Discretizing Group-Convolutional Neural Networks for 3D Geometry in Feature Space

TL;DR

This paper introduces a method to discretize group-convolutional neural networks for 3D geometry by sampling in feature space, significantly reducing computational costs while maintaining accuracy.

Contribution

It proposes a novel feature-space sampling approach that decouples geometric resolution from computational costs in GCNNs for 3D data.

Findings

Coarse feature-space sampling preserves classification accuracy.

Precomputation based on geometric similarity accelerates training.

Decoupling geometric resolution from memory reduces costs.

Abstract

Group-convolutional neural networks (GCNNs) are among the most important methods for introducing symmetry as an inductive bias in deep learning: In each linear layer, GCNNs sample a transformation group $G$ densely and correlate data and filters in different poses (with suitable anti-aliasing for steerable GCNNs) to maintain equivariance with respect to $G$. Unfortunately, applying filters to many data items resulting from this sampling is expensive (even for translations alone, i.e., in ordinary CNNs), and costs grow exponentially with increasing degrees of freedom (such as translations and rotations in 3D), which often hinders practical applications. In this paper, we propose sampling in feature space, i.e., replacing geometrically dense samples with representative samples selected by feature similarity. This decouples geometric resolution from memory and processing costs during training and inference, providing a novel way to trade off computational effort and accuracy. Our main empirical finding is that a coarse feature-space sampling already preserves classification accuracy remarkably well, which permits precomputation based on geometric similarity, accelerating the training of equivariant 3D classifiers substantially.