Title: Implicit Convolutional Kernels for Steerable CNNs
Venue: NeurIPS 2023
Accepted: yes
Keywords: equivariance; group convolutions; implicit kernels; physical simulations
Abstract: Steerable convolutional neural networks (CNNs) provide a general framework for building neural networks equivariant to translations and transformations of an origin-preserving group $G$, such as reflections and rotations. They rely on standard convolutions with $G$-steerable kernels obtained by analytically solving the group-specific equivariance constraint imposed onto the kernel space. As the solution is tailored to a particular group $G$, implementing a kernel basis does not generalize to other symmetry transformations, complicating the development of general group equivariant models. We propose using implicit neural representation via multi-layer perceptrons (MLPs) to parameterize $G$-steerable kernels. The resulting framework offers a simple and flexible way to implement Steerable CNNs and generalizes to any group $G$ for which a $G$-equivariant MLP can be built. We prove the effectiveness of our method on multiple tasks, including N-body simulations, point cloud classification and molecular property prediction.

Implicit Convolutional Kernels for Steerable CNNs

Abstract

Steerable convolutional neural networks (CNNs) provide a general framework for building neural networks equivariant to translations and transformations of an origin-preserving group G , such as reflections and rotations. They rely on standard convolutions with G -steerable kernels obtained by analytically solving the groupspecific equivariance constraint imposed onto the kernel space. As the solution is tailored to a particular group G , implementing a kernel basis does not generalize to other symmetry transformations, complicating the development of general group equivariant models. We propose using implicit neural representation via multilayer perceptrons (MLPs) to parameterize G -steerable kernels. The resulting framework offers a simple and flexible way to implement Steerable CNNs and generalizes to any group G for which a G -equivariant MLP can be built. We prove the effectiveness of our method on multiple tasks, including N-body simulations, point cloud classification and molecular property prediction.

1 Introduction

Equivariant deep learning is a powerful tool for high-dimensional problems with known data domain symmetry. By incorporating this knowledge as inductive biases into neural networks, the hypothesis class of functions can be significantly restricted, leading to improved data efficiency and generalization performance [6]. Convolutional neural networks [28] (CNNs) are a prominent example as they are equivariant to translations. Group-equivariant CNNs [9] (G-CNNs) generalize CNNs to exploit a larger number of symmetries via group convolutions, making them equivariant to the desired symmetry group, such as the Euclidean group E ( n ) that encompasses translations, rotations, and reflections in n -dimensional Euclidean space. In physics and chemistry, many important problems, such as molecular modelling or point clouds, rely on the Euclidean group. Objects defined in physical space have properties that are invariant or equivariant to Euclidean transformations, and respecting this underlying symmetry is often desired for the model to perform as expected.

Neural networks can be parameterized in various ways to incorporate equivariance to the Euclidean group. One option is to use a message-passing neural network as a backbone and compute/update messages equivariantly via convolutions. This approach generalizes well to point clouds and graphs and offers the high expressivity of graph neural networks. Equivariant convolutional operators can be further categorized as regular [3, 9, 14, 26] or steerable group convolutions [10, 11, 52]. The latter recently proved to be especially suitable for incorporating physical and geometric quantities into a model [5]. The key idea behind Steerable CNNs is using standard convolution - which guarantees translation equivariance - with G -steerable kernels that ensure commutativity with the transformations of another group G , such as rotations. The commutation requirement imposes a constraint onto the kernel space that must be solved analytically for each group G . This, in turn, does not allow generalizing a convolution operator tailored to a specific group to other symmetry transformations. In

the case of the Euclidean group, Cesa et al. [7] proposed a generally applicable way of parameterizing steerable convolutions for sub-groups of E ( n ) . The method relies on adapting a pre-defined kernel basis explicitly developed for the group E ( n ) to an arbitrary sub-group by using group restriction .

However, because only a finite basis can be chosen, a basis tailored for E ( n ) can be sub-optimal in terms of expressiveness for its sub-groups; see Section 3.6 more details. Hence, we propose an alternative way of building steerable convolutions based on implicit neural kernels, i.e. convolutional kernels implemented as continuous functions parameterized by MLPs [38, 39]. We demonstrate how G -steerable convolutions with implicit neural kernels can be implemented from scratch for any sub-group G of the orthogonal group O ( n ) . The method allows us to ultimately minimize requirements to implement equivariance to new groups; see Section 3.3. The flexibility of neural functions also permits the injection of geometric and physical quantities in point convolutions, increasing their expressiveness [5]; see Section 3.2. We validate our framework on synthetic N-body simulation problem, point-cloud data (ModelNet-40 [56]) and molecular data (QM9 [55]) and demonstrate key bene-

Flexible & simple strategy ∀ G ≤ O ( n )

Allows for edge attributes

• Task specific

steer.

kernels

9

fout

fin

k(ri)

X

values

aggregate

ri

G-equivariantMLP

C

fits of our approach such as flexibility and generalizability. Besides, we demonstrate that implicit kernels allow Steerable CNNs to achieve performance competitive with state-of-the-art models and surpass them with the correct choice of the task-specific symmetry group.

2 Background: Steerable Convolutions

In this work, we propose a general solution to easily build Steerable CNNs equivariant to translations and any compact group G 3 . In Section 2.1, we provide some necessary prerequisites from group theory and group representation theory [43]. Then, we review the framework of Steerable CNNs and discuss the constraint it induces on the convolutional kernels in Section 2.2.

2.1 Groups, Representations and Equivariance

Definition 1 (Group action) . An action of a group G on a set X is a mapping ( g, x ) ↦→ g.x associating a group element g ∈ G and a point x ∈ X with some other point on X such that the following holds:

Definition 2 (Group representation) . A linear representation ρ of a group G is a map ρ : G → R d × d that assigns an invertible matrix ρ ( g ) ∀ g ∈ G and satisfies the following condition

A group representation ρ ( g ) : V → V furthermore can be seen as a linear action of G on a vector space V . Additionally, if two (or more) vectors v 1 ∈ R d 1 and v 2 ∈ R d 2 belong to vector spaces transforming under representations ρ 1 and ρ 2 , their concatenation v 1 ⊕ v 2 ∈ R d 1 + d 2 transforms under the direct sum representation ρ 1 ⊕ ρ 2 . ( ρ 1 ⊕ ρ 2 )( g ) is a d 1 + d 2 dimensional block-diagonal matrix containing ρ 1 ( g ) and ρ 2 ( g ) in its diagonal blocks.

There are three types of group representations that are important for the definition of our method:

Trivial representation: all elements of G act on V as the identity mapping of V , i.e. ρ ( g ) = I .

Standard representation: the group O ( n ) of all orthogonal n × n matrices has a natural action on V = R n ; similarly, if G is a subgroup of O ( n ) , elements of G can act on V = R n via the inclusion mapping, i.e. ρ st ( g ) = g ∈ R n × n .

Irreducible representations is a collection of generally known representations that can be used as a building block for larger representations via direct sum . As argued in [49], Steerable CNNs can be parameterized without loss of generality solely in terms of irreducible representations.

