We introduce two constructions in geometric deep learning for 1) transporting orientation-dependent convolutional filters over a manifold in a continuous way and thereby defining a convolution operator that naturally incorporates the rotational effect of holonomy; and 2) allowing efficient evaluation of manifold convolution layers by sampling manifold valued random variables that center around a weighted diffusion mean. Both methods are inspired by stochastics on manifolds and geometric statistics, and provide examples of how stochastic methods – here horizontal frame bundle flows and non-linear bridge sampling schemes, can be used in geometric deep learning. We outline the theoretical foundation of the two methods, discuss their relation to Euclidean deep networks and existing methodology in geometric deep learning, and establish important properties of the proposed constructions.

中文翻译：

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几何深度学习中的水平流和流形随机

我们在几何深度学习中引入了两种结构：1）以连续方式在流形上传输依赖于方向的卷积滤波器，从而定义一个自然地结合了完整旋转效应的卷积算子；2) 通过采样以加权扩散平均值为中心的流形值随机变量，允许对流形卷积层进行有效评估。这两种方法都受到流形和几何统计的随机性的启发，并提供了如何将随机方法（此处为水平框架束流和非线性桥采样方案）用于几何深度学习的示例。我们概述了这两种方法的理论基础，讨论了它们与欧几里得深度网络的关系以及几何深度学习中的现有方法，