prespeciﬁed transformation law.
Revised: The kernels of our model are thus designed to be equivariant under gauge transformations , that is, to guarantee that the responses for different kernel orientations are related by a prespeciﬁed transformation law.

Original: Such features are identiﬁed as geometric objects like scalars, vectors, tensors, etc., depending on the speciﬁc choice of transformation law.
Revised: Such features are identiﬁed as geometric objects like scalars, vectors, tensors, etc., depending on the speciﬁc choice of transformation law.

Original: In order to compare such geometric features at neighbouring vertices, they need to be parallel transported along the connecting edge.
Revised: In order to compare such geometric features at neighbouring vertices, they need to be parallel transported along the connecting edge.

Original: In our implementation we ﬁrst specify the transformation laws of the feature spaces and compute a space of gauge equivariant kernels.
Revised: In our implementation we ﬁrst specify the transformation laws of the feature spaces and compute a space of gauge equivariant kernels.

Original: Then we pick arbitrary reference orientations at each node, relative to which we compute neighbour orientations and compute the corresponding edge transporters.
Revised: Then we pick arbitrary reference orientations at each node, relative to which we compute neighbour orientations and compute the corresponding edge transporters.

Original: Given these quantities, we deﬁne the forward pass as a message passing step via edge transporters followed by a contraction with the equivariant kernels evaluated at the neighbour orientations.
Revised: Given these quantities, we deﬁne the forward pass as a message passing step via edge transporters followed by a contraction with the equivariant kernels evaluated at the neighbour orientations.

Original: Algorithmically,
Revised: Algorithmically,

Original: Gauge Equivariant Mesh CNNs are therefore just GCNs with anisotropic, gauge equivariant kernels and message passing via parallel transporters.
Revised: Gauge Equivariant Mesh CNNs are therefore just GCNs with anisotropic, gauge equivariant kernels and message passing via parallel transporters.

Original: Conventional GCNs are covered in this framework for the speciﬁc choice of isotropic kernels and trivial edge transporters, given by identity maps.
Revised: Conventional GCNs are covered in this framework for the speciﬁc choice of isotropic kernels and trivial edge transporters, given by identity maps.

Original: In Sec. 2, we will give an outline of our method, deferring details to Secs.
Revised: In Sec. 2, we will give an outline of our method, deferring details to Secs.

Original: 3 and 4.
Revised: 3 and 4.

Original: In Sec. 3.2, we describe how to compute general geometric quantities, not speciﬁc to our method, used for the computation of the convolution.
Revised: In Sec. 3.2, we describe how to compute general geometric quantities, not speciﬁc to our method, used for the computation of the convolution.

Original: In our experiments in Sec. 6, we ﬁnd that the enhanced expressiveness of Gauge Equivariant Mesh CNNs enables them to outperform conventional GCNs and other prior work in a shape correspondence task.
Revised: In our experiments in Sec. 6.1, we ﬁnd that the enhanced expressiveness of Gauge Equivariant Mesh CNNs enables them to outperform conventional GCNs and other prior work in a shape correspondence task.

Original:  C ONVOLUTIONS ON G RAPHS WITH G EOMETRY
Revised:  C ONVOLUTIONS ON G RAPHS WITH G EOMETRY

Original: We consider the problem of processing signals on discrete 2-dimensional manifolds, or meshes M .
Revised: We consider the problem of processing signals on discrete 2-dimensional manifolds, or meshes M .

Original: Such meshes are described by a set V of vertices in R 3 together with a set F of tuples, each consisting of the vertices at the corners of a face.
Revised: Such meshes are described by a set V of vertices in R 3 together with a set F of tuples, each consisting of the vertices at the corners of a face.

Original: For a mesh to describe a proper manifold, each edge needs to be connected to two faces, and the neighbourhood of each vertex needs to be homeomorphic to a disk.
Revised: For a mesh to describe a proper manifold, each edge needs to be connected to two faces, and the neighbourhood of each vertex needs to be homeomorphic to a disk.

Original: Mesh M induces a graph G by forgetting the coordinates of the vertices while preserving the edges.
Revised: Mesh M induces a graph G by forgetting the coordinates of the vertices while preserving the edges.

Original: A conventional graph convolution between kernel K and signal f , evaluated at a vertex p , can be deﬁned by
Revised: A conventional graph convolution between kernel K and signal f , evaluated at a vertex p , can be deﬁned by

Original: [Equation]
Revised: [Equation]

Original:  where N p is the set of neighbours of p in G , and K self ∈ R C in × C out and K neigh ∈ R C in × C out are two linear maps which model a self interaction and the neighbour contribution, respectively.
Revised: (1) where N p is the set of neighbours of p in G , and K self ∈ R C in × C out and K neigh ∈ R C in × C out are two linear maps which model a self interaction and the neighbour contribution, respectively.

Original: Importantly, graph convolution does not distinguish different neighbours, because each feature vector f q is multiplied by the same matrix K neigh and then summed.
Revised: Importantly, graph convolution does not distinguish different neighbours, because each feature vector f q is multiplied by the same matrix K neigh and then summed.

Original: For this reason we say the kernel is isotropic .
Revised: For this reason we say the kernel is isotropic .

Original: Consider the example in ﬁgure 1, where on the left and right, the neighbourhood of one vertex p , containing neighbours
Revised: Consider the example in ﬁgure 1, where on the left and right, the neighbourhood of one vertex p , containing neighbours

Original: q ∈ N p , is visualized.
Revised: q ∈ N p , is visualized.

Original: An isotropic kernel would propagate the signal from the neighbours to p in exactly the same way in both neighbourhoods, even though the neighbourhoods are geometrically distinct.
Revised: An isotropic kernel would propagate the signal from the neighbours to p in exactly the same way in both neighbourhoods, even though the neighbourhoods are geometrically distinct.

Original: For this reason, our method uses direction sensitive ( anisotropic ) kernels instead of isotropic kernels.
Revised: For this reason, our method uses direction sensitive ( anisotropic ) kernels instead of isotropic kernels.

Original: Anisotropic kernels are inherently more expressive than isotropic ones which is why they are used universally in conventional planar CNNs.
Revised: Anisotropic kernels are inherently more expressive than isotropic ones which is why they are used universally in conventional planar CNNs.

Original:  We propose the Gauge Equivariant Mesh Convolution, a minimal modiﬁcation of graph convolution that allows for anisotropic kernels K ( θ ) whose value depends on an orientation θ ∈ [0 , 2 π ) .
Revised: [Table] We propose the Gauge Equivariant Mesh Convolution, a minimal modiﬁcation of graph convolution that allows for anisotropic kernels K ( θ ) whose value depends on an orientation θ ∈ [0 , 2 π ) .

Original: 1 To deﬁne the orientations θ pq of neighbouring vertices q ∈ N p of p , we ﬁrst map them to the tangent plane T p M at p , as visualized in ﬁgure 1.
Revised: 1 To deﬁne the orientations θ pq of neighbouring vertices q ∈ N p of p , we ﬁrst map them to the tangent plane T p M at p , as visualized in ﬁgure 1.

Original: We then pick an arbitrary reference neighbour q p 0 to determine a reference orientation 2 θ pq p 0 :
Revised: We then pick an arbitrary reference neighbour q p 0 to determine a reference orientation 2 θ pq p 0 :

Original: = 0 , marked orange in ﬁgure 1.
Revised: = 0 , marked orange in ﬁgure 1.

Original: This induces a basis on the tangent plane, which, when expressed in polar coordinates, deﬁnes the angles θ pq of the other neighbours.
Revised: This induces a basis on the tangent plane, which, when expressed in polar coordinates, deﬁnes the angles θ pq of the other neighbours.

Original: As we will motivate in the next section, features in a Gauge Equivariant CNN are coefﬁcients of geometric quantities.
Revised: As we will motivate in the next section, features in a Gauge Equivariant CNN are coefﬁcients of geometric quantities.

Original: For example, a tangent vector at vertex p can be described either geometrically by a 3 dimensional vector orthogonal to the normal at p or by two coefﬁcients in the basis on the tangent plane.
Revised: For example, a tangent vector at vertex p can be described either geometrically by a 3 dimensional vector orthogonal to the normal at p or by two coefﬁcients in the basis on the tangent plane.

