Implicit Convolutional Kernels for Steerable CNNs

TL;DR

This paper introduces a flexible method for constructing steerable CNNs using implicit neural representations, enabling easy generalization to various symmetry groups and demonstrating effectiveness across multiple scientific tasks.

Contribution

It proposes using MLPs to parameterize G-steerable kernels, simplifying implementation and broadening applicability to any group G with an equivariant MLP.

Findings

Effective on N-body simulations

Improves point cloud classification accuracy

Enhances molecular property prediction

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.