Problem
Research questions and friction points this paper is trying to address.
Group-Convolutional Neural Networks
3D Geometry
Equivariance
Computational Cost
Feature Space
Discretizing Group-Convolutional Neural Networks for 3D Geometry in Feature Space
🤖 AI Summary
This work addresses the high computational and memory costs of group convolutional neural networks when processing 3D geometric data, which arise from dense sampling over transformation groups. To mitigate this, the authors propose a feature-space sparse sampling strategy that selects representative samples based on feature similarity, replacing conventional dense geometric sampling. This approach strictly preserves equivariance while substantially reducing computational overhead. By decoupling geometric resolution from computational cost, the method enables flexible trade-offs between accuracy and efficiency and further accelerates training through precomputed geometric similarities. Experimental results demonstrate that even with coarse-grained sampling, the model maintains high classification accuracy and significantly speeds up the training of 3D equivariant classifiers.
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
feature-space sampling
group-convolutional neural networks
equivariance
3D geometry
computational efficiency
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