  Recent works show that group equivariance as an inductive bias improves
neural network performance for both classification and generation. However,
designing group-equivariant neural networks is challenging when the group of
interest is large and is unknown. Moreover, inducing equivariance can
significantly reduce the number of independent parameters in a network with
fixed feature size, affecting its overall performance. We address these
problems by proving a new group-theoretic result in the context of equivariant
neural networks that shows that a network is equivariant to a large group if
and only if it is equivariant to smaller groups from which it is constructed.
Using this result, we design a novel fast group equivariant construction
algorithm, and a deep Q-learning-based search algorithm in a reduced search
space, yielding what we call autoequivariant networks (AENs). AENs find the
right balance between equivariance and network size when tested on new
benchmark datasets, G-MNIST and G-Fashion-MNIST, obtained via group
transformations on MNIST and Fashion-MNIST respectively that we release.
Extending these results to group convolutional neural networks, where we
optimize between equivariances, augmentations, and network sizes, we find group
equivariance to be the most dominating factor in all high-performing GCNNs on
several datasets like CIFAR10, SVHN, RotMNIST, ASL, EMNIST, and KMNIST.
