流形上的几何散射
Geometric Scattering on Manifolds
摘要：
欧几里德散射变换是近十年前引入的，旨在提高对卷积神经网络（ConvNets）在图像数据分析和其他任务中成功的数学理解。受最近对几何深度学习的兴趣的启发，其目的是将ConvNets推广到流形和图形结构域，我们将散射变换推广到紧致流形。类似于欧几里德散射变换，我们的几何散射变换是基于级联的设计过滤器和逐点非线性，这使得散射层提供的特征提取的严格分析。我们在这里的主要重点是对这个几何散射网络的理论理解，而搁置实施方面，虽然我们注意到，应用类似的变换图形数据分析最近已在相关工作中进行了研究。我们的研究结果建立的条件下，几何散射提供本地化的等距不变的描述的流形信号，这也是稳定的家庭制定的内在流形方面的同构。这些结果不仅推广了欧几里得散射的变形稳定性和局部旋转平移不变性，而且还证明了将所使用的滤波器结构（例如，在几何深度学习中）到底层流形几何或其表示的数据几何。
英文摘要：
The Euclidean scattering transform was introduced nearly a decade ago to improve the mathematical understanding of the success of convolutional neural networks (ConvNets) in image data analysis and other tasks. Inspired by recent interest in geometric deep learning, which aims to generalize ConvNets to manifold and graph-structured domains, we generalize the scattering transform to compact manifolds. Similar to the Euclidean scattering transform, our geometric scattering transform is based on a cascade of designed filters and pointwise nonlinearities, which enables rigorous analysis of the feature extraction provided by scattering layers. Our main focus here is on theoretical understanding of this geometric scattering network, while setting aside implementation aspects, although we remark that application of similar transforms to graph data analysis has been studied recently in related work. Our results establish conditions under which geometric scattering provides localized isometry invariant descriptions of manifold signals, which are also stable to families of diffeomorphisms formulated in intrinsic manifolds terms. These results not only generalize the deformation stability and local roto-translation invariance of Euclidean scattering, but also demonstrate the importance of linking the used filter structures (e.g., in geometric deep learning) to the underlying manifold geometry, or the data geometry it represents.
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