Definition 3 (Equivariance) . Let us have two spaces X , Y endowed with a symmetry group G , i.e. with an action defined on them. A function φ : X → Y is called G -equivariant , if it commutes with the action of G on the two spaces, i.e. φ ( g.x ) = g. φ ( x ) for all g ∈ G , x ∈ X .

As we have discussed, the layers of conventional CNNs are translation equivariant by design; however, they do not commute with transformations of other groups, such as rotations and reflections.

2.2 Steerable CNNs

Steerable CNNs provide a more general framework that allows building convolutions that are equivariant to a group of isometries of R n , i.e. ( R n , +) /multicloseright G ≤ E ( n ) . Those groups are decomposable as a semi-direct 4 product of the translations group ( R n , +) and an origin-preserving compact 5 group G ≤ O ( n ) , where O ( n ) is the group of n -dimensional rotations and reflections. As translation equivariance is guaranteed [52] by the convolution operator itself, one only has to ensure equivariance to G . See [51] for a more in-depth description of Steerable CNNs.

The feature spaces of Steerable CNNs are described as collections of feature fields . A feature field of type ρ is a feature map f : R n → R d endowed with a group representation ρ : G → R d × d that defines how an element g ∈ G transforms the feature:

Furthermore, each convolutional layer is a map between feature fields. For the map to be equivariant, it must preserve the transformation laws of its input and output feature fields. In practice, it means that the following constraint onto the space of convolution kernels must be applied [52]:

where k : R n → R d out × d in , and ρ in : G → R d in × d in , ρ out : G → R d out × d out are respective representations of input and output feature fields.

To parameterize the kernel k , the constraint in equation 2 needs to be solved analytically for each specific group G of interest. This renders a general solution challenging to obtain and limits the applicability of steerable convolutions.

3 Implicit neural kernels

Instead of deriving a steerable kernel basis for each particular group G , we propose parameterizing the kernel k : R n → R d out × d in with an MLP satisfying the constraint in Eq. 2. The approach only requires the G -equivariance of the MLP and suggests a flexible framework of implicit steerable convolutions that generalizes to arbitrary groups G ≤ O ( n ) . We argue about the minimal requirements of this approach in Section 3.3.

We first define the kernel as an equivariant map between vector spaces that we model with an MLP (see Section 3.1). Then, we demonstrate that G -equivariance of an MLP is a sufficient condition for building the implicit representation of steerable kernels for compact groups. We indicate that the flexibility of neural representation allows expanding the input of a steerable kernel in Section 3.2. Next, we describe how a G -equivariant MLP can be implemented in section 3.3. Later, we describe how one can implement G -steerable point convolution in the form of equivariant message passing [5, 40] in Section 3.4 and its generalization to dense convolution in Section 3.5. Finally, we compare our method with the solution strategy proposed in [7] in Section 3.6.

3.1 Kernel vectorization and equivariance

Our goal is to implement the kernel k : R n → R d out × d in of a G -steerable convolution that maps between spaces of feature fields with representations ρ in and ρ out . The kernel itself is a function whose input in R n transforms under the standard representation ρ st (as defined in Section 2.1) and which we will model with an MLP. Since MLPs typically output vectors, it is convenient to vectorize the d out × d in output of the kernel. We denote the column-wise vectorization of a matrix M ∈ R d 1 × d 2 as vec ( M ) ∈ R d 1 d 2 . Henceforth, we will consider kernel's vector form vec ( k ( · )) : R n → R d out d in .

Let ⊗ denote the Kronecker product between two matrices. Then, ρ ⊗ ( g ) := ρ in ( g ) ⊗ ρ out ( g ) is also a representation 6 of G . We suggest an implicit representation of the vectorized kernel k using an G -equivariant MLP φ : R n → R d out d in based on the following lemma (see A.2 for the proof):

Lemma 1. If a kernel k is parameterized by a G -equivariant MLP φ with input representation ρ st and output representation ρ ⊗ := ρ in ⊗ ρ out , i.e. vec ( k )( x ) := φ ( x ) , then the kernel satisfies the equivariance constraint in Equation 2 for a compact group G .

In other words, G -equivariance of MLP is a sufficient condition for G -equivariance of the convolutional layer whose kernel it parameterizes. Using implicit kernels also has a very favourable property - it allows arbitrary steerable features as its input, which we discuss in the following section.

3.2 Expanding the input

Note that in the case of standard steerable convolutions, the input x ∈ R n of a kernel is usually only the difference between the spatial position of two points. However, there is no requirement that would disallow the expansion of the input space except for practical reasons. Hence, here we augment steerable kernels with an additional feature vector z ∈ R d z . This formulation allows us to incorporate relevant information in convolutional layers, such as physical and geometric features. For example, when performing convolutions on molecular graphs, z can encode the input and output atoms' types and yield different responses for different atoms. If introducing additional arguments into a kernel, the steerability constraint in equation 2 should be adapted to account for transformations of z :

which must hold for all g ∈ G,x ∈ R n , z ∈ R d z , where ρ z : G → R d z × d z is the representation of G acting on the additional features.

Again, analytically solving the constraint 3 to find a kernel basis for arbitrary ρ z is generally unfeasible. Note also that the solution strategy proposed in [7] requires a basis for functions over R n + d z , whose size tends to grow exponentially with d z and, therefore, is not suitable. Alternatively, we can now use the flexibility of neural representation and introduce additional features into a kernel at no cost.

3.3 Implementing a G -equivariant MLP

We are now interested in how to build an MLP that is equivariant to the transformations of the group G , i.e. a sequence of equivariant linear layers alternated with equivariant non-linearities [15, 37, 44]. It is important to say that our approach does not rely on a specific implementation of G -MLPs, and any algorithm of preference might be used (e.g. [15] or enforcing via an additional loss term).

The approach we employed in our experiments is described below. Since the irreducible representations of G are typically known 7 , one can always rely on the following properties: 1) any representation ρ of a compact group G can be decomposed as a direct sum of irreducible representations ρ ( g ) = Q T (⊕ i ∈ I ψ i ( g ) ) Q with a change of basis Q 8 and 2) Schur's Lemma , which states that there exist equivariant linear maps only between the irreducible representation of the same kind 9 . Hence, one can apply the right change of basis to the input and output of a linear layer and then learn only maps between input and output channels associated with the same irreducible representations. In the context of implicit kernels, the tensor product representation ρ in ⊗ ρ out in the last layer is

decomposed by a matrix containing the Clebsch-Gordan coefficients, which often appears in the analytical solutions of the kernel constraint in the related works. Note that the non-linearity σ used in the Steerable CNNs are G -equivariant and, therefore, can be used for the MLP as well.

3.4 G -steerable point convolutions

Point convolutions operate on point clouds - sets of N points endowed with spatial information X = { x i } N -1 i =0 ∈ R n × N . A point cloud thus provides a natural discretization of the data domain, which renders a convolution operator as follows:

To reduce the computational cost for large objects, one can induce connectivity onto x ∈ X and represent a point cloud as a graph G = ( V , E ) with nodes v i ∈ V and edges e ij ∈ E ⊆ V × V , where each node v i has a spatial location x i , node features z i and a corresponding feature map f in ( x i ) (see Figure 1). Additionally, each edge e ij can have an attribute vector z ij assigned to it (as in Section 3.2). This allows for a learnable message-passing point convolution whose computational cost scales linearly with the number of edges:

where N ( i ) = { j : ( v i , v j ) ∈ E} and the kernel k ( · ) is parameterized by a G -equivariant MLP.

