几何深度学习中的水平流和流形随机性
Horizontal Flows and Manifold Stochastics in Geometric Deep Learning
摘要：
我们在几何深度学习中引入了两种构造：1）以连续的方式在流形上传输方向相关的卷积滤波器，从而定义一个自然结合完整性旋转效应的卷积算子; 2）通过对以加权扩散平均值为中心的流形值随机变量进行采样，允许对流形卷积层进行有效评估。这两种方法都受到流形上的随机学和几何统计学的启发，并提供了随机方法的例子-这里是水平框架束流和非线性桥梁采样方案，可以用于几何深度学习。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和几何深度学习中现有方法的关系，并建立了所提出的构造的重要属性。
英文摘要：
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.
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