Neural Information Processing Systems http://nips.cc/

The big empirical success of group equivariant networks has led in recent years to the sprouting of a great variety of equivariant network architectures. A particular focus has thereby been on rotation and reflection equivariant CNNs for planar images. Here we give a general description of E(2)-equivariant convolutions in the framework of Steerable CNNs. The theory of Steerable CNNs thereby yields constraints on the convolution kernels which depend on group representations describing the transformation laws of feature spaces. We show that these constraints for arbitrary group representations can be reduced to constraints under irreducible representations. A general solution of the kernel space constraint is given for arbitrary representations of the Euclidean group E(2) and its subgroups. We implement a wide range of previously proposed and entirely new equivariant network architectures and extensively compare their performances. E(2)-steerable convolutions are further shown to yield remarkable gains on CIFAR-10, CIFAR-100 and STL-10 when used as drop in replacement for non-equivariant convolutions.

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.

We present a convolutional network that is equivariant to rigid body motions.

Work done while at Helmholtz-Zentrum Dresden-Rossendorf. Qualcomm AI Research is an initiative of Qualcomm Technologies, Inc. 37th Conference on Neural Information Processing Systems (NeurIPS 2023).

Classical image based CNNs are equivariant to translations (modulo pooling) and this is perhaps a major reason for their immense success. A image recognition task however also contains various other types of symmetries, that are usually incorporated by means of data augmentation. In order to incorporate more symmetries in a principled manner, such that they obviate data augmentation for those symmetries, group equivariant convolutional networks were proposed. Originally they incorporated simple symmetries such as 90 degree rotations, reflections in addition to translations. This was followed by work incorporating 360 degree rotations as in harmonic networks, gated harmonic networks and so on.

We had a discussion about this paper with the reviewers raising the question of novelty since the general topic of steerable neural networks (i.e., equivariant neural networks where the activations are defined as a function on a homogeneous space) has now been thoroughly explored in the literature. Also, Weiler et al have a paper on SE(3) equivariant nets, so the E(2) is arguably just a simpler variant. There are three reasons why I nonetheless recommend this paper for acceptance: 1. The kernel constraint conditions are meticulously worked out for a range of specific subgroups of O(2). One wonders whether this could be done more generally and whether it really requires so much tedious algebra.

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.