3.5 Extension to G -steerable CNNs

Straightforwardly, the proposed method can be extended to dense convolutions. In such case, the kernel is defined as k : R n → R c out × c in × K n - that is, a continuous function that returns a collection of K × K × ... kernels given a relative position (and, optionally, arbitrary steerable features). In this case, the vectorized kernel is parameterized in exactly the same way as described above but is estimated at the center of each pixel.

3.6 Comparison with the analytical solution

A general basis for any G -steerable kernel is described in [7]. Essentially, it relies on two ingredients: i) a (pre-defined) finite G -steerable basis [17] for scalar filters and ii) a learnable equivariant linear map. Let us look at those in detail. Firstly, a finite G -steerable basis is essentially a collection of B orthogonal functions, i.e. Y : R n → R B , with the following equivariant property: Y ( g.x ) = ρ Y ( g ) Y ( x ) , for some representation ρ Y of G . The linear map in ii) , then, is a general equivariant linear layer, whose input and output transform as ρ Y and ρ in ⊗ ρ out .

In practice, it means that one has to provide a pre-defined basis Y for a group of interest G . Since the design might not be straightforward, [7] suggest a way to reuse an already derived O ( n ) steerable basis for any subgroup G ⊂ O ( n ) . While general, such a solution can be sub-optimal. For example, if n = 3 , an O (3) -steerable basis has local support inside a sphere, which is suitable for the group of 3D rotations SO (3) but not ideal for cylindrical symmetries, i.e. when G is the group SO (2) of planar rotations around the Z axis.

In comparison, the solution proposed in Section 3.1 replaces the pre-defined basis Y with a learnable G -MLP. It allows us to learn G -specific kernels without relying on a pre-derived basis for a larger group, which in turn means that we can theoretically obtain G -optimal kernel basis via learning (see A.3 for further details). Furthermore, the kernels defined in Section 3.1 can now be conditioned on arbitrary steerable features, which makes them more task-specific and expressive. In the context of a general basis, one can interpret the last linear layer of an implicit kernel as the map in ii) , and the activations before this layer as a learnable version of the basis Y in i) .

4 Related works

Group convolutions. Multiple architectures were proposed to achieve equivariance to a certain symmetry group. It has been proven [26] that the convolutional structure is a sufficient condition for

building a model that is equivariant to translations and actions of a compact group. One can separate group convolutions into two classes depending on the space on which the convolution operates: regular [3, 4, 9, 14, 26] and steerable [7, 10, 11, 46, 49, 52, 54] group convolutions. In the first case, the input signal is represented in terms of scalar fields on a group G , and the convolution relies on a discretization of the group space. Steerable convolutions are a class of G -equivariant convolutions that operate on feature fields over homogeneous spaces and achieve equivariance via constraining the kernel space. They further avoid the discretization of the group space and can reduce the equivariance error in the case of continuous groups.

G -steerable kernels. In the case of rigid body motions G = SO (3) , the solution of the equivariance constraint is given by spherical harmonics modulated by an arbitrary continuous radial function, which was analytically obtained by Weiler et al. [52]. Lang and Weiler [27] then applied the WignerEckart theorem to parametrize G -steerable kernel spaces over orbits of a compact G . The approach was later generalized by Cesa et al. [7], who proposed a solution for any compact sub-group of O (3) based on group restriction. Using this approach, one can obtain a kernel basis for a group G ≤ H if the basis for H is known. Despite its generalizability, the method still requires a pre-defined basis for the group H that is further adapted to G . The resulting solution is not guaranteed to be optimal for G ; see Section 3.6. We note that the practical value of steerable kernels is also high as they can be used for convolution over arbitrary manifolds in the framework of Gauge CNNs [8, 21, 50].

Implicit kernels. Using the implicit representation of convolution kernels for regular CNNs is not novel. It was used, for example, to model long-range dependencies in sequential data [39] or for signal representation [45]. Romero et al. [39] demonstrated that such parametrization allows building shallower networks, thus requiring less computational resources to capture global information about the system. Continuous kernels were recently used to build an architecture [25] for processing data of arbitrary resolution, dimensionality and length, yet equivariant solutions are scarce [47]. Finzi et al. [14] proposed parametrizing convolutions on Lie groups as continuous scalar functions in the group space. The method relies on discretizing a continuous group, which might lead to undesirable stochasticity of the model's output. Instead, we use Steerable CNNs that define the kernel as a function on a homogeneous space. While the discretization of this space is still required, in most cases, it is naturally given by the data itself, e.g. for point clouds; hence, no sampling error arises. It is also important to mention the key difference between implicit kernels and applying G -MLPs [15] directly - the latter is incapable of processing image/volumetric data as convolutions do. Henceforth, we focus on CNNs with consideration for potential extensions to various data modalities.

Equivariant point convolutions. A particular momentum has been gained by point convolutions in the form of equivariant message-passing [5, 40, 41, 46] specifically for problems where symmetry provides a strong inductive bias such as molecular modelling [1] or physical simulations [18]. Thomas et al. [46] pioneered SE (3) -equivariant steerable convolutions whose kernels are based on spherical harmonics modulated by a radial function. The approach was further generalized by Batzner et al. [2], who uses an MLP conditioned on the relative location to parameterize the radial function, although the basis of spherical harmonics is preserved. Brandstetter et al. [5] demonstrated that introducing geometric and physical information into an equivariant message-passing model improves the expressivity on various tasks, which we also observe in this work. Note that, for G = SO (3) or O (3) and without additional edge features z , our MLP can only learn a function of the radius and, therefore, is equivalent to the models proposed in [2].

5 Experiments

In this section, we implement Steerable CNNs with implicit kernels and apply them to various tasks 10 . First, we indicate the importance of correctly choosing the symmetry group on a synthetic N-body simulation problem where an external axial force breaks the rotational symmetry (see Section 5.2). Then, we prove the generalizability of the proposed approach as well as the gain in performance compared to the method proposed in [7] on ModelNet-40 (see Section 5.3). Afterwards, we show that one can introduce additional physical information into a kernel and significantly improve the performance of steerable convolutions on molecular data (see Section 5.4). Code and data to reproduce all experiments are available on GitHub.

5.1 Implementation

Implicit kernels. To parameterize implicit kernels, we employ linear layers (see Section 3.3) followed by quotient ELU non-linearities [7]. The last layer generates a steerable vector, which we reshape to yield a final convolutional kernel. The G -MLP takes as input a steerable vector obtained via direct sum (concatenation) of batch-normalized harmonic polynomial representation of the relative location x , edge features z ij and input node features z i . See details on pre-processing and task-specific inputs in Appendix B.1.

