用偏微分方程和变分方法设计旋转不变神经网络
Designing Rotationally Invariant Neural Networks from PDEs and Variational Methods
摘要：
偏微分方程（PDE）模型及其相关的变分能量公式通常是旋转不变的设计。这确保了输入的旋转导致输出的相应旋转，这在诸如图像分析的应用中是期望的。卷积神经网络（CNN）不具有这种特性，现有的补救措施通常很复杂。本文的目标是研究扩散和变分模型如何实现旋转不变性，并将这些想法转移到神经网络。作为一个核心的新颖性，我们提出了激活函数，耦合网络通道，结合信息从几个定向过滤器。这保证了网络的基本构建块内的旋转不变性，同时仍然允许方向滤波。由此产生的神经架构本质上是旋转不变的。只有几个小的过滤器，他们可以实现相同的不变性现有的技术，需要一个细粒度的采样方向。我们的研究结果有助于将扩散和变分模型转化为数学上有根据的网络架构，并为基于模型的CNN设计提供了新的概念。
英文摘要：
Partial differential equation (PDE) models and their associated variational energy formulations are often rotationally invariant by design. This ensures that a rotation of the input results in a corresponding rotation of the output, which is desirable in applications such as image analysis. Convolutional neural networks (CNNs) do not share this property, and existing remedies are often complex. The goal of our paper is to investigate how diffusion and variational models achieve rotation invariance and transfer these ideas to neural networks. As a core novelty we propose activation functions which couple network channels by combining information from several oriented filters. This guarantees rotation invariance within the basic building blocks of the networks while still allowing for directional filtering. The resulting neural architectures are inherently rotationally invariant. With only a few small filters, they can achieve the same invariance as existing techniques which require a fine-grained sampling of orientations. Our findings help to translate diffusion and variational models into mathematically well-founded network architectures, and provide novel concepts for model-based CNN design.
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