Original: In order to perform convolution, geometric features at different vertices need to be linearly combined, for which it is required to ﬁrst “parallel transport” the features to the same vertex.
Revised: In order to perform convolution, geometric features at different vertices need to be linearly combined, for which it is required to ﬁrst “parallel transport” the features to the same vertex.

Original: This is done by applying a matrix ρ ( g q → p ) ∈ R C in × C in to the coefﬁcients of the feature at q , in order to obtain the coefﬁcients of the feature vector transported to p , which can be used for the convolution at p .
Revised: This is done by applying a matrix ρ ( g q → p ) ∈ R C in × C in to the coefﬁcients of the feature at q , in order to obtain the coefﬁcients of the feature vector transported to p , which can be used for the convolution at p .

Original: The transporter depends on the geometric type (group representation) of the feature, denoted by ρ and described in more detail below.
Revised: The transporter depends on the geometric type (group representation) of the feature, denoted by ρ and described in more detail below.

Original: Details of how the tangent space is deﬁned, how to compute the map to the tangent space, angles θ pq , and the parallel transporter are given in Appendix A.
Revised: Details of how the tangent space is deﬁned, how to compute the map to the tangent space, angles θ pq , and the parallel transporter are given in Appendix A.

Original: In combination, this leads to the GEM-CNN convolution
Revised: In combination, this leads to the GEM-CNN convolution

Original: [Equation]
Revised: [Equation]

Original: which differs from the conventional graph convolution, deﬁned in Eq.
Revised: which differs from the conventional graph convolution, deﬁned in Eq.

Original: 1 only by the use of an anisotropic kernel and the parallel transport message passing.
Revised: 1 only by the use of an anisotropic kernel and the parallel transport message passing.

Original: We require the outcome of the convolution to be equivalent for any choice of reference orientation.
Revised: We require the outcome of the convolution to be equivalent for any choice of reference orientation.

Original: This is not the case for any anisotropic kernel but only for those which are equivariant under changes of reference orientations (gauge transformations).
Revised: This is not the case for any anisotropic kernel but only for those which are equivariant under changes of reference orientations (gauge transformations).

Original: Equivariance imposes a linear constraint on the kernels.
Revised: Equivariance imposes a linear constraint on the kernels.

Original: We therefore solve for complete sets of “basis-kernels” K i self and K i neigh satisfying this constraint and linearly combine them with parameters w i self and w i neigh such that K self =
Revised: We therefore solve for complete sets of “basis-kernels” K i self and K i neigh satisfying this constraint and linearly combine them with parameters w i self and w i neigh such that K self =

Original: (cid:80)
Revised: (cid:80)

Original: i w i
Revised: i w i

Original: self K i self and K neigh
Revised: self K i self and K neigh

Original: =
Revised: =

Original: (cid:80)
Revised: (cid:80)

Original: i w i neigh K i neigh .
Revised: i w i neigh K i neigh .

Original: Details on the computation of basis kernels are given in section 3.
Revised: Details on the computation of basis kernels are given in section 3.

Original: The full algorithm for initialisation and forward pass, which is of time and space complexity linear in the number of vertices, for a GEM-CNN layer are listed in algorithm 1.
Revised: The full algorithm for initialisation and forward pass, which is of time and space complexity linear in the number of vertices, for a GEM-CNN layer are listed in algorithm 1.

Original: Gradients can be computed by automatic differentiation.
Revised: Gradients can be computed by automatic differentiation.

Original: The GEM-CNN is gauge equivariant, but furthermore satisﬁes two important properties.
Revised: The GEM-CNN is gauge equivariant, but furthermore satisﬁes two important properties.

Original: Firstly, it depends only on the intrinsic shape of the 2D mesh, not on the embedding of the mesh in R 3 .
Revised: Firstly, it depends only on the intrinsic shape of the 2D mesh, not on the embedding of the mesh in R 3 .

Original: Secondly, whenever a map from the mesh to itself exists that preserves distances and orientation, the convolution is equivariant to moving the signal along such transformations.
Revised: Secondly, whenever a map from the mesh to itself exists that preserves distances and orientation, the convolution is equivariant to moving the signal along such transformations.

Original: These properties are proven in Appendix D and empirically shown in Appendix F.2.
Revised: These properties are proven in Appendix D and empirically shown in Appendix F.2.

Original: G AUGE E QUIVARIANCE & G EOMETRIC F EATURES

Original: On a general mesh, the choice of the reference neighbour, or gauge, which deﬁnes the orientation of the kernel, can only be made arbitrarily.

Original: However, this choice should not arbitrarily affect the outcome of the convolution, as this would impede the generalization between different locations and different meshes.

Original: Instead, Gauge Equivariant Mesh CNNs have the property that their output transforms according to a known rule as the gauge changes. Consider the left hand side of ﬁgure 2(a). Given a neighbourhood of vertex p , we want to express each neighbour q in terms of its polar coordinates ( r q , θ q ) on the tangent plane, so that the kernel p q A q B q C p q A q B q C p q A q B q C p q A q B q C pickgauge q 0 =
Revised: p q A q B q C p q A q B q C p q A q B q C p q A q B q C pickgauge q 0 =

Original: q A mapbacktomesh pickgauge q 0 =
Revised: q A mapbacktomesh pickgauge q 0 =

Original: q B mapbacktomesh geometricconv convingauge A convingauge B g a ug e t r a n s f o r m a ti on A → B g a ug e t r a n s f o r m a ti on A → B
Revised: q B mapbacktomesh geometricconv convingauge A convingauge B g a ug e t r a n s f o r m a ti on A → B g a ug e t r a n s f o r m a ti on A → B

Original: [Figure]
Revised: [Figure]

Original: p q A q B q C p q A q B q C p q A q B q C p q A q B q C pickgauge q 0 =
Revised: p q A q B q C p q A q B q C p q A q B q C p q A q B q C pickgauge q 0 =

Original: q A mapbacktomesh pickgauge q 0 =
Revised: q A mapbacktomesh pickgauge q 0 =

Original: q B mapbacktomesh geometricconv convingauge A convingauge B g a ug e t r a n s fr o m a ti on A → B g a ug e t r a n s fr o m a ti on A → B
Revised: q B mapbacktomesh geometricconv convingauge A convingauge B g a ug e t r a n s fr o m a ti on A → B g a ug e t r a n s fr o m a ti on A → B

Original: [Figure]
Revised: [Figure]

Revised:  G AUGE E QUIVARIANCE & G EOMETRIC F EATURES

Revised: On a general mesh, the choice of the reference neighbour, or gauge, which deﬁnes the orientation of the kernel, can only be made arbitrarily.

Original:  value at that neighbour K neigh ( θ q ) is well deﬁned.
Revised: However, this choice should not arbitrarily affect the outcome of the convolution, as this would impede the generalization between different locations and different meshes. Instead, Gauge Equivariant Mesh CNNs have the property that their output transforms according to a known rule as the gauge changes. Consider the left hand side of ﬁgure 2(a). Given a neighbourhood of vertex p , we want to express each neighbour q in terms of its polar coordinates ( r q , θ q ) on the tangent plane, so that the kernel value at that neighbour K neigh ( θ q ) is well deﬁned.

Original: This requires choosing a basis on the tangent plane, determined by picking a neighbour as reference neighbour (denoted q 0 ), which has the zero angle θ q 0 = 0 .
Revised: This requires choosing a basis on the tangent plane, determined by picking a neighbour as reference neighbour (denoted q 0 ), which has the zero angle θ q 0 = 0 .

Original: In the top path, we pick q A as reference neighbour.
Revised: In the top path, we pick q A as reference neighbour.

Original: Let us call this gauge A, in which neighbours have angles θ Aq .
Revised: Let us call this gauge A, in which neighbours have angles θ Aq .

Original: In the bottom path, we instead pick neighbour q B as reference point and are in gauge
Revised: In the bottom path, we instead pick neighbour q B as reference point and are in gauge

Original: B.
Revised: B.