Steerable convolutions. We employ steerable point convolutional layers as described in Equation 5. For each task, we tune hyperparameters of all models on validation data: the number of layers, the number of channels in each layer, the depth and the width of implicit kernels, and the number of training epochs. For all the experiments, except the one reported in Table 2, we yield a number of parameters similar to the baselines with a deviation of less than 10%. For ModelNet-40 experiments, our architecture is partially inspired by the model introduced in [34], which uses gradual downsampling of a point cloud 11 . For N-body and QM9 experiments, we add residual connections to each steerable layer to learn higher-frequency features [20]. For QM9 and ModelNet-40 experiments, the last steerable layer returns a vector with invariant features for each node, to which we apply global pooling to obtain a global representation of an object, which is further passed to a classification MLP. In the case of Nbody experiments, we output the standard representation corresponding to the particles' coordinates. Details about optimization and model implementation for each task can be found in Appendix B.2.

5.2 The relevance of smaller G < O ( n ) : N-body simulation

Dataset. We conduct experiments using the N-body system [23], where particles are connected through springs, and their interaction is governed by Hooke's law. Similar to previous studies [5, 40], we modify the original trajectory prediction task to calculate the position of each particle in a 3dimensional space, given its initial position and velocity. We attach each particle to the XY plane using strings with random equilibrium lengths and pre-defined stiffness, which can slide freely over the plane (see Figure 2, upper left corner). This additional force term breaks the rotational symmetry of the system into only azimuthal symmetry; this system resembles a simplified version

of the problem of molecule binding to the surface of a larger molecule. We choose the model's hyperparameters to have a similar parameter budget to the SEGNN model [5], which has shown stateof-the-art performance on a similar task (we also compare against a non-equivariant baseline; see Table A2). For all models, we use the highest frequency of hidden representations in both G -MLP and Steerable CNNs equal to 1 . We use the velocity of a particle and the equilibrium length of the attached XY spring as input; the model's output transforms under the standard representation. We train a separate model for each value of plane strings' stiffness (3000 training points) to measure the impact of symmetry breaks on performance.

Results. As can be seen in Fig. 2, a Steerable CNN with azimuthal symmetry SO (2) significantly outperforms SEGNN, which is equivariant to a larger group O (3) . Since we introduced a force term that now breaks the rotational symmetry, SEGNN struggles to learn it. Furthermore, while in the default setting (plane strings' stiffness is 0), models achieve roughly the same performance, the absolute difference grows exponentially once the plane strings are introduced.

The synthetic problem is meant to show the importance of choosing the correct symmetry when designing a model. While the Euclidean group E(3) is often enough for N body systems in a vacuum, it is important to be careful when an external influence or underlying structure can disrupt global symmetries and negatively impact a model with a larger symmetry group. This can be relevant in scenarios like molecular docking simulations [12], where the system can align with the larger molecule, or material science [22], where the arrangement of atoms and crystal lattice structures yield a discrete group of symmetries smaller than O ( n ) .

5.3 Generalizability of implicit kernels: ModelNet-40

Dataset. The ModelNet-40 [56] dataset contains 12311 CAD models from the 40 categories of furniture with the orientation of each object aligned. The task is to predict the category of an object based on its point cloud model. 2468 models are reserved for the test partition. From the remaining objects, we take 80% for training and 20% for validation. We augment each partition with random rotations around the Z-axis. We induce connectivity on point clouds with a k-nearest neighbour search with k = 10 at each model layer and use normals as input node features.

Results. In this section, we demonstrate how one can build a Steerable CNN that is equivariant to an arbitrary subgroup of the Euclidean group G /multicloseright ( R 3 , +) ≤ E (3) . We compare the performance of implicit kernels with the standard steerable kernels obtained by group restriction [7] and keep the number of parameters similar. The results are shown in Figure 3. Implicit kernels achieve significant improvement in accuracy on test data for the majority of groups. The only statistically significant negative difference is presented for G = SO (3) , for which a tailored and hence optimal kernel basis is already available. When a custom solution is unknown, implicit kernels often significantly outperform the previously proposed method. There-

±

fore, they pose an efficient toolkit for building a kernel basis for an arbitrary subgroup of E (3) . We report the result of the best-performing model in Table 1. Although we do not outperform more taskspecific and involved approaches such as PointMLP [31], our model is on par with other group equivariant models. We emphasize that our model is a simple stack of convolutional layers and hypothesize that a smarter downsampling strategy, residual connections and hierarchical representations would significantly improve the overall performance of our model, which we leave for further research.

5.4 Flexibility of implicit kernels: QM9

Dataset. The QM9 dataset [55] is a public dataset consisting of about 130 k molecules with up to 29 atoms per molecule. Each molecule is represented by a graph with nodes denoting atoms and edges indicating covalent bonds. Each node is assigned a feature vector consisting of one-hot encoding of the type of the respective atom (H, C, N, O, F) and its spatial information corresponding to a low energy conformation. Additionally, each molecule is described by 19 properties from which we select

12 commonly taken in literature [5] for regression tasks. Different from common practice [5], we perform convolution on molecular graphs, i.e. with connectivity pre-defined by molecular structure instead of inducing it. The design choice is motivated by the presence of edge features, which we include in an implicit kernel to display the flexibility of neural representation.

Results. Wefirst demonstrate how one can use the flexibility of neural representation to introduce additional features of choice into a steerable convolutional layer. As each pair of atoms connected by a covalent bond is assigned a onehot encoding of the bond type z ij , we use it as a condition for O (3) -equivariant implicit kernels. Additionally, we follow Musaelian et al. [33] and embed one-hot encoding of the center and neighbour atom types z i and z j into the MLP. We include each property one by one and indicate corresponding performance gain in Figure 4. First, we observed a sharp improvement by switching from standard steerable kernels to implicit ones. We attribute it to the higher expressivity of implicit kernels and their ability to learn more complex interactions. Furthermore, injecting edge attributes into the kernel computation reduced MAE even further. Introducing both atom types and an edge type significantly influenced the per-

formance, corresponding to the model learning how to process each specific combination differently, thus adding to its expressivity. It is not surprising since a similar result was obtained by Brandstetter et al. [5], who used non-linear message aggregation conditioned on physical information, yielding state-of-the-art performance.

Scaling the model up. Table 2 shows the results of an E (3) -equivariant Steerable CNN with implicit kernels on the QM9 dataset. Both models we reported here have approx. 2 · 10 6 parameters and only differ in length and width of the convolutional layers. For non-energy regression tasks ( α , ∆ ε , ε HOMO , ε LUMO , µ and C ν ), we obtain results that are on par with the best-performing message-passing based approaches (we do not compare against transformers). We also indicate that Steerable CNNs with implicit kernels significantly outperform steerable linear convolutions (TFN, LieConv, L1Net) on most tasks. This is consistent with the observation of Brandstetter et al. [5], who pointed out that non-linear convolutions generally perform better than linear ones. For the remaining energy variables (G, H, U, U 0 , ZPVE) and R 2 , our model significantly falls behind the task-specific benchmark approaches. We theorize that it can be attributed to two factors. First, steerable convolutions generally do not perform well on these tasks compared to problem-tailored

frameworks (PaiNN [42], DimeNet++ [24], SphereNet [29]), also in the non-linear case (e.g. SEGNN [5]). Second, we hypothesize that molecular connectivity does not produce a sufficient number of atom-atom interactions, which is crucial for the performance of a message-passing-based model [5]. However, as the goal of the section was to demonstrate the flexibility of implicit kernels that can be conditioned on features of graph edges, we leave developing more involved architectures (e.g. with induced connectivity) for further work.

