PDE-Based Group Equivariant Convolutional Neural Networks

We present a PDE-based framework that generalizes Group equivariant Convolutional Neural Networks (G-CNNs). In this framework, network layer is seen as set of PDE-solvers where geometrically meaningful PDE-coefficients become the layer's trainable weights. Formulating our PDEs on homogeneous spaces allows these networks to be designed with built-in symmetries such rotation in addition standard translation equivariance CNNs. Having all desired included design obviates need include them by means costly techniques data augmentation. will discuss G-CNNs (PDE-G-CNNs) general space setting while also going into specifics primary case interest: roto-translation equivariance. solve PDE interest combination linear group convolutions and non-linear morphological analytic kernel approximations we underpin formal theorems. Our allow for fast GPU-implementation PDE-solvers, release implementation article form LieTorch extension PyTorch, available at https://gitlab.com/bsmetsjr/lietorch . Just like convolution specified train PDE-G-CNNs. PDE-G-CNNs do not use non-linearities max/min-pooling ReLUs they are already subsumed convolutions. experiments demonstrate strength proposed increasing performance deep learning based imaging applications far fewer parameters than traditional