Original: We get a different basis for the tangent plane and different angles θ Bq for each neighbour.
Revised: We get a different basis for the tangent plane and different angles θ Bq for each neighbour.

Original: Comparing the two gauges, we see that they are related by a rotation, so that θ Bq = θ
Revised: Comparing the two gauges, we see that they are related by a rotation, so that θ Bq = θ

Original: Aq − θ Aq B .
Revised: Aq − θ Aq B .

Original: This change of gauge is called a gauge transformation of angle g := θ Aq B .
Revised: This change of gauge is called a gauge transformation of angle g := θ Aq B .

Original: In ﬁgure 2(a), we illustrate a gauge equivariant convolution that takes input and output features such as gray scale image values on the mesh, which are called scalar features.
Revised: In ﬁgure 2(a), we illustrate a gauge equivariant convolution that takes input and output features such as gray scale image values on the mesh, which are called scalar features.

Original: The top path represents the convolution in gauge A, the bottom path in gauge B.
Revised: The top path represents the convolution in gauge A, the bottom path in gauge B.

Original: In either case, the convolution can be interpreted as consisting of three steps.
Revised: In either case, the convolution can be interpreted as consisting of three steps.

Original: First, for each vertex p , the value of the scalar features on the mesh at each neighbouring vertex q , represented by colors, is mapped to the tangent plane at p at angle θ q deﬁned by the gauge.
Revised: First, for each vertex p , the value of the scalar features on the mesh at each neighbouring vertex q , represented by colors, is mapped to the tangent plane at p at angle θ q deﬁned by the gauge.

Original: Subsequently, the convolutional kernel sums for each neighbour q , the product of the feature at q and kernel K ( θ q ) .
Revised: Subsequently, the convolutional kernel sums for each neighbour q , the product of the feature at q and kernel K ( θ q ) .

Original: Finally the output is mapped back to the mesh.
Revised: Finally the output is mapped back to the mesh.

Original: These three steps can be composed into a single step, which we could call a geometric convolution , mapping from input features on the mesh to output features on the mesh.
Revised: These three steps can be composed into a single step, which we could call a geometric convolution , mapping from input features on the mesh to output features on the mesh.

Original: The convolution is gauge equivariant if this geometric convolution does not depend on the gauge we pick in the interim, so in ﬁgure 2(a), if the convolution in the top path in gauge A has same result the convolution in the bottom path in gauge B, making the diagram commute.
Revised: The convolution is gauge equivariant if this geometric convolution does not depend on the gauge we pick in the interim, so in ﬁgure 2(a), if the convolution in the top path in gauge A has same result the convolution in the bottom path in gauge B, making the diagram commute.

Original: In this case, however, we see that the convolution output needs to be the same in both gauges, for the convolution to be equivariant.
Revised: In this case, however, we see that the convolution output needs to be the same in both gauges, for the convolution to be equivariant.

Original: Hence, we must have that
Revised: Hence, we must have that

Original: K ( θ q ) =
Revised: K ( θ q ) =

Original: K ( θ q − g ) , as the orientations of the neighbours differ by some angle g , and the kernel must be isotropic.
Revised: K ( θ q − g ) , as the orientations of the neighbours differ by some angle g , and the kernel must be isotropic.

Original: As we aim to design an anisotropic convolution, the output feature of the convolution at p can, instead of a scalar, be two numbers v ∈ R 2 , which can be interpreted as coefﬁcients of a tangent featurevector in the tangent space at p , visualized in ﬁgure 2(b).
Revised: As we aim to design an anisotropic convolution, the output feature of the convolution at p can, instead of a scalar, be two numbers v ∈ R 2 , which can be interpreted as coefﬁcients of a tangent feature

Original: As shown on the right hand side, different gauges induce a different basis of the tangent plane, so that the same tangent vector (shown on the middle right on the mesh), is represented by different coefﬁcients in the gauge (shown on the top and bottom on the right).

Original: This gauge equivariant convolution must be anisotropic: going from the top row to the bottom row, if we change orientations of the neighbours by − g , the coefﬁcients of the output vector v ∈ R 2 of the kernel must be also rotated by − g .

Original: This is written as R ( − g ) v , where

Original: R ( − g ) ∈

Original: R 2 × 2 is the matrix that rotates by angle − g .

Original: Vectors and scalars are not the only type of geometric features that can be inputs and outputs of a GEM-CNN layer.
Revised: Vectors and scalars are not the only type of geometric features that can be inputs and outputs of a GEM-CNN layer.

Original: In general, the coefﬁcients of a geometric feature of C dimensions changes by an invertible linear transformation ρ ( − g ) ∈ R C × C if the gauge is rotated by angle g .
Revised: In general, the coefﬁcients of a geometric feature of C dimensions changes by an invertible linear transformation ρ ( − g ) ∈ R C × C if the gauge is rotated by angle g .

Original: The map ρ : [0 , 2 π ) → R C × C is called the type of the geometric quantity and is formally known as a group representation of the planar rotation group SO(2) .
Revised: The map ρ : [0 , 2 π ) → R C × C is called the type of the geometric quantity and is formally known as a group representation of the planar rotation group SO(2) .

Original: Group representations have the property that ρ ( g + h )
Revised: Group representations have the property that ρ ( g + h )

Original: =
Revised: =

Original: ρ ( g ) ρ ( h ) (they are group homomorphisms), which implies in particular that ρ (0)
Revised: ρ ( g ) ρ ( h ) (they are group homomorphisms), which implies in particular that ρ (0)

Original: =
Revised: =

Original: and ρ ( − g )
Revised: and ρ ( − g )

Original: =
Revised: =

Original: ρ ( g ) − 1 .
Revised: ρ ( g ) − 1 .

Original: For more background on group representation theory, we refer the reader to (Serre, 1977) and, speciﬁcally in the context of equivariant deep learning, to (Lang & Weiler, 2020).
Revised: For more background on group representation theory, we refer the reader to (Serre, 1977) and, speciﬁcally in the context of equivariant deep learning, to (Lang & Weiler, 2020).

Original: From the theory of group representations, we know that any feature type can be composed from “irreducible representations” (irreps).
Revised: From the theory of group representations, we know that any feature type can be composed from “irreducible representations” (irreps).

Original: For SO(2) , these are the one dimensional invariant scalar representation ρ 0 and for all n ∈ N > 0 , a two dimensional representation ρ n , 
Revised: For SO(2) , these are the one dimensional invariant scalar representation ρ 0 and for all n ∈ N > 0 , a two dimensional representation ρ n , ρ 0 ( g )

Original: [Equation]
Revised: = 1 , ρ n ( g ) =

Original: [Equation] [Equation] cos ng (cid:19) .
Revised: (cid:18) cos ng (cid:57) sin ng sin ng cos ng (cid:19) .

Revised: where we write, for example, ρ = ρ 0 ⊕ ρ 1 ⊕ ρ 1 to denote that representation ρ ( g ) is the direct sum (i.e. block-diagonal stacking) of the matrices

Revised: ρ 0 ( g ) , ρ 1 ( g ) , ρ 1 ( g ) .

Original: Scalars and tangent vector features correspond to ρ 0 and ρ 1 respectively and we have R ( g )
Revised: Scalars and tangent vector features correspond to ρ 0 and ρ 1 respectively and we have R ( g )

Original: =
Revised: =

Original: ρ 1 ( g ) .
Revised: ρ 1 ( g ) .

Original: The type of the feature at each layer in the network can thus be fully speciﬁed (up to a change of basis) by the number of copies of each irrep.
Revised: The type of the feature at each layer in the network can thus be fully speciﬁed (up to a change of basis) by the number of copies of each irrep.

Original: Similar to the dimensionality in a conventional CNN, the choice of type is a hyperparameter that can be freely chosen to optimize performance.
Revised: Similar to the dimensionality in a conventional CNN, the choice of type is a hyperparameter that can be freely chosen to optimize performance.

Original: We use a notation, such that, for example, ρ = ρ 0 ⊕ ρ 1 ⊕

Original: ρ 1 means that the feature contains one ρ 0 irrep, which is a scalar, and two ρ 1 irreps, which are vectors.