6 Conclusion

We propose a novel approach for implementing convolutional kernels of Steerable CNNs, allowing for the use of smaller groups and easy integration of additional features under the same framework with minor changes. To avoid analytically solving the group G -specific equivariance constraint, we use a G -equivariant MLP to parameterize a G -steerable kernel basis. We theoretically prove that MLP equivariance is sufficient for building an equivariant steerable convolutional layer. Our implicit representation outperforms a previous general method, offering a way to implement equivariance to various groups for which a custom kernel basis has not been developed. The N-body experiment suggests that this method will be particularly applicable in scenarios where rotational symmetry is disturbed, such as in material science or computational chemistry. The critical force term, which violated rotational symmetry, was effectively captured by our model, while the state-of-the-art model struggled with it. Additionally, our flexible neural representation enables the introduction of arbitrary features into a convolutional kernel, enhancing the expressivity of Steerable CNNs. We validate this advantage by applying Steerable CNNs to point cloud and molecular graph data and achieving competitive performance with state-of-the-art approaches. In conclusion, we present a simple yet efficient solution for constructing a general kernel basis equivariant to an arbitrary compact group.

Limitations

Steerable CNNs generally have high computational complexity and higher memory requirements compared to traditional CNNs. Implicit neural kernels do not help to mitigate the issue, yet they provide additional control over the kernel complexity. We found that when the kernels are parameterized by a single linear layer, the run time slightly decreases compared to the analytical solution 12 , while a relative performance gain remains. We suggest that more efficient ways to implement G -MLPs would significantly contribute to the acceleration of the method and leave it for further research. From the implementation point of view, the most troublesome was achieving initialization similar to the non-implicit convolutions. We expect the problem to manifest even stronger when dealing with dense convolutions. This, combined with the discretization error of matrix filters, might negatively affect the performance. We are, however, convinced that it is a matter of time before a robust way of initializing G -equivariant implicit kernels will be obtained.

Acknowledgments and Disclosure of Funding

All the experiments were performed using the Hemera compute cluster of Helmholtz-Zentrum Dresden-Rossendorf and the IvI cluster of the University of Amsterdam. This research results from a collaboration initiated at the London Geometry and Machine Learning Summer School 2022 (LOGML). The authors thank Anna Mészáros, Chen Cai and Ahmad Hammoudeh for their help at the initial stage of the project. They also thank Rob Hesselink for his assistance with visualizations.

References

Brandon M. Anderson, Truong Son Hy, and Risi Kondor. Cormorant: Covariant molecular neural networks. In NeurIPS , 2019. 6, 9

Simon Batzner, Tess E. Smidt, Lixin Sun, Jonathan P. Mailoa, Mordechai Kornbluth, Nicola Molinari, and Boris Kozinsky. E(3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials. Nature Communications , 13, 2022. 6

Erik J. Bekkers. B-spline cnns on lie groups. ArXiv , abs/1909.12057, 2020. 1, 6

Erik J. Bekkers, Maxime W Lafarge, Mitko Veta, Koen A.J. Eppenhof, Josien P.W. Pluim, and Remco Duits. Roto-translation covariant convolutional networks for medical image analysis. In MICCAI , 2018. 6

Johannes Brandstetter, Rob Hesselink, Elise van der Pol, Erik J. Bekkers, and Max Welling. Geometric and physical quantities improve e(3) equivariant message passing. ArXiv , abs/2110.02905, 2022. 1, 2, 3, 6, 7, 8, 9, 10, 16

Michael M. Bronstein, Joan Bruna, Taco Cohen, and Petar Velivckovi'c. Geometric deep learning: Grids, groups, graphs, geodesics, and gauges. ArXiv , abs/2104.13478, 2021. 1

Gabriele Cesa, Leon Lang, and Maurice Weiler. A program to build e(n)-equivariant steerable cnns. In ICLR , 2022. 2, 3, 4, 5, 6, 7, 8, 10, 14, 15, 18

Taco Cohen, Maurice Weiler, Berkay Kicanaoglu, and Max Welling. Gauge equivariant convolutional networks and the icosahedral cnn. In ICML , pages 1321-1330. PMLR, 2019. 6

Taco Cohen and Max Welling. Group equivariant convolutional networks. In ICML , 2016. 1, 6

Taco S. Cohen, Mario Geiger, and Maurice Weiler. A general theory of equivariant CNNs on homogeneous spaces. NeurIPS , 32, 2019. 1, 6

Taco S. Cohen and Max Welling. Steerable CNNs. In ICLR , Nov. 2016. 1, 6

Gabriele Corso, Hannes Stärk, Bowen Jing, Regina Barzilay, and T. Jaakkola. Diffdock: Diffusion steps, twists, and turns for molecular docking. ArXiv , abs/2210.01776, 2022. 8

Carlos Esteves, Christine Allen-Blanchette, Ameesh Makadia, and Kostas Daniilidis. Learning so(3) equivariant representations with spherical cnns. International Journal of Computer Vision , 128:588-600, 2017. 8

Marc Finzi, Samuel Stanton, Pavel Izmailov, and Andrew Gordon Wilson. Generalizing convolutional neural networks for equivariance to lie groups on arbitrary continuous data. In ICML , 2020. 1, 6, 9

Marc Finzi, Max Welling, and Andrew Gordon Wilson. A practical method for constructing equivariant multilayer perceptrons for arbitrary matrix groups. ArXiv , abs/2104.09459, 2021. 4, 6

Daniel Franzen and Michael Wand. Nonlinearities in steerable so(2)-equivariant cnns. ArXiv , abs/2109.06861, 2021. 8

William T. Freeman and Edward H. Adelson. The design and use of steerable filters. IEEE Transactions on Pattern Analysis & Machine Intelligence , (9):891-906, 1991. 5

Fabian B. Fuchs, Daniel E. Worrall, Volker Fischer, and Max Welling. Se(3)-transformers: 3d roto-translation equivariant attention networks. ArXiv , abs/2006.10503, 2020. 6, 9

Justin Gilmer, Samuel S. Schoenholz, Patrick F. Riley, Oriol Vinyals, and George E. Dahl. Neural message passing for quantum chemistry. ArXiv , abs/1704.01212, 2017. 9

Francesco Di Giovanni, James R. Rowbottom, Benjamin Paul Chamberlain, Thomas Markovich, and Michael Bronstein. Graph neural networks as gradient flows: understanding graph convolutions via energy. 2022. 7

Pim De Haan, Maurice Weiler, Taco Cohen, and Max Welling. Gauge equivariant mesh {cnn}s: Anisotropic convolutions on geometric graphs. In ICLR , 2021. 6

Sekouba Kaba and Siamak Ravanbakhsh. Equivariant networks for crystal structures. ArXiv , abs/2211.15420, 2022. 8

Thomas Kipf, Ethan Fetaya, Kuan-Chieh Wang, Max Welling, and Richard Zemel. Neural relational inference for interacting systems. ArXiv , 2018. 7

Johannes Klicpera, Shankari Giri, Johannes T. Margraf, and Stephan Gunnemann. Fast and uncertainty-aware directional message passing for non-equilibrium molecules. ArXiv , abs/2011.14115, 2020. 9, 10

