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Affine Invariance in Continuous-Domain Convolutional Neural Networks

The notion of group invariance helps neural networks in recognizing patterns and features under geometric transformations. Group convolutional neural networks enhance traditional convolutional neural networks by incorporating group-based geometric structures into their design. 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 group of all invertible $2 \times 2$ real matrices (generalized linear group $\mathrm{GL}_2(\mathbb{R})$). We introduce a new criterion to assess the invariance of two signals under affine transformations. The input image is embedded into the affine Lie group $G_2 = \mathbb{R}^2 \ltimes \mathrm{GL}_2(\mathbb{R})$ to facilitate group convolution operations that respect affine invariance. Then, we analyze the convolution of embedded signals over $G_2$. In sum, our research could eventually extend the scope of geometrical transformations that usual deep-learning pipelines can handle.

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