Original: This feature is ﬁve dimensional ( 1 × dim ρ 0 + 2 × dim ρ 1 = 1 × 1 + 2 × 2 = 5 ) and transforms with block diagonal matrix:

Original: [Equation]

Original:  3. K ERNEL C ONSTRAINT
Revised: 3. K ERNEL C ONSTRAINT

Original: Given an input type ρ in and output type ρ out of dimensions C in and C out , the kernels are K self ∈ R C out × C in and K neigh : [0 , 2 π ) → R C out × C in .
Revised: Given an input type ρ in and output type ρ out of dimensions C in and C out , the kernels are K self ∈ R C out × C in and K neigh : [0 , 2 π ) → R C out × C in .

Original: However, not all such kernels are equivariant.
Revised: However, not all such kernels are equivariant.

Original: Consider again examples ﬁgure 2(a) and ﬁgure 2(b).
Revised: Consider again examples ﬁgure 2(a) and ﬁgure 2(b).

Original: If we map from a scalar to a scalar, we get that K neigh ( θ − g )
Revised: If we map from a scalar to a scalar, we get that K neigh ( θ − g )

Original: =
Revised: =

Original: K neigh ( θ ) for all angles θ, g and the convolution is isotropic.
Revised: K neigh ( θ ) for all angles θ, g and the convolution is isotropic.

Original: If we map from a scalar to a vector, we get that rotating the angles θ q results in the same tangent vector as rotating the output vector coefﬁcients, so that K neigh ( θ − g )
Revised: If we map from a scalar to a vector, we get that rotating the angles θ q results in the same tangent vector as rotating the output vector coefﬁcients, so that K neigh ( θ − g )

Original: =
Revised: =

Original: R ( − g )
Revised: R ( − g )

Original: K neigh ( θ ) .
Revised: K neigh ( θ ) .

Original: [Table]
Revised: [Equation]

Revised: in →

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: out linearly independent solutions for K neigh ( θ )

Original: [Equation]
Revised: [Equation]

Revised: 0 →

Original: [Table] [Equation]
Revised: [Equation] [Equation]

Original: [Table] [Equation] [Table]
Revised: n

Original: [Equation][Table] [Equation]
Revised: [Equation] [Equation]

Original: [Table] [Equation] [Table] [Equation]
Revised: m (cid:16)

Original: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation] [Table] [Equation] [Table]
Revised: solutions

Original: [Table]

Original: (cid:16)

Original: [Table]
Revised: [Table]

Original: In general, as derived by Cohen et al. (2019b) and in appendix B, the kernels must satisfy for any gauge transformation g ∈ [0 , 2 π ) and angle θ ∈ [0 , 2 π ) , that
Revised: In general, as derived by Cohen et al. (2019b) and in appendix B, the kernels must satisfy for any gauge transformation g ∈ [0 , 2 π ) and angle θ ∈ [0 , 2 π ) , that

Original: [Equation]
Revised: [Equation]

Original: The kernel can be seen as consisting of multiple blocks, where each block takes as input one irrep and outputs one irrep.
Revised: The kernel can be seen as consisting of multiple blocks, where each block takes as input one irrep and outputs one irrep.

Original: For example if ρ in would be of type ρ 0 ⊕ ρ 1 ⊕ ρ 1 and ρ out of type ρ 1 ⊕ ρ 3 , we have 4 × 5 matrix
Revised: For example if ρ in would be of type ρ 0 ⊕ ρ 1 ⊕ ρ 1 and ρ out of type ρ 1 ⊕ ρ 3 , we have 4 × 5 matrix

Original: [Equation]
Revised: [Equation]

Original: where e.g. K 31 ( θ ) ∈ R 2 × 2 is a kernel that takes as input irrep ρ 1 and as output irrep ρ 3 and needs to satisfy Eq.
Revised: where e.g. K 31 ( θ ) ∈ R 2 × 2 is a kernel that takes as input irrep ρ 1 and as output irrep ρ 3 and needs to satisfy Eq.

Original: As derived by Weiler & Cesa (2019) and in Appendix C, the kernels K neigh ( θ ) and
Revised: As derived by Weiler & Cesa (2019) and in Appendix C, the kernels K neigh ( θ ) and

Original: K self mapping from irrep ρ n to irrep ρ m can be written as a linear combination of the basis kernels listed in Table 1.
Revised: K self mapping from irrep ρ n to irrep ρ m can be written as a linear combination of the basis kernels listed in Table 1.

Original: The table shows that equivariance requires the self-interaction to only map from one irrep to the same irrep.
Revised: The table shows that equivariance requires the self-interaction to only map from one irrep to the same irrep.

Original: Hence, we have K self 
Revised: Hence, we have K self =

Original: [Equation]
Revised: [Equation]

Original: All basis-kernels of all pairs of input irreps and output irreps can be linearly combined to form an arbitrary equivariant kernel from feature of type ρ in to ρ out .
Revised: All basis-kernels of all pairs of input irreps and output irreps can be linearly combined to form an arbitrary equivariant kernel from feature of type ρ in to ρ out .

Original: In the above example, we have 2 × 2 + 4 × 4 = 20 basis kernels for K neigh and 4 basis kernels for K self .
Revised: In the above example, we have 2 × 2 + 4 × 4 = 20 basis kernels for K neigh and 4 basis kernels for K self .

Original: The layer thus hasparameters.
Revised: The layer thus hasparameters.

Original: As proven in (Weiler & Cesa, 2019) and (Lang & Weiler, 2020), this parameterization of the equivariant kernel space is complete , that is, more general equivariant kernels do not exist.
Revised: As proven in (Weiler & Cesa, 2019) and (Lang & Weiler, 2020), this parameterization of the equivariant kernel space is complete , that is, more general equivariant kernels do not exist.

Original: 3. G EOMETRY AND P ARALLEL T RANSPORT
Revised: 3. G EOMETRY AND P ARALLEL T RANSPORT

Original: In order to implement gauge equivariant mesh CNNs, we need to make the abstract notion of tangent spaces, gauges and transporters concrete.
Revised: In order to implement gauge equivariant mesh CNNs, we need to make the abstract notion of tangent spaces, gauges and transporters concrete.

Original: As the mesh is embedded in R 3 , a natural deﬁnition of the tangent spaces T p M is as two dimensional subspaces that are orthogonal to the normal vector at p .
Revised: As the mesh is embedded in R 3 , a natural deﬁnition of the tangent spaces T p M is as two dimensional subspaces that are orthogonal to the normal vector at p .

Original: We follow the common deﬁnition of normal vectors at mesh vertices as the area weighted average of the adjacent faces’ normals.
Revised: We follow the common deﬁnition of normal vectors at mesh vertices as the area weighted average of the adjacent faces’ normals.

Original: The
Revised: The

Original: Riemannian logarithm map log p : N p
Revised: Riemannian logarithm map log p : N p

Original: → T p M represents the one-ring neighborhood of each point p on their tangent spaces as visualized in ﬁgure 1.
Revised: → T p M represents the one-ring neighborhood of each point p on their tangent spaces as visualized in ﬁgure 1.

Original: Speciﬁcally, neighbors q ∈ N p are mapped to log p ( q ) ∈ T p M by ﬁrst projecting them to T p M and then rescaling the projection such that the norm is preserved, i.e. | log p ( q ) | = | q − p | ; see Eq. 6.
Revised: Speciﬁcally, neighbors q ∈ N p are mapped to log p ( q ) ∈ T p M by ﬁrst projecting them to T p M and then rescaling the projection such that the norm is preserved, i.e. | log p ( q ) | = | q − p | ; see Eq. 6.

Original: A choice of reference neighbor q p ∈ N uniquely determines a right handed, orthonormal reference frame ( e p, 1 , e p, 2 ) of T p M by setting e p, 1 := log p ( q 0 ) / | log p ( q 0 ) | and e p, 2 := n × e p, 1 .
Revised: A choice of reference neighbor q p ∈ N uniquely determines a right handed, orthonormal reference frame ( e p, 1 , e p, 2 ) of T p M by setting e p, 1 := log p ( q 0 ) / | log p ( q 0 ) | and e p, 2 := n × e p, 1 .