David M. Knigge, David W. Romero, Albert Gu, Efstratios Gavves, Erik J. Bekkers, Jakub M. Tomczak, Mark Hoogendoorn, and Jan-Jakob Sonke. Modelling long range dependencies in n-d: From task-specific to a general purpose cnn. ArXiv , abs/2301.10540, 2023. 6

Risi Kondor and Shubhendu Trivedi. On the generalization of equivariance and convolution in neural networks to the action of compact groups. ArXiv , abs/1802.03690, 2018. 1, 3, 5, 6

Leon Lang and Maurice Weiler. A Wigner-Eckart theorem for group equivariant convolution kernels. In ICLR , 2020. 6

Yann LeCun and Yoshua Bengio. Convolutional networks for images, speech, and time series. 1998. 1

Yi Liu, Limei Wang, Meng Liu, Xuan Zhang, Bora Oztekin, and Shuiwang Ji. Spherical message passing for 3d graph networks. ArXiv , abs/2102.05013, 2021. 9, 10

Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization. In ICLR , 2019. 17

Xu Ma, Can Qin, Haoxuan You, Haoxi Ran, and Yun Raymond Fu. Rethinking network design and local geometry in point cloud: A simple residual mlp framework. ArXiv , abs/2202.07123, 2022. 8

Benjamin Kurt Miller, Mario Geiger, Tess E. Smidt, and Frank No'e. Relevance of rotationally equivariant convolutions for predicting molecular properties. ArXiv , abs/2008.08461, 2020. 9

Albert Musaelian, Simon Batzner, Anders Johansson, Lixin Sun, Cameron J. Owen, Mordechai Kornbluth, and Boris Kozinsky. Learning local equivariant representations for large-scale atomistic dynamics. ArXiv , abs/2204.05249, 2022. 9

Adrien Poulenard and Leonidas J. Guibas. A functional approach to rotation equivariant nonlinearities for tensor field networks. CVPR , pages 13169-13178, 2021. 7, 8

Charles R Qi, Li Yi, Hao Su, and Leonidas J Guibas. Pointnet++: Deep hierarchical feature learning on point sets in a metric space. ArXiv , 2017. 8

Yongming Rao, Jiwen Lu, and Jie Zhou. Spherical fractal convolutional neural networks for point cloud recognition. CVPR , pages 452-460, 2019. 8

Siamak Ravanbakhsh. Universal equivariant multilayer perceptrons. ArXiv , 2020. 4

David W Romero, Robert-Jan Bruintjes, Jakub M Tomczak, Erik J Bekkers, Mark Hoogendoorn, and Jan C van Gemert. Flexconv: Continuous kernel convolutions with differentiable kernel sizes. ArXiv , 2021. 2

David W. Romero, Anna Kuzina, Erik J. Bekkers, Jakub M. Tomczak, and Mark Hoogendoorn. Ckconv: Continuous kernel convolution for sequential data. ArXiv , abs/2102.02611, 2022. 2, 6, 17, 18

Victor Garcia Satorras, Emiel Hoogeboom, and Max Welling. E(n) equivariant graph neural networks. In ICML , 2021. 3, 6, 7, 9

Kristof Schütt, Pieter-Jan Kindermans, Huziel Enoc Sauceda Felix, Stefan Chmiela, Alexandre Tkatchenko, and Klaus-Robert Müller. Schnet: A continuous-filter convolutional neural network for modeling quantum interactions. In NeurIPS , 2017. 6, 9

Kristof T. Schütt, Oliver T. Unke, and Michael Gastegger. Equivariant message passing for the prediction of tensorial properties and molecular spectra. In ICML , 2021. 9, 10

Jean-Pierre Serre. Linear representations of finite groups. 1977. 2

J. Shawe-Taylor. Building symmetries into feedforward networks. In 1989 First IEE International Conference on Artificial Neural Networks, (Conf. Publ. No. 313) , page 158-162, Oct 1989. 4

Vincent Sitzmann, Julien N. P. Martel, Alexander W. Bergman, David B. Lindell, and Gordon Wetzstein. Implicit neural representations with periodic activation functions. ArXiv , abs/2006.09661, 2020. 6

Nathaniel Thomas, Tess E. Smidt, Steven M. Kearnes, Lusann Yang, Li Li, Kai Kohlhoff, and Patrick F. Riley. Tensor field networks: Rotation- and translation-equivariant neural networks for 3d point clouds. ArXiv , abs/1802.08219, 2018. 6, 9

Tycho F. A. van der Ouderaa, David W. Romero, and Mark van der Wilk. Relaxing equivariance constraints with non-stationary continuous filters. ArXiv , abs/2204.07178, 2022. 6

Yue Wang, Yongbin Sun, Ziwei Liu, Sanjay E. Sarma, Michael M. Bronstein, and Justin M. Solomon. Dynamic graph cnn for learning on point clouds. ACM TOG , 38:1 - 12, 2019. 17

Maurice Weiler and Gabriele Cesa. General E (2) -Equivariant Steerable CNNs. In NeurIPS , 2019. 3, 6

Maurice Weiler, Patrick Forré, Erik Verlinde, and Max Welling. Coordinate Independent Convolutional Networks - Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds. ArXiv , 2021. 6

Maurice Weiler, Patrick Forré, Erik Verlinde, and Max Welling. Equivariant and Coordinate Independent Convolutional Networks . 2023. 3, 14

Maurice Weiler, Mario Geiger, Max Welling, Wouter Boomsma, and Taco Cohen. 3d steerable cnns: Learning rotationally equivariant features in volumetric data. In NeurIPS , 2018. 1, 3, 6

Maurice Weiler, Fred A. Hamprecht, and Martin Storath. Learning steerable filters for rotation equivariant CNNs. In CVPR , 2018. 16, 18

Daniel E. Worrall, Stephan J. Garbin, Daniyar Turmukhambetov, and Gabriel J. Brostow. Harmonic networks: Deep translation and rotation equivariance. In CVPR , 2017. 6

Zhenqin Wu, Bharath Ramsundar, Evan N. Feinberg, Joseph Gomes, Caleb Geniesse, Aneesh S. Pappu, Karl Leswing, and Vijay S. Pande. Moleculenet: A benchmark for molecular machine learning. ArXiv , 2017. 2, 8

Zhirong Wu, Shuran Song, Aditya Khosla, Fisher Yu, Linguang Zhang, Xiaoou Tang, and Jianxiong Xiao. 3d shapenets: A deep representation for volumetric shapes. CVPR , pages 19121920, 2015. 2, 8

