We introduce a general method for achieving robust group-invariance in
group-equivariant convolutional neural networks ($G$-CNNs), which we call the
$G$-triple-correlation ($G$-TC) layer. The approach leverages the theory of the
triple-correlation on groups, which is the unique, lowest-degree polynomial
invariant map that is also complete. Many commonly used invariant maps - such
as the max - are incomplete: they remove both group and signal structure. A
complete invariant, by contrast, removes only the variation due to the actions
of the group, while preserving all information about the structure of the
signal. The completeness of the triple correlation endows the $G$-TC layer with
strong robustness, which can be observed in its resistance to invariance-based
adversarial attacks. In addition, we observe that it yields measurable
improvements in classification accuracy over standard Max $G$-Pooling in
$G$-CNN architectures. We provide a general and efficient implementation of the
method for any discretized group, which requires only a table defining the
group's product structure. We demonstrate the benefits of this method for
$G$-CNNs defined on both commutative and non-commutative groups - $SO(2)$,
$O(2)$, $SO(3)$, and $O(3)$ (discretized as the cyclic $C8$, dihedral $D16$,
chiral octahedral $O$ and full octahedral $O_h$ groups) - acting on
$\mathbb{R}^2$ and $\mathbb{R}^3$ on both $G$-MNIST and $G$-ModelNet10
datasets.

我们引入了一种在群等变卷积神经网络（$G$-CNNs）中实现鲁棒群不变性的通用方法，称之为 $G$-triple-correlation ($G$-TC) 层。该方法利用了群上的三重相关性理论，即唯一的、最低次多项式不变映射，也是完备的。通过与标准 Max $G$-Pooling 在 $G$-CNN 体系结构中相比较，我们观察到 $G$-TC 层在分类精度上具有显著的改进。

G - 对称网络中健壮的 G - 不变性的通用框架

A General Framework for Robust G-Invariance in G-Equivariant Networks

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。

变形鲁棒的旋转 - 缩放 - 平移等变卷积神经网络

Deformation Robust Roto-Scale-Translation Equivariant CNNs

Convolutional neural networks or standard CNNs (StdCNNs) are
translation-equivariant models that achieve translation invariance when trained
on data augmented with sufficient translations. Recent work on equivariant
models for a given group of transformations (e.g., rotations) has lead to
group-equivariant convolutional neural networks (GCNNs). GCNNs trained on data
augmented with sufficient rotations achieve rotation invariance. Recent work by
authors arXiv:2002.11318 studies a trade-off between invariance and robustness
to adversarial attacks. In another related work arXiv:2005.08632, given any
model and any input-dependent attack that satisfies a certain spectral
property, the authors propose a universalization technique called SVD-Universal
to produce a universal adversarial perturbation by looking at very few test
examples. In this paper, we study the effectiveness of SVD-Universal on GCNNs
as they gain rotation invariance through higher degree of training
augmentation. We empirically observe that as GCNNs gain rotation invariance
through training augmented with larger rotations, the fooling rate of
SVD-Universal gets better. To understand this phenomenon, we introduce
universal invariant directions and study their relation to the universal
adversarial direction produced by SVD-Universal.

本文研究使用 CNN 以及其衍生的 GCNN 在旋转不变性训练方面的成果，以及在特定谱属性下所提出的通用性对抗性扰动技术 SVD-Universal 的效果与普适性。