论文标题
小组神经网络的理论方面
Theoretical Aspects of Group Equivariant Neural Networks
论文作者
论文摘要
在过去的几年中，探索了群体的神经网络，从理论和实践的角度来看很有趣。他们利用群体表示理论，非共同的谐波分析和差异几何形状来利用概念，这些几何形状经常出现在机器学习中。实际上，它们已被证明可以减少样本和建模复杂性，特别是在存在诸如任意旋转之类的输入转换的挑战性任务中。我们以群体代表理论的阐述以及定义和评估群体积分和卷积所必需的机制开始这项工作。然后，我们显示了最近的SO（3）和SE（3）e（3）eproivariant网络的应用程序，即球形CNN，Clebsch-Gordan网络和3D Stopoble CNNS。我们开始讨论两个最新的理论结果。首先，由Kondor和Trivedi（ICML'18）撰写，显示神经网络在且仅当它具有卷积结构时是群体的。第二，Cohen等人。 （Neurips'19），将第一个概括为较大类的网络，其中特征地图是均匀空间上的字段。
Group equivariant neural networks have been explored in the past few years and are interesting from theoretical and practical standpoints. They leverage concepts from group representation theory, non-commutative harmonic analysis and differential geometry that do not often appear in machine learning. In practice, they have been shown to reduce sample and model complexity, notably in challenging tasks where input transformations such as arbitrary rotations are present. We begin this work with an exposition of group representation theory and the machinery necessary to define and evaluate integrals and convolutions on groups. Then, we show applications to recent SO(3) and SE(3) equivariant networks, namely the Spherical CNNs, Clebsch-Gordan Networks, and 3D Steerable CNNs. We proceed to discuss two recent theoretical results. The first, by Kondor and Trivedi (ICML'18), shows that a neural network is group equivariant if and only if it has a convolutional structure. The second, by Cohen et al. (NeurIPS'19), generalizes the first to a larger class of networks, with feature maps as fields on homogeneous spaces.
