Incorporating group symmetry directly into the learning process has proved to
be an effective guideline for model design. By producing features that are
guaranteed to transform covariantly to the group actions on the inputs,
group-equivariant convolutional neural networks (G-CNNs) achieve significantly
improved generalization performance in learning tasks with intrinsic symmetry.
General theory and practical implementation of G-CNNs have been studied for
planar images under either rotation or scaling transformation, but only
individually. We present, in this paper, a roto-scale-translation equivariant
CNN (RST-CNN), that is guaranteed to achieve equivariance jointly over these
three groups via coupled group convolutions. Moreover, as symmetry
transformations in reality are rarely perfect and typically subject to input
deformation, we provide a stability analysis of the equivariance of
representation to input distortion, which motivates the truncated expansion of
the convolutional filters under (pre-fixed) low-frequency spatial modes. The
resulting model provably achieves deformation-robust RST equivariance, i.e.,
the RST symmetry is still "approximately" preserved when the transformation is
"contaminated" by a nuisance data deformation, a property that is especially
important for out-of-distribution generalization. Numerical experiments on
MNIST, Fashion-MNIST, and STL-10 demonstrate that the proposed model yields
remarkable gains over prior arts, especially in the small data regime where
both rotation and scaling variations are present within the data.

该研究提出了一种基于耦合群卷积的旋转、缩放和平移等变卷积神经网络 RST-CNN，该网络通过稳定性分析可证明具有变形鲁棒性，能在旋转、缩放和平移等输入畸变的情况下保持等变性，从而在 MNIST、Fashion-MNIST 和 STL-10 数据集上实现了显著提升。