The notion of group invariance helps neural networks in recognizing patterns
and features under geometric transformations. Indeed, it has been shown that
group invariance can largely improve deep learning performances in practice,
where such transformations are very common. This research studies affine
invariance on continuous-domain convolutional neural networks. Despite other
research considering isometric invariance or similarity invariance, we focus on
the full structure of affine transforms generated by the generalized linear
group $\mathrm{GL}_2(\mathbb{R})$. We introduce a new criterion to assess the
similarity of two input signals under affine transformations. Then, unlike
conventional methods that involve solving complex optimization problems on the
Lie group $G_2$, we analyze the convolution of lifted signals and compute the
corresponding integration over $G_2$. In sum, our research could eventually
extend the scope of geometrical transformations that practical deep-learning
pipelines can handle.

本研究探讨连续域卷积神经网络中的仿射不变性，并引入一种新准则评估仿射变换下两个输入信号的相似性。通过分析抬升信号的卷积并计算相应的广义线性群 $G_2$ 上的积分，与解决复杂优化问题的传统方法不同，本研究为实际深度学习流程处理几何变换的范围提供了扩展。

连续域卷积神经网络中的仿射不变性

Affine Invariance in Continuous-Domain Convolutional Neural Networks

Invariance and equivariance to geometrical transformations have proven to be
very useful inductive biases when training (convolutional) neural network
models, especially in the low-data regime. Much work has focused on the case
where the symmetry group employed is compact or abelian, or both. Recent work
has explored enlarging the class of transformations used to the case of Lie
groups, principally through the use of their Lie algebra, as well as the group
exponential and logarithm maps. The applicability of such methods to larger
transformation groups is limited by the fact that depending on the group of
interest $G$, the exponential map may not be surjective. Further limitations
are encountered when $G$ is neither compact nor abelian. Using the structure
and geometry of Lie groups and their homogeneous spaces, we present a framework
by which it is possible to work with such groups primarily focusing on the Lie
groups $G = \text{GL}^{+}(n, \mathbb{R})$ and $G = \text{SL}(n, \mathbb{R})$,
as well as their representation as affine transformations $\mathbb{R}^{n}
\rtimes G$. Invariant integration as well as a global parametrization is
realized by decomposing the `larger` groups into subgroups and submanifolds
which can be handled individually. Under this framework, we show how
convolution kernels can be parametrized to build models equivariant with
respect to affine transformations. We evaluate the robustness and
out-of-distribution generalisation capability of our model on the standard
affine-invariant benchmark classification task, where we outperform all
previous equivariant models as well as all Capsule Network proposals.

使用李群和李代数的结构与几何学，提出了一个框架，用来在大多数情况下处理几何变换的不规则群，重点关注李群 GL+(n, R) 和 SL (n, R)，以及它们作为仿射变换的表示。通过将 `较大的` 群分解为子群和子流形来实现不变积分和全局参数化。在这个框架下，我们展示了如何参数化卷积核来构建关于仿射变换等变的模型，并在标准的仿射不变基准分类任务上评估了我们模型的鲁棒性和越域泛化能力，结果表明我们的模型优于所有先前的等变模型以及所有胶囊网络提议。