Original: The polar angle θ pq of any neighbor q ∈ N relative to the ﬁrst frame axis is then given by θ pq :=
Revised: The polar angle θ pq of any neighbor q ∈ N relative to the ﬁrst frame axis is then given by θ pq :=

Original: atan2 (cid:0) e (cid:62) p, 2 log p ( q ) , e (cid:62) p, 1 log p ( q )) (cid:1) .
Revised: atan2 (cid:0) e (cid:62) p, 2 log p ( q ) , e (cid:62) p, 1 log p ( q )) (cid:1) .

Original: Given the reference frame ( e p, 1 , e p, 2 ) , a 2-tuple of coefﬁcients ( v 1 , v 2 ) ∈ R 2 speciﬁes an (embedded) tangent vector v 1 e p, 1 + v 2 e p, 2 ∈ T p M ⊂ R 3 .
Revised: Given the reference frame ( e p, 1 , e p, 2 ) , a 2-tuple of coefﬁcients ( v 1 , v 2 ) ∈ R 2 speciﬁes an (embedded) tangent vector v 1 e p, 1 + v 2 e p, 2 ∈ T p M ⊂ R 3 .

Original: This assignment is formally given by the gauge map
Revised: This assignment is formally given by the gauge map

Original: E p : R 2 → T p M ⊂ R 3 which is a vector space isomorphism.
Revised: E p : R 2 → T p M ⊂ R 3 which is a vector space isomorphism.

Original: In our case, it can be identiﬁed with the matrix
Revised: In our case, it can be identiﬁed with the matrix

Original: [Equation]
Revised: [Equation]

Original: Feature vectors f p and f q at neighboring (or any other) vertices p ∈ M and q ∈ N p ⊆
Revised: Feature vectors f p and f q at neighboring (or any other) vertices p ∈ M and q ∈ N p ⊆

Original: M live in different vector spaces and are expressed relative to independent gauges, which makes it invalid to sum them directly.
Revised: M live in different vector spaces and are expressed relative to independent gauges, which makes it invalid to sum them directly.

Original: Instead, they have to be parallel transported along the mesh edge that connects the two vertices.
Revised: Instead, they have to be parallel transported along the mesh edge that connects the two vertices.

Original: As explained above, this transport is given by group elements
Revised: As explained above, this transport is given by group elements

Original: g q → p ∈ [0 , 2 π ) , which determine the transformation of tangent vector coefﬁcients as v q (cid:55)→ R ( g q → p ) v q ∈ R 2 and, analogously, for feature vector coefﬁcients as f q (cid:55)→ ρ ( g q → p ) f q .
Revised: g q → p ∈ [0 , 2 π ) , which determine the transformation of tangent vector coefﬁcients as v q (cid:55)→ R ( g q → p ) v q ∈ R 2 and, analogously, for feature vector coefﬁcients as f q (cid:55)→ ρ ( g q → p ) f q .

Original: Figure 4 in the appendix visualizes the deﬁnition of edge transporters for ﬂat spaces and meshes.
Revised: Figure 4 in the appendix visualizes the deﬁnition of edge transporters for ﬂat spaces and meshes.

Original: On a ﬂat space, tangent vectors are transported by keeping them parallel in the usual sense on Euclidean spaces.
Revised: On a ﬂat space, tangent vectors are transported by keeping them parallel in the usual sense on Euclidean spaces.

Original: However, if the source and target frame orientations disagree, the vector coefﬁcients relative to the source frame need to be transformed to the target frame.
Revised: However, if the source and target frame orientations disagree, the vector coefﬁcients relative to the source frame need to be transformed to the target frame.

Original: This coordinate transformation from polar angles ϕ q of v to ϕ p of
Revised: This coordinate transformation from polar angles ϕ q of v to ϕ p of

Original: R ( g q → p ) v deﬁnes the transporter g q → p = ϕ
Revised: R ( g q → p ) v deﬁnes the transporter g q → p = ϕ

Original: p − ϕ q .
Revised: p − ϕ q .

Original: On meshes, the source and target tangent spaces
Revised: On meshes, the source and target tangent spaces

Original: T q M and T p M are not longer parallel.
Revised: T q M and T p M are not longer parallel.

Original: It is therefore additionally necessary to rotate the source tangent space and its vectors parallel to the target space, before transforming between the frames.
Revised: It is therefore additionally necessary to rotate the source tangent space and its vectors parallel to the target space, before transforming between the frames.

Original: Since transporters effectively make up for differences in the source and target frames, the parallel transporters transform under gauge transformations g p and g q according to g q → p (cid:55)→ g p + g q → p − g q .
Revised: Since transporters effectively make up for differences in the source and target frames, the parallel transporters transform under gauge transformations g p and g q according to g q → p (cid:55)→ g p + g q → p − g q .

Original: Note that this transformation law cancels with the transformation law of the coefﬁcients at q and lets the transported coefﬁcients transform according to gauge transformations at p .
Revised: Note that this transformation law cancels with the transformation law of the coefﬁcients at q and lets the transported coefﬁcients transform according to gauge transformations at p .

Original: It is therefore valid to sum vectors and features that are parallel transported into the same gauge at p .
Revised: It is therefore valid to sum vectors and features that are parallel transported into the same gauge at p .

Original: A more detailed discussion of the concepts presented in this section can be found in Appendix A.
Revised: A more detailed discussion of the concepts presented in this section can be found in Appendix A.

Original:  N ON - LINEARITY
Revised:  N ON - LINEARITY

Original: Besides convolutional layers, the GEM-CNN contains non-linear layers, which also need to be gauge equivariant, for the entire network to be gauge equivariant.
Revised: Besides convolutional layers, the GEM-CNN contains non-linear layers, which also need to be gauge equivariant, for the entire network to be gauge equivariant.

Original: The coefﬁcients of features built out of irreducible representaions, as described in section 3, do not commute with point-wise nonlinearities (Worrall et al., 2017; Thomas et al., 2018; Weiler et al., 2018a;
Revised: The coefﬁcients of features built out of irreducible representaions, as described in section 3, do not commute with point-wise nonlinearities (Worrall et al., 2017; Thomas et al., 2018; Weiler et al., 2018a;

Original: Kondor et al., 2018).
Revised: Kondor et al., 2018).

Original: Norm non-linearities and gated non-linearities (Weiler & Cesa, 2019) can be used with such features, but generally perform worse in practice compared to point-wise non-linearities (Weiler & Cesa,2019).
Revised: Norm non-linearities and gated non-linearities (Weiler & Cesa, 2019) can be used with such features, but generally perform worse in practice compared to point-wise non-linearities (Weiler & Cesa,

Original: Hence, we propose the RegularNonlinearity , which uses point-wise non-linearities and is approximately gauge equivariant.
Revised: Hence, we propose the RegularNonlinearity , which uses point-wise non-linearities and is approximately gauge equivariant.

Original: This non-linearity is built on Fourier transformations.
Revised: This non-linearity is built on Fourier transformations.

Original: Consider a continuous periodic signal, on which we perform a band-limited Fourier transform with band limit b , obtaining 2 b + 1 Fourier coefﬁcients.
Revised: Consider a continuous periodic signal, on which we perform a band-limited Fourier transform with band limit b , obtaining 2 b + 1 Fourier coefﬁcients.

Original: If this continuous signal is shifted by an arbitrary angle g , then the corresponding Fourier components transform with linear transformation ρ 0: b ( − g ) , for 2 b + 1 dimensional representation ρ 0: b :=
Revised: If this continuous signal is shifted by an arbitrary angle g , then the corresponding Fourier components transform with linear transformation ρ 0: b ( − g ) , for 2 b + 1 dimensional representation ρ 0: b :=

Original: ρ 

Revised: ρ

Original: ⊕ ρ 1 ⊕ ...
Revised: ⊕ ρ 1 ⊕ ...

Original: ⊕ ρ b .
Revised: ⊕ ρ b .

