Chew Joyce, Hirn Matthew, Krishnaswamy Smita, Needell Deanna, Perlmutter Michael, Steach Holly, Viswanath Siddharth, Wu Hau-Tieng
Appl Comput Harmon Anal. 2024 May;70. doi: 10.1016/j.acha.2024.101635. Epub 2024 Feb 6.
The scattering transform is a multilayered, wavelet-based transform initially introduced as a mathematical model of convolutional neural networks (CNNs) that has played a foundational role in our understanding of these networks' stability and invariance properties. In subsequent years, there has been widespread interest in extending the success of CNNs to data sets with non-Euclidean structure, such as graphs and manifolds, leading to the emerging field of geometric deep learning. In order to improve our understanding of the architectures used in this new field, several papers have proposed generalizations of the scattering transform for non-Euclidean data structures such as undirected graphs and compact Riemannian manifolds without boundary. Analogous to the original scattering transform, these works prove that these variants of the scattering transform have desirable stability and invariance properties and aim to improve our understanding of the neural networks used in geometric deep learning. In this paper, we introduce a general, unified model for geometric scattering on measure spaces. Our proposed framework includes previous work on compact Riemannian manifolds without boundary and undirected graphs as special cases but also applies to more general settings such as directed graphs, signed graphs, and manifolds with boundary. We propose a new criterion that identifies to which groups a useful representation should be invariant and show that this criterion is sufficient to guarantee that the scattering transform has desirable stability and invariance properties. Additionally, we consider finite measure spaces that are obtained from randomly sampling an unknown manifold. We propose two methods for constructing a data-driven graph on which the associated graph scattering transform approximates the scattering transform on the underlying manifold. Moreover, we use a diffusion-maps based approach to prove quantitative estimates on the rate of convergence of one of these approximations as the number of sample points tends to infinity. Lastly, we showcase the utility of our method on spherical images, a directed graph stochastic block model, and on high-dimensional single-cell data.
散射变换是一种基于小波的多层变换，最初作为卷积神经网络（CNN）的数学模型引入，在我们对这些网络的稳定性和不变性特性的理解中发挥了基础性作用。在随后的几年里，人们广泛关注将CNN的成功扩展到具有非欧几里得结构的数据集，如图和流形，从而催生了几何深度学习这一新兴领域。为了增进我们对这个新领域中使用的架构的理解，几篇论文提出了针对非欧几里得数据结构（如无向图和无边界紧致黎曼流形）的散射变换的推广。与原始散射变换类似，这些工作证明了散射变换的这些变体具有理想的稳定性和不变性特性，旨在增进我们对几何深度学习中使用的神经网络的理解。在本文中，我们为测度空间上的几何散射引入了一个通用的统一模型。我们提出的框架将先前关于无边界紧致黎曼流形和无向图的工作作为特殊情况包含在内，但也适用于更一般的设置，如有向图、带符号图和有边界流形。我们提出了一个新的准则，该准则确定一个有用的表示应该对哪些群不变，并表明这个准则足以保证散射变换具有理想的稳定性和不变性特性。此外，我们考虑从对未知流形进行随机采样得到的有限测度空间。我们提出了两种方法来构建一个数据驱动的图，在该图上相关的图散射变换近似于基础流形上的散射变换。此外，我们使用基于扩散映射的方法来证明当样本点数量趋于无穷时这些近似之一的收敛速率的定量估计。最后，我们在球面图像、有向图随机块模型和高维单细胞数据上展示了我们方法的实用性。
