Many applications require the robustness, or ideally the invariance, of a neural network to certain transformations of input data. Most commonly, this requirement is addressed by either augmenting the training data, using adversarial training, or defining network architectures that include the desired invariance automatically. Unfortunately, the latter often relies on the ability to enlist all possible transformations, which make such approaches largely infeasible for infinite sets of transformations, such as arbitrary rotations or scaling. In this work, we propose a method for provably invariant network architectures with respect to group actions by choosing one element from a (possibly continuous) orbit based on a fixed criterion. In a nutshell, we intend to 'undo' any possible transformation before feeding the data into the actual network. We analyze properties of such approaches, extend them to equivariant networks, and demonstrate their advantages in terms of robustness as well as computational efficiency in several numerical examples. In particular, we investigate the robustness with respect to rotations of images (which can possibly hold up to discretization artifacts only) as well as the provable rotational and scaling invariance of 3D point cloud classification.

翻译：许多应用都需要神经网络的坚固性,或者理想地说,神经网络对于输入数据的某些变异性需要一定的坚固性。最常见的情况是,这一要求是通过增加培训数据,使用对抗性培训,或者自动界定网络结构,包括所期望的变异性。不幸的是,后者往往依赖于吸收所有可能的变异的能力,这使得这些变异性对于无限的变异性基本不可行,例如任意旋转或缩放。在这项工作中,我们提出了一个在基于固定标准的轨道上从一个(可能的连续)轨道上选择一个元素来组合动作的变异性网络结构的方法。在坚固的轨道上,我们打算“在将数据输入到实际网络之前不做任何可能的变异性。我们分析这些变异性方法的特性,将其扩大到电子变异性网络,并在若干数字实例中显示其在稳健性和计算效率方面的优势。特别是,我们研究了图像变异性图的转换性(可能只保持离散的手工艺品)以及3级的云性旋转和升缩放。