Original: It would be exactly equivariant to take a feature of type ρ 0: b , take a continuous inverse Fourier transform to a continuous periodic signal, then apply a point-wise non-linearity to that signal, and take the continuous Fourier transform, to recover a feature of type ρ 0: b .
Revised: It would be exactly equivariant to take a feature of type ρ 0: b , take a continuous inverse Fourier transform to a continuous periodic signal, then apply a point-wise non-linearity to that signal, and take the continuous Fourier transform, to recover a feature of type ρ 0: b .

Original: However, for implementation, we use N intermediate samples and the discrete Fourier transform.
Revised: However, for implementation, we use N intermediate samples and the discrete Fourier transform.

Original: This is exactly gauge equivariant for gauge transformation of angles multiple of 2 π/N , but only approximately equivariant for other angles.
Revised: This is exactly gauge equivariant for gauge transformation of angles multiple of 2 π/N , but only approximately equivariant for other angles.

Original: In App.
Revised: In App.

Original: G we prove that as N → ∞ , the non-linearity is exactly gauge equivariant.
Revised: G we prove that as N → ∞ , the non-linearity is exactly gauge equivariant.

Original: The run-time cost per vertex of the (inverse) Fourier transform implemented as a simple linear transformation is O ( bN ) , which is what we use in our experiments.
Revised: The run-time cost per vertex of the (inverse) Fourier transform implemented as a simple linear transformation is O ( bN ) , which is what we use in our experiments.

Original: The pointwise non-linearity scales linearly with N , so the complexity of the RegularNonLineariy is also O ( bN ) .
Revised: The pointwise non-linearity scales linearly with N , so the complexity of the RegularNonLineariy is also O ( bN ) .

Original: However, one can also use a fast Fourier transform, achieving a complexity of O ( N log N ) .
Revised: However, one can also use a fast Fourier transform, achieving a complexity of O ( N log N ) .

Original: Concrete memory and run-time cost of varying N are shown in appendix F.1.
Revised: Concrete memory and run-time cost of varying N are shown in appendix F.1.

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

Original: The irregular structure of meshes leads to a variety of approaches to deﬁne convolutions.
Revised: The irregular structure of meshes leads to a variety of approaches to deﬁne convolutions.

Original: Closely related to our method are graph based methods which are often based on variations of graph convolutional networks (Kipf & Welling, 2017; Defferrard et al., 2016).
Revised: Closely related to our method are graph based methods which are often based on variations of graph convolutional networks (Kipf & Welling, 2017; Defferrard et al., 2016).

Original: GCNs have been applied on spherical meshes (Perraudin et al., 2019) and cortical surfaces (Cucurull et al., 2018;
Revised: GCNs have been applied on spherical meshes (Perraudin et al., 2019) and cortical surfaces (Cucurull et al., 2018;

Original: Zhao et al., 2019a).
Revised: Zhao et al., 2019a).

Original: Verma et al.
Revised: Verma et al.

Original: (2018) augment GCNs with anisotropic kernels which are dynamically computed via an attention mechanism over graph neighbours.
Revised: (2018) augment GCNs with anisotropic kernels which are dynamically computed via an attention mechanism over graph neighbours.

Original: Instead of operating on the graph underlying a mesh, several approaches leverage its geometry by treating it as a discrete manifold.
Revised: Instead of operating on the graph underlying a mesh, several approaches leverage its geometry by treating it as a discrete manifold.

Original: Convolution kernels can then be deﬁned in geodesic polar coordinates which corresponds to a projection of kernels from the tangent space to the mesh via the exponential map.
Revised: Convolution kernels can then be deﬁned in geodesic polar coordinates which corresponds to a projection of kernels from the tangent space to the mesh via the exponential map.

Original: This allows for kernels that are larger than the immediate graph neighbourhood and message passing over faces but does not resolve the issue of ambiguous kernel orientation.
Revised: This allows for kernels that are larger than the immediate graph neighbourhood and message passing over faces but does not resolve the issue of ambiguous kernel orientation.

Original: Masci et al.
Revised: Masci et al.

Original: ; Monti et al.
Revised: ; Monti et al.

Original: and Sun et al.
Revised: and Sun et al.

Original: (2018) address this issue by restricting the network to orientation invariant features which are computed by applying anisotropic kernels in several orientations and pooling over the resulting responses.
Revised: (2018) address this issue by restricting the network to orientation invariant features which are computed by applying anisotropic kernels in several orientations and pooling over the resulting responses.

Original: The models proposed in (Boscaini et al., 2016) and (Schonsheck et al., 2018) are explicitly gauge dependent with preferred orientations chosen via the principal curvature direction and the parallel transport of kernels, respectively.
Revised: The models proposed in (Boscaini et al., 2016) and (Schonsheck et al., 2018) are explicitly gauge dependent with preferred orientations chosen via the principal curvature direction and the parallel transport of kernels, respectively.

Original: Poulenard & Ovsjanikov (2018) proposed a non-trivially gauge equivariant network based on geodesic convolutions, however, the model parallel transports only partial information of the feature vectors, corresponding to certain kernel orientations.
Revised: Poulenard & Ovsjanikov (2018) proposed a non-trivially gauge equivariant network based on geodesic convolutions, however, the model parallel transports only partial information of the feature vectors, corresponding to certain kernel orientations.

Original: In concurrent work, Wiersma et al.
Revised: In concurrent work, Wiersma et al.

Original: (2020) also deﬁne convolutions on surfaces equivariantly to the orientation of the kernel, but differ in that they use norm non-linearities instead of regular ones and that they apply the convolution along longer geodesics, which adds complexity to the geometric pre-computation - as partial differential equations need to be solved, but may result in less susceptibility to the particular discretisation of the manifold.
Revised: (2020) also deﬁne convolutions on surfaces equivariantly to the orientation of the kernel, but differ in that they use norm non-linearities instead of regular ones and that they apply the convolution along longer geodesics, which adds complexity to the geometric pre-computation - as partial differential equations need to be solved, but may result in less susceptibility to the particular discretisation of the manifold.

Original: Another class of approaches deﬁnes spectral convolutions on meshes.
Revised: Another class of approaches deﬁnes spectral convolutions on meshes.

Original: However, as argued in (Bronstein et al., 2017), the Fourier spectrum of a mesh depends heavily on its geometry, which makes such methods instable under deformations and impedes the generalization between different meshes.
Revised: However, as argued in (Bronstein et al., 2017), the Fourier spectrum of a mesh depends heavily on its geometry, which makes such methods instable under deformations and impedes the generalization between different meshes.

Original: Spectral convolutions further correspond to isotropic kernels.
Revised: Spectral convolutions further correspond to isotropic kernels.

Revised: Kostrikov et al.

Revised: overcomes isotropy of the Laplacian by decomposing it into two applications of the ﬁrst-order Dirac operator.

Original: A construction based on toric covering maps of topologically spherical meshes was presented in (Maron et al., 2017).
Revised: A construction based on toric covering maps of topologically spherical meshes was presented in (Maron et al., 2017).

Original: An entirely different approach to mesh convolutions is to apply a linear map to a spiral of neighbours (Bouritsas et al., 2019;
Revised: An entirely different approach to mesh convolutions is to apply a linear map to a spiral of neighbours (Bouritsas et al., 2019;

Original: Gong et al., 2019), which works well only for meshes with a similar graph structure.
Revised: Gong et al., 2019), which works well only for meshes with a similar graph structure.

Original: The abovementioned methods operate on the intrinsic, 2 -dimensional geometry of the mesh.
Revised: The above-mentioned methods operate on the intrinsic, 2 -dimensional geometry of the mesh.

Original: A popular alternative for embedded meshes is to deﬁne convolutions in the embedding space R 3 .
Revised: A popular alternative for embedded meshes is to deﬁne convolutions in the embedding space R 3 .

Original: This can for instance be done by voxelizing space and representing the mesh in terms of an occupancy grid (Wu et al., 2015; Tchapmi et al., 2017; Hanocka et al., 2018).
Revised: This can for instance be done by voxelizing space and representing the mesh in terms of an occupancy grid (Wu et al., 2015; Tchapmi et al., 2017; Hanocka et al., 2018).