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper tackles one problem of Steerable CNN: one needs to analytically solve a group G-specific equivariant constraint (eq 2 in the paper) in order to obtain the basics for the kernel.\n\nThe authors propose to avoid this analytical solution by using a G-equivariant MLP to parameterize a G-steerable kernel basis (hence called implicit kernel). By doing so, they obtain a framework that allows one to achieve equivariance to various groups even when an analytical solution for the kernel basis is not know.\n\nThe authors provide a theoretical proof that MLP equivariance is sufficient for building an equivariant steerable convolutional layer and empirically show that the suggested framework is useful in practice. Specifically, they show the relevance of the model being able to adapt to smaller G < O(n)  (as opposed to assuming O(3) as in SEGNN) using the N-Body simulation system. They show the generalizability of their method on the ModelNet-40 dataset where the proposed technique works better than Standard Steerable CNNs when the analytical solution for the basis is not available. And finally they show the flexibility of their method that allows one to introduce additional constraints (such as bonds between molecules or atoms) on QM9 dataset. In both the last two tasks the authors compare their method with an extensive list of SOTA methods.\n\nSTRENGTHS:\n- The paper tackles an important limitation of steerable CCNs.\n- The proposed solution is shown to be effective, at least when the analytical solution is unknown. Hence constitutes a detectable and positive improvement for the community.\n- The paper reads well.\n\nWEAKNESSES:\nIn my opinion there is an opportunity to strengthen the paper with a bit more empirical analysis. Since the kernel is learnt, it might be insightful to know to which extent the learnt solution is equivariant. Currently accuracy on downstream tasks is used as a proxy for that, but a specific analysis on equivariance violation could further enhance the paper. For example, is there a relation with the MLP size and the ability to learn the equivariance? Etc...\n\nSome parts could be more clear or more details could be provided. There are missing details about some experimental tasks, or how the model size used compares to the SOTA, etc… Please see the questions below for a more concrete list.\n\nThe paper is not accessible to the wide ML community. Specifically the paper makes a lot of assumptions about the reader’s knowledge. I don’t think this is a reason for rejection but it would be great to make an effort, whenever possible, to make the paper more accessible. Just a few examples but there are many through the paper: \n- not all readers might be familiar with the concept of natural action (line 82). I’d recommend to clarify it.\n- \"matrix containing the Clebsch-Gordan coefficients\". Could explain what they are or add a reference.\n- Some of the notation used could be clarified, for example not everyone is familiar with the outer semidirect product used. It would be great to define it.\n\n**Minor notes:**\n- Definition 3: I’ would consider adding “for all g in G in x in X”\n- Line 289 “it’s” ==> it is (to be consistent with the rest of the paper)\n- Line 314 refers to Table 5.3 but I think it might be Table 1?\n- In the Bibliography the name of conferences are not consistent: sometimes it uses the full name of the conference  and the abbreviation, some other just the abbreviation, other again just the full name.\n\nQUESTIONS:\nI think the paper is good. There are opportunities to strengthen it by addressing some of the weaknesses mentioned above and by addressing the questions here.\n\n- Do the authors think it’s useful and possible to provide the empirical analysis mentioned in the weaknesses.\n- Can the author make an effort to make the paper more accessible?\n- In section 2 the authors write “Trivial representation: all elements of G act on V as the identity mapping of V , i.e. ⇢(g) = I. “\nThis does not seem to satisfy the definition 2, specifically this mapping does not seem to be invertible. Is this statement correct?\n- Can you describe maybe with an example how you would build the G-equivariant MLP? Section 3.4 would benefit from a more concrete explanation (even if in the appendix). Even better if the authors would consider to publicly releasing the code.\n- “We choose the model’s hyperparameters to have a similar parameter budget to the SEGNN”. Can one be more specific? How many parameters for example were trained in SEGNN compared to the proposed model?\n- In the ModelNet-40 experiment there is a sampling step involved. The Figure reports results with error bars using 5 runs but it’s unclear if the 5 runs are with different initial seeds or different sampling. Also is 5 runs sufficient?\n- In Figure 2: Do the authors have any intuition about why there is such a large STD for G=M but not for other groups?\n- 5.3: The task for this experiment could be explained better. The dataset is clear but what is the task?\n- QM9 task: “Both models we reported here have approx. 2 · 10^6”  How does this compare to other models? \n- For the QM9 tasks more details about the meaning of the regressed energy values would help. Not everyone if familiar with this dataset and task.\n\n**Appendix B2**\n- For the N-Body “The run-time, on average, was 5 min. “ On which machine? And what is “run-time”? Per Single point? Per Bach? For the whole training?\n- For  ModelNet-40 the machine is specified but the what is ”Runtime” should still be clarified.\n\nLIMITATIONS:\nThe papers states the limitations in within the results discussion. For example :\n- ”The only statistically significant negative difference is presented for G = SO(3), for which a tailored and hence optimal kernel basis is already available.”\nor\n- “Although we do not outperform more task-specific and involved approaches such as PointMLP [31],” \nor\n- \"For the remaining energy variables (G, H, U, U0, ZPVE) and R2, our model significantly falls behind the task-specific benchmark approaches such as DimeNet or PaiNN.” \n\nAdditionally there is a limitation section in Appendix C. I would encourage the authors to bring the limitation section from the appendix into the paper and summarize the limitations found during all the experiments in the same section.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work considers the setting of steerable networks, in which a particular kind of group-equivariant (alternatively, G-steerable) kernel is translationally convolved with an input vector (or matrix) field. The group-specific design of such a network focuses on the derivation of the group-equivariant kernel. Although prior work laid out a procedure for deriving an analytic, orthogonal basis of functions whose span encompasses all such G-steerable kernels, this paper presents a conceptually simpler method: parametrizing the kernel with an equivariant map. This method is flexible and allows conditioning the kernel on other invariant or equivariant attributes. They test their method on datasets of point cloud classification, synthetic N-body simulations, and molecules (QM9).\n\nSTRENGTHS:\nOriginality: Using an equivariant network to parametrize a group-steerable kernel is a natural and elegant idea. Although implicit kernels have been used in other settings, their application to steerable kernels is novel and well-founded. Conceptually, it is simpler than previous analytical approaches to deriving explicit, orthogonal bases. \n\nClarity: The proposal is clearly described, and the paper provides a helpful consolidation of related work overall.\n\nQuality: The proposal is indeed equivariant, and experiments show good results over a sufficiently wide range of settings.\n\nSignificance: Group-steerable networks are widely used, and this formulation seems to apply to all compact groups of interest.\n\nWEAKNESSES:\n1. It seems to me that the vast majority of application problems involving subgroups of $E(3)$ acting on $\\mathbb{R}^3$ simply involve $SO(3)$ or $O(3)$. In that sense, the scope of this paper may be limited. Nonetheless, this does not preclude other groups of interest from arising in future applications, but the paper could be strengthened by providing additional motivating examples.\n2. The tradeoffs between this method and the analytical method of [7] are subtle, and indeed they require many of the same tools (for example, knowing the irrep structure of the group of interest, so that one can construct equivariant functions over the input space, even if they are not required to be orthogonal). As noted in the questions, I do not fully understand the benefits of this method as compared to [7], although it is clearly more general and simpler to articulate.\n\nQUESTIONS:\n1. My biggest question is that I do not understand the second paragraph of Section 3.2, comparing this approach with that of [7]. The paragraph seems to claim that the basis produced by [7] might be too large, but I thought [7] would produce an orthogonal equivariant basis — and if it is orthogonal, how can the “size” of the basis be suboptimal (as it is determined the dimension of the vector space of steerable functions)?  \n2. On a related note, both this paper and [7] seem to require knowledge of the irreps of the group of interest. Can the authors elaborate on which of the ingredients (a) through (d) listed at the bottom of page 5 of [7], are troublesome? Section 3.2 seems to imply not that the subgroup method for obtaining a steerable basis is computationally intensive, but that it produces a lower quality output — is that the case?\n3. If already provided with a steerable basis in the style of [7], could one still “condition” on physical info (at least invariants) by just storing a bank of coefficients, and choosing different coefficients based on endpoint features? If so, this could make more sense as a baseline experiment. Of course, such a method may not choose coefficients smoothly as a function of the endpoint features, which is one potential benefit of the authors’ proposed conditioning method.\n4. To clarify, if you had a G-equivariant MLP, is the primary barrier to using this G-MLP to derive a G-steerable orthogonal basis the orthogonalization? In other words, one could set random weights to the G-MLP, and then try to orthogonalized to get a basis of equivariant functions? (I think the problem is this: orthogonalization can be done via Gram-Schmidt, but only if one can compute inner products exactly — with $\\mathbb{R}^3$ as the base space, it does not seem like one could do this exactly. However, I would be happy to hear the authors’ thoughts on this, as if there were an easy way to reduce a G-equivariant MLP directly to a steerable basis, it would perhaps weaken the argument that their proposed method is easier than computing an explicit G-steerable basis.)\n5. In Section 5.2, why is the baseline an $O(3)$-equivariant net? This is an unfair comparison for a problem that is not $O(3)$-equivariant, since such a network can never express the ground truth solution. Can the authors report performance relative to a generic net instead (e.g. only translation invariant), or an approximately equivariant network?\n6. What do the authors hypothesize causes the difference in performance shown in Figure 3? For example, is it conditioning on radial vectors? \n\nAs a quick note, here is one other piece of slightly related work at the intersection of equivariance and implicit convolutional kernels: “Relaxing equivariance constraints with non-stationary continuous filters” by Ouderaa et al 2022.\n\nLIMITATIONS:\nThere are not potential negative societal impacts.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposed implicit neural representations via MLPs to parameterize G-steerable kernels. Additionally, the performance of the approach is extensively empirically tested against alternative methods on multiple tasks.\n\nSTRENGTHS:\n- Conceptually\n\nIn many cases, steerable group equivariant neural networks yield more accurate equivariance in practice compared to counterparts with regular representations that use sampling. Implicit kernels have shown to be beneficial in the space of regular group equivariant networks but have not yet been explored in the context of steerable equivariance. This paper demonstrates how steerable equivariant networks can also be extended to use implicit kernels.\n\n- Scientifically\n\nThe paper extensively validates the method on actual datasets in which more complex symmetries play an important role. The paper extensively validates the approach on both graph structures and dynamical systems, demonstrating relevance in real-world applications and also includes extensive comparison with existing baselines.\n\nWEAKNESSES:\n- Implicit kernels do not alleviate the need to solve Eq.2 analytically.\n\nThe method relies on steerable kernels, which require analytically solving Eq. 2 for specific group G of interest. Implicit kernels do not alleviate this, as the constraint of Eq. 2 must still be solved in order to build G-equivariant MLPs. Although these are shortcomings of steerable equivariance and not of the implicit kernels, it should be stressed that implicit kernels do not mitigate these shortcomings.\n\n- High computational complexity and memory cost\n\nThe method only seems to increase computational and memory complexity over already more expensive steerable kernels.\n\n- No benefit with increased depth of implicit kernels\n\nThe paper relies on using multi-layer MLPs as implicit kernels but notices that deeper MLPs do not necessarily lead to improved performance (App. B.3 and App. B.4). The paper does argue that the increased flexibility that implicit kernels offer may benefit long-range dependencies but does not conduct any experiments on this. It would be interesting to demonstrate the benefit of implicit kernels on tasks where deeper kernels help.\n\n- Class of implicit functions\n\nIs there a reason why only multi-layer perceptrons are considered to parameterise implicit functions? Since the depth of implicit functions has not been found to be beneficial, it could also be interesting to consider different simpler non-linear bases. Even more so, because they appear to be effective in the context of regular group equivariant networks (e.g. sinusoidal/fourier [1], b-splines [2] or simple radial basis features [3]). Or is there a reason why this would not be applicable or less interesting in this setting?\n\n[1] Sitzmann, Vincent, et al. \"Implicit neural representations with periodic activation functions.\" Advances in neural information processing systems 33 (2020): 7462-7473.\n\n[2] Bekkers, Erik J. \"B-spline cnns on lie groups.\" arXiv preprint arXiv:1909.12057 (2019).\n\n[3] van der Ouderaa, Tycho FA, and Mark van der Wilk. \"Sparse Convolutions on Lie Groups.\" NeurIPS Workshop on Symmetry and Geometry in Neural Representations. PMLR, 2023.\n\nQUESTIONS:\na) Am I correct in the assessment that the method does not alleviate the need to solve Eq. 2 analytically and thus does not directly improve the efficiency of steerable kernels in that regard?\n\nb) What are computational runtimes? (and what is the explicit overhead of implicit kernels)\n\nc) What are memory costs? (and what is the explicit overhead of implicit kernels)\n\nLIMITATIONS:\nThe paper extensively compares (externally) with alternative approaches. However, the (intrinsic) analysis of how different types of implicit kernels may affect performance remains limited. One of the main benefits of implicit kernels is the increased flexibility it provides when designing the model. It would therefore be very interesting to get a better idea of how different choices in the functional form of the implicit kernels affect the output or model performance. This point becomes especially relevant since the paper includes negative results on deeper implicit kernels (which nevertheless are interesting findings), as this would help demonstrate how implicit kernels can be beneficial in practice.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-08"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work proposes a novel method for achieving a G-equivariant neural network by implicitly parameterizing the steerable filters by G-equivariant MLP instead of learning them with a steerable basis function. The proposed frame is flexible, and the implicit kernel can also consider the problem context by expanding input to the kernel with problem-related parameters. The work provides theoretical justification for the proposed method and shows the model's effectiveness via empirical study.\n\nSTRENGTHS:\n1. The work proposes a novel method to achieve G-equivariant by implicit parameterization, which provides more flexibility to the G-equivariant model. And empirically shows the benefit of the flexible kernel that can consider the additional problem-dependent features.\n2. The text is easy to follow and clearly written\n\nWEAKNESSES:\n1. Evaluation: The empirical evaluations are performed on N-body simulations, ModelNet, and the QM9 dataset. Which is different from the previous works [1,2,3]. And this makes it difficult for readers to make a direct comparison between the approaches. And raises the question of whether the performance of the approaches is significantly dependent on the chosen problem.\n2. The proposed method utilized G-equivariant MLP for implicit parameterization of the kernel. But the rationale for choosing an implicit steerable kernel over EMLP[4] is not addressed nor included in the evaluations.\n\n1.Steerable CNNs\n\n2.3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data\n\n3.Clebsch–Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network\n\n4.A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups\n\nQUESTIONS:\n1. The G-equivariant MLPs also require us to solve constraints [1]. And we need to do the same for the steerable methods. How are these two different?\n2. The result of Table 1: What is the rationale for choosing a problem where we have non-G-equivariant models outperforming G-equivariant methods? \n3. Line: 105, equation 2 seems a little out of context. Even though the relevant work is cited and a little background on how we reach that constraint will make the work self-contained.\n\n1.A Practical Method for Constructing Equivariant Multilayer Perceptrons for\nArbitrary Matrix Group\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2023-07-09"}