Geometric wavelet scattering on graphs and manifolds
Geometric wavelet scattering on graphs and manifolds
复制标题
图和流形上的几何小波散射
DOI：
10.1117/12.2529615
复制
发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Wolf, Guy
中科院分区：
文献类型：
--
作者：
Gao, Feng;Hirn, Matthew;Perlmutter, Michael;Wolf, Guy
卷积神经网络(CNN)正在为欧几里德区域上的二维和三维图像成像科学带来革命性的变化。然而，许多数据集本质上是非欧几里得的，并且通过其他数学结构(如图或流形)可以更好地建模。这种情况导致了几何深度学习的发展，几何深度学习指的是一系列旨在将CNN的原理转化为这些非欧几里德结构的研究。在这个过程中，出现了各种各样的挑战，包括如何定义这样的几何网络，如何有效地计算和训练它们，以及它们的数学性质是什么。本文描述了几何小波散射变换，它是由交替的多尺度几何小波变换和非线性激活函数组成的图和流形的一种几何CNN。顾名思义，几何小波散射变换是由S.Mallat首先引入的欧几里德小波散射变换的改编，适用于图形和流形数据。与欧氏对应的几何小波散射变换一样，几何小波散射变换也具有几个理想的性质。在流形环境中，这些性质包括直到用户指定的尺度的等距不变性和对小的微分同胚稳定性。在流形和图数据集上的数值结果，包括图和流形分类任务等，说明了该方法的实用价值。
Convolutional neural networks (CNNs) are revolutionizing imaging science for two- and three-dimensional images over Euclidean domains. However, many data sets are intrinsically non-Euclidean and are better modeled through other mathematical structures, such as graphs or manifolds. This state of affairs has led to the development of geometric deep learning, which refers to a body of research that aims to translate the principles of CNNs to these non-Euclidean structures. In the process, various challenges have arisen, including how to define such geometric networks, how to compute and train them efficiently, and what are their mathematical properties.In this letter we describe the geometric wavelet scattering transform, which is a type of geometric CNN for graphs and manifolds consisting of alternating multiscale geometric wavelet transforms and nonlinear activation functions. As the name suggests, the geometric wavelet scattering transform is an adaptation of the Euclidean wavelet scattering transform, first introduced by S. Mallat, to graph and manifold data. Like its Euclidean counterpart, the geometric wavelet scattering transform has several desirable properties. In the manifold setting these properties include isometric invariance up to a user specified scale and stability to small diffeomorphisms. Numerical results on manifold and graph data sets, including graph and manifold classification tasks as well as others, illustrate the practical utility of the approach.
关于 STD_k/3[k ;
DOI：
--
发表时间：
2008
期刊：
Discrete Mathematics 308
影响因子：
--
作者：
K. Akiyama;C. Suetake;Kenzi Akiyama and Chihiro Suetake
通讯作者：
Kenzi Akiyama and Chihiro Suetake