We present a PDE-based framework that generalizes Group equivariant Convolutional Neural Networks (G-CNNs). In this framework, a network layer is seen as a 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 as rotation in addition to the standard translation equivariance of CNNs. Having all the desired symmetries included in the design obviates the need to include them by means of costly techniques such as data augmentation. We will discuss our PDE-based G-CNNs (PDE-G-CNNs) in a general homogeneous space setting while also going into the specifics of our primary case of interest: roto-translation equivariance. We solve the PDE of interest by a combination of linear group convolutions and non-linear morphological group convolutions with analytic kernel approximations that we underpin with formal theorems. Our kernel approximations allow for fast GPU-implementation of the PDE-solvers, we release our implementation with this article in the form of the LieTorch extension to PyTorch, available at https://gitlab.com/bsmetsjr/lietorch . Just like for linear convolution a morphological convolution is specified by a kernel that we train in our PDE-G-CNNs. In PDE-G-CNNs we do not use non-linearities such as max/min-pooling and ReLUs as they are already subsumed by morphological convolutions. We present a set of experiments to demonstrate the strength of the proposed PDE-G-CNNs in increasing the performance of deep learning based imaging applications with far fewer parameters than traditional CNNs.
翻译:我们提出了一个基于 PDE 的框架, 使 Group Qevarial Convolutional Neal 网络( G- CNNs) 普遍化为基于 PDE 的框架。 在这个框架中, 网络层被视为一组 PDE- Soltvers, 其中以具有几何意义的 PDE 系数作为该层的可训练重量。 在同质空间上制定我们的 PDE 格式, 使得这些网络能够以内在的对称来设计, 例如, 除了CNN 的标准翻译平衡之外, 还可以进行旋转。 由于设计中包含所有想要的对数, 不需要通过数据增强等昂贵的技术来包括它们。 我们将讨论基于 PDE G- 的 GNNN( PE- G- CNs) 的 G- GN( 具有几何意义的PDE- G- G- GNs) 组合, 同时也可以进入我们主要感兴趣的案例: 恒定的变现变数。 我们的变数和变数组之间的变数和变数组的变数组合, 我们的变数的变数和变数的变数的变数的变数, 我们的变数的变数的变数的变数, 的变数的变数, 的变数的变数的变数, 变数的变数, 变数的变数的变数已经的变数的变数的变数的变数的变数, 的变数的变数, 变数的变数, 变数的变数, 变数的变数的变数的变数的变数, 变数的变数的变数, 变数的变数的变数的变数的变数的变数的变数, 通过的变数, 变数的变数的变数的变数的变数的变数的变数的变数, 变数, 变数的变数的变数的变数的变数的变数的变数的变数的变数的变数的变数的变数的变数的变数的变数的变数的变数, 变数的变数的变数的变数的变数的变数, 变数的变数的变数的变数的变数的变数