Downsampling layers are crucial building blocks in CNN architectures, which
help to increase the receptive field for learning high-level features and
reduce the amount of memory/computation in the model. In this work, we study
the generalization of the uniform downsampling layer for group equivariant
architectures, e.g., G-CNNs. That is, we aim to downsample signals (feature
maps) on general finite groups with anti-aliasing. This involves the following:
(a) Given a finite group and a downsampling rate, we present an algorithm to
form a suitable choice of subgroup. (b) Given a group and a subgroup, we study
the notion of bandlimited-ness and propose how to perform anti-aliasing.
Notably, our method generalizes the notion of downsampling based on classical
sampling theory. When the signal is on a cyclic group, i.e., periodic, our
method recovers the standard downsampling of an ideal low-pass filter followed
by a subsampling operation. Finally, we conducted experiments on image
classification tasks demonstrating that the proposed downsampling operation
improves accuracy, better preserves equivariance, and reduces model size when
incorporated into G-equivariant networks
