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.

翻译：群不变性概念有助于神经网络在几何变换下识别模式和特征。事实上，研究表明群不变性可以极大提升深度学习在实际应用中的性能，而这类变换在实际场景中非常常见。本研究探讨了连续域卷积神经网络中的仿射不变性。与其他研究考虑等距不变性或相似不变性不同，我们聚焦于由广义线性群 $\mathrm{GL}_2(\mathbb{R})$ 生成的仿射变换的完整结构。我们引入了一个新准则来评估输入信号在仿射变换下的相似性。随后，与涉及在李群 $G_2$ 上求解复杂优化问题的传统方法不同，我们分析了提升信号的卷积并计算了在 $G_2$ 上的相应积分。总之，我们的研究有望拓展实际深度学习流水线所能处理的几何变换范围。