  Group-equivariant convolutional neural networks (G-CNN) heavily rely on
parameter sharing to increase CNN's data efficiency and performance. However,
the parameter-sharing strategy greatly increases the computational burden for
each added parameter, which hampers its application to deep neural network
models. In this paper, we address these problems by proposing a
non-parameter-sharing approach for group equivariant neural networks. The
proposed methods adaptively aggregate a diverse range of filters by a weighted
sum of stochastically augmented decomposed filters. We give theoretical proof
about how the continuous group convolution can be approximated by our methods.
Our method applies to both continuous and discrete groups, where the
augmentation is implemented using Monte Carlo sampling and bootstrap
resampling, respectively. We demonstrate that our methods serve as an efficient
extension of standard CNN. Experiments on group equivariance tests show how our
methods can achieve superior performance to parameter-sharing group equivariant
networks. Experiments on image classification and image denoising tasks show
that in certain scenarios, with a suitable set of filter bases, our method
helps improve the performance of standard CNNs and build efficient lightweight
image denoising networks. The code will be available at
https://github.com/ZhaoWenzhao/MCG_CNN.