Original: A downside of this approach are the high memory and compute requirements of voxel representations.
Revised: A downside of this approach are the high memory and compute requirements of voxel representations.

Original: If the grid occupancy is low, this can partly be addressed by resorting to an inhomogeneous grid density (Riegler et al., 2017).
Revised: If the grid occupancy is low, this can partly be addressed by resorting to an inhomogeneous grid density (Riegler et al., 2017).

Original: Instead of voxelizing space, one may interpret the set of mesh vertices as a point cloud and run a convolution on those (Qi et al., 2017a;b).
Revised: Instead of voxelizing space, one may interpret the set of mesh vertices as a point cloud and run a convolution on those (Qi et al., 2017a;b).

Original: Point cloud based methods can be made equivariant w.r.t.
Revised: Point cloud based methods can be made equivariant w.r.t.

Original: the isometries of R 3 (Zhao et al., 2019b; Thomas et al., 2018), which implies in particular the isometry equivariance on the embedded mesh.
Revised: the isometries of R 3 (Zhao et al., 2019b; Thomas et al., 2018), which implies in particular the isometry equivariance on the embedded mesh.

Original: In general, geodesic distances within the manifold differ usually substantially from the distances in the embedding space.
Revised: In general, geodesic distances within the manifold differ usually substantially from the distances in the embedding space.

Original: Which approach is more suitable depends on the particular application.
Revised: Which approach is more suitable depends on the particular application.

Original: On ﬂat Euclidean spaces our method corresponds to Steerable CNNs (Cohen & Welling, 2017; Weiler et al., 2018a; Weiler & Cesa, 2019; Cohen et al., 2019a; Lang & Weiler, 2020).
Revised: On ﬂat Euclidean spaces our method corresponds to Steerable CNNs (Cohen & Welling, 2017; Weiler et al., 2018a; Weiler & Cesa, 2019; Cohen et al., 2019a; Lang & Weiler, 2020).

Original: As our model, these networks process geometric feature ﬁelds of types ρ and are equivariant under gauge transformations, however, due to the ﬂat geometry, the parallel transporters become trivial.
Revised: As our model, these networks process geometric feature ﬁelds of types ρ and are equivariant under gauge transformations, however, due to the ﬂat geometry, the parallel transporters become trivial.

Original: Regular nonlinearities are on ﬂat spaces used in group convolutional networks (Cohen & Welling, 2016;
Revised: Regular nonlinearities are on ﬂat spaces used in group convolutional networks (Cohen & Welling, 2016;

Original: Weiler et al., 2018b;
Revised: Weiler et al., 2018b;

Original: Hoogeboom et al., 2018;
Revised: Hoogeboom et al., 2018;

Original: Bekkers et al., 2018; Winkels & Cohen, 2018; Worrall & Brostow, 2018;
Revised: Bekkers et al., 2018; Winkels & Cohen, 2018; Worrall & Brostow, 2018;

Original: Worrall & Welling, 2019; Sosnovik et al., 2020).
Revised: Worrall & Welling, 2019; Sosnovik et al., 2020).

Original:  E XPERIMENTS 6. E MBEDDED MNIST
Revised:  E XPERIMENTS 6. E MBEDDED MNIST

Original: We ﬁrst investigate how Gauge Equivariant Mesh CNNs perform on, and generalize between, different mesh geometries.
Revised: We ﬁrst investigate how Gauge Equivariant Mesh CNNs perform on, and generalize between, different mesh geometries.

Original: For this purpose we conduct simple MNIST digit classiﬁcation experiments on embedded rectangular meshes of 28 × 28 vertices.
Revised: For this purpose we conduct simple MNIST digit classiﬁcation experiments on embedded rectangular meshes of 28 × 28 vertices.

Original: As a baseline geometry we consider a ﬂat mesh as visualized in ﬁgure 5(a).
Revised: As a baseline geometry we consider a ﬂat mesh as visualized in ﬁgure 5(a).

Original: A second type of geometry is deﬁned as different isometric embeddings of the ﬂat mesh, see ﬁgure 5(b).
Revised: A second type of geometry is deﬁned as different isometric embeddings of the ﬂat mesh, see ﬁgure 5(b).

Original: Note that this implies that the intrinsic geometry of these isometrically embedded meshes is indistinguishable from that of the ﬂat mesh.
Revised: Note that this implies that the intrinsic geometry of these isometrically embedded meshes is indistinguishable from that of the ﬂat mesh.

Original: To generate geometries which are intrinsically curved, we add random normal displacements to the ﬂat mesh.
Revised: To generate geometries which are intrinsically curved, we add random normal displacements to the ﬂat mesh.

Original: We control the amount of curvature by smoothing the resulting displacement ﬁelds with Gaussian kernels of different widths σ and deﬁne the roughness of the resulting mesh as 3 − σ .
Revised: We control the amount of curvature by smoothing the resulting displacement ﬁelds with Gaussian kernels of different widths σ and deﬁne the roughness of the resulting mesh as 3 − σ .

Original: Figures 5(c)-5(h) show the results for roughnesses of 0.5, 1, 1.5, 2, 2.25 and 2.5.
Revised: Figures 5(c)-5(h) show the results for roughnesses of 0.5, 1, 1.5, 2, 2.25 and 2.5.

Original: For each of the considered settings we generatedifferent train and 32 test geometries.
Revised: For each of the considered settings we generatedifferent train and 32 test geometries.

Original: To test the performance on, and generalization between, different geometries, we train equivalent
Revised: To test the performance on, and generalization between, different geometries, we train equivalent

Original: GEM-CNN models on a ﬂat mesh and meshes with a roughness of 1, 1.5, 2, 2.25 and 2.5.
Revised: GEM-CNN models on a ﬂat mesh and meshes with a roughness of 1, 1.5, 2, 2.25 and 2.5.

Original: Each model is tested individually on each of the considered test geometries, which are the ﬂat mesh, isometric embeddings and curved embeddings with a roughness of 0.5, 1, 1.25, 1.5, 1.75, 2, 2. and 2.5.
Revised: Each model is tested individually on each of the considered test geometries, which are the ﬂat mesh, isometric embeddings and curved embeddings with a roughness of 0.5, 1, 1.25, 1.5, 1.75, 2, 2. and 2.5.

Original: Figure 3 shows the test errors of the GEM-CNNs on the different train geometries (different curves) for all test geometries (shown on the x-axis).
Revised: Figure 3 shows the test errors of the GEM-CNNs on the different train geometries (different curves) for all test geometries (shown on the x-axis).

Original: Since our model is purely deﬁned in terms of the intrinsic geometry of a mesh, it is expected to be insensitive to isometric changes in the embeddings.
Revised: Since our model is purely deﬁned in terms of the intrinsic geometry of a mesh, it is expected to be insensitive to isometric changes in the embeddings.

Original: This is empirically conﬁrmed by the fact that the test performances on ﬂat and isometric embeddings are exactly equal.
Revised: This is empirically conﬁrmed by the fact that the test performances on ﬂat and isometric embeddings are exactly equal.

Original: As expected, the test error increases for most models with the surface roughness.
Revised: As expected, the test error increases for most models with the surface roughness.

Original: Models trained on more rough surfaces are hereby more robust to deformations.
Revised: Models trained on more rough surfaces are hereby more robust to deformations.

Original: The models generalize well from a rough training to smooth test geometry up to a training roughness of 1.5.
Revised: The models generalize well from a rough training to smooth test geometry up to a training roughness of 1.5.

Original: Beyond that point, the test performances on smooth meshes degrades up to the point of random guessing at a training roughness of 2.5.
Revised: Beyond that point, the test performances on smooth meshes degrades up to the point of random guessing at a training roughness of 2.5.

Original: As a baseline, we build an isotropic graph CNN with the same network topology and number of parameters ( ≈ 163 k ).
Revised: As a baseline, we build an isotropic graph CNN with the same network topology and number of parameters ( ≈ 163 k ).

Original: This model is insensitive to the mesh geometry and therefore performs exactly equal on all surfaces.
Revised: This model is insensitive to the mesh geometry and therefore performs exactly equal on all surfaces.

