Affine Invariance in Continuous-Domain Convolutional Neural Networks
Ali Mohaddes Johannes Lederer
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.