## Deformation Robust Roto-Scale-Translation Equivariant CNNs

### Liyao Gao, Guang Lin, Wei Zhu

04 Apr 2022, 21:10 (modified: 20 Jul 2022, 19:56)Accepted by TMLREveryone
Abstract: 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 ($\mathcal{RST}$-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 $\mathcal{RST}$ equivariance, i.e., the $\mathcal{RST}$ 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.
License: Creative Commons Attribution 4.0 International (CC BY 4.0)
Submission Length: Regular submission (no more than 12 pages of main content)
Changes Since Last Submission: School title corrected. Name corrected following university guidance.
Code: https://github.com/gaoliyao/Roto-scale-translation-Equivariant-CNN
Assigned Action Editor: ~Simon_Kornblith1
Submission Number: 29