Original: While this enhances its robustness on very rough meshes, its test error of 19 . 80 ± 3 .
Revised: While this enhances its robustness on very rough meshes, its test error of 19 . 80 ± 3 .

Original: 43% is an extremely bad result on MNIST.
Revised: 43% is an extremely bad result on MNIST.

Original: In contrast, the use of anisotropic ﬁlters of
Revised: In contrast, the use of anisotropic ﬁlters of

Original: GEM-CNN allows it to reach a test error of only 0 .
Revised: GEM-CNN allows it to reach a test error of only 0 .

Original: 60 ± 0 .
Revised: 60 ± 0 .

Original: 05% on the ﬂat geometry.
Revised: 05% on the ﬂat geometry.

Original: It is therefore competitive with conventional CNNs on pixel grids, which apply anisotropic kernels as well.
Revised: It is therefore competitive with conventional CNNs on pixel grids, which apply anisotropic kernels as well.

Original: More details on the datasets, models and further experimental setup are given in appendix E.1. 
Revised: More details on the datasets, models and further experimental setup are given in appendix E.1. [Figure] [Table] [Figure]

Revised: [Table]

Original: 6. S HAPE C ORRESPONDENCE
Revised: [Figure] [Table] [Figure] [Table]

Original:  As a second experiment, we perform non-rigid shape correspondence on the FAUST dataset (Bogo et al., 2014), following Masci et al.
Revised: [Figure] [Table] 6. S HAPE C ORRESPONDENCE As a second experiment we perform non-rigid shape correspondence on the FAUST dataset (Bogo et al., 2014), following Masci et al.

Original: The data consists of 100 meshes of human bodies in various positions, split into 80 train and 20 test meshes.
Revised: The data consists of 100 meshes of human bodies in various positions, split into 80 train and 20 test meshes.

Original: The vertices are registered, such that vertices[Figure] [Table] on the same position on the body, such as the tip of the left thumb, have the same identiﬁer on all meshes.
Revised: The vertices are registered, such that vertices on the same position on the body, such as the tip of the left thumb, have the same identiﬁer on all meshes.

Original: All meshes have 6890 vertices, making this a 6890 -class segmentation problem.
Revised: All meshes have 6890 vertices, making this a 6890 -class segmentation problem.

Original: We use a simple architecture, which transforms the vertices’ XY Z coordinates, which are of type 3 ρ 0 , via 6 convolutional layers to a feature of type 64 ρ 0 , with intermediate featuresof type 16( ρ 0 ⊕ ρ 1 ⊕ ρ 2 ) .
Revised: The architecture transforms the vertices’ XY Z coordinates (of type 3 ρ 0 ), via 6 convolutional layers to features 64 ρ 0 , with intermediate features 16( ρ 0 ⊕ ρ 1 ⊕ ρ 2 ) , with residual connections and the

Original: The convolutional layers use residual connections and the RegularNonlinearity with N = 5 samples.
Revised: RegularNonlinearity with N = 5 samples.

Original: Afterwards, we use two 1 × 1 convolutions with ReLU to map ﬁrst to 256channels and ﬁnally tochannels, after which a softmax predicts the registration probabilities.
Revised: Afterwards, we use two 1 × 1 convolutions with ReLU to map ﬁrst to 256 and then 6980 channels, after which a softmax predicts the registration probabilities.

Original: The 1 × 1 convolutions use a dropout of 50% and 1E-4 weight decay.
Revised: The 1 × 1 convolutions use a dropout of 50% and 1E-4 weight decay.

Original: The network is trained with a cross entropy loss with an initial learning rate of 0.01, which is halved when training loss reaches a plateau.
Revised: The network is trained with a cross entropy loss with an initial learning rate of 0.01, which is halved when training loss reaches a plateau.

Original: As all meshes in the FAUST dataset share the same topology, breaking the gauge equivariance in higher layers can actually be beneﬁcial.
Revised: As all meshes in the FAUST dataset share the same topology, breaking the gauge equivariance in higher layers can actually be beneﬁcial.

Original: As shown in (Weiler & Cesa, 2019), symmetry can be broken by treating non-invariant features as invariant features as input to the ﬁnal 1 × 1 convolution.
Revised: As shown in (Weiler & Cesa, 2019), symmetry can be broken by treating non-invariant features as invariant features as input to the ﬁnal 1 × 1 convolution.

Original: Such architectures are equivariant on lower levels, while allowing orientation sensitivity at higher layers.

Original: As baselines, we compare to various models, some of which use more complicated pipelines, such as (1) the computation of geodesics over the mesh, which requires solving partial differential equations, (2) pooling, which requires ﬁnding a uniform sub-selection of vertices, (3) the pre-computation of
Revised: As baselines, we compare to various models, some of which use more complicated pipelines, such as (1) the computation of geodesics over the mesh, which requires solving partial differential equations, (2) pooling, which requires ﬁnding a uniform sub-selection of vertices, (3) the pre-computation of

Original: SHOT features which locally describe the geometry (Tombari et al., 2010), or (4) post-processing reﬁnement of the predictions.
Revised: SHOT features which locally describe the geometry (Tombari et al., 2010), or (4) post-processing reﬁnement of the predictions.

Original: The GEM-CNN requires none of these additional steps.
Revised: The GEM-CNN requires none of these additional steps.

Original: In addition, we compare to SpiralNet++ (Gong et al., 2019), which requires all inputs to be similarly meshed.
Revised: In addition, we compare to SpiralNet++ (Gong et al., 2019), which requires all inputs to be similarly meshed.

Original: Finally, we compare to an isotropic version of the GEM-CNN, which reduces to a conventional graph
Revised: Finally, we compare to an isotropic version of the GEM-CNN, which reduces to a conventional graph

Original: CNN, as well as a non-gauge-equivariant CNN based on SHOT frames.
Revised: CNN, as well as a non-gauge-equivariant CNN based on SHOT frames.

Original: The results in table 2 show that the GEM-CNN outperforms prior works and a non-gauge-equivariant CNN, that isotropic graph
Revised: The results in table 2 show that the GEM-CNN outperforms prior works and a non-gauge-equivariant CNN, that isotropic graph

Original: CNNs are unable to solve the task and that for this data set breaking gauge symmetry in the ﬁnal layers of the network is beneﬁcial.
Revised: CNNs are unable to solve the task and that for this data set breaking gauge symmetry in the ﬁnal layers of the network is beneﬁcial.

Original: More experimental details are given in appendix E.2.
Revised: More experimental details are given in appendix E.2.

Original:  C ONCLUSIONS
Revised:  C ONCLUSIONS

Original: Convolutions on meshes are commonly performed as a convolution on their underlying graph, by forgetting geometry, such as orientation of neighbouring vertices.
Revised: Convolutions on meshes are commonly performed as a convolution on their underlying graph, by forgetting geometry, such as orientation of neighbouring vertices.

Original: In this paper we propose Gauge
Revised: In this paper we propose Gauge

Original: Equivariant Mesh CNNs, which endow Graph Convolutional Networks on meshes with anisotropic kernels and parallel transport.
Revised: Equivariant Mesh CNNs, which endow Graph Convolutional Networks on meshes with anisotropic kernels and parallel transport.

Original: Hence, they are sensitive to the mesh geometry, and result in equivalent outputs regardless of the arbitrary choice of kernel orientation.
Revised: Hence, they are sensitive to the mesh geometry, and result in equivalent outputs regardless of the arbitrary choice of kernel orientation.

Original: We demonstrate that the inference of GEM-CNNs is invariant under isometric deformations of meshes and generalizes well over a range of non-isometric deformations.
Revised: We demonstrate that the inference of GEM-CNNs is invariant under isometric deformations of meshes and generalizes well over a range of non-isometric deformations.

Original: On the FAUST shape correspondence task, we show that Gauge equivariance, combined with symmetry breaking in the ﬁnal layer, leads to state of the art performance.
Revised: On the FAUST shape correspondence task, we show that Gauge equivariance, combined with symmetry breaking in the ﬁnal layer, leads to state of the art performance.