The Euclidean scattering transform was introduced nearly a decade ago to
improve the mathematical understanding of convolutional neural networks.
Inspired by recent interest in geometric deep learning, which aims to
generalize convolutional neural networks to manifold and graph-structured
domains, we define a geometric scattering transform on manifolds. Similar to
the Euclidean scattering transform, the geometric scattering transform is based
on a cascade of wavelet filters and pointwise nonlinearities. It is invariant
to local isometries and stable to certain types of diffeomorphisms. Empirical
results demonstrate its utility on several geometric learning tasks. Our
results generalize the deformation stability and local translation invariance
of Euclidean scattering, and demonstrate the importance of linking the used
filter structures to the underlying geometry of the data.

本文介绍了一种基于仿射不变性和微分同胚稳定性的 Geometric Scattering Transform 方法，可以使卷积神经网络在流形和图结构领域中更具有普适性和稳定性。

紧致黎曼流形上的几何小波散射网络

Geometric Wavelet Scattering Networks on Compact Riemannian Manifolds

Dictionary learning algorithms or supervised deep convolution networks have
considerably improved the efficiency of predefined feature representations such
as SIFT. We introduce a deep scattering convolution network, with predefined
wavelet filters over spatial and angular variables. This representation brings
an important improvement to results previously obtained with predefined
features over object image databases such as Caltech and CIFAR. The resulting
accuracy is comparable to results obtained with unsupervised deep learning and
dictionary based representations. This shows that refining image
representations by using geometric priors is a promising direction to improve
image classification and its understanding.

本文介绍一种基于波浪变换滤波器的深度散射卷积网络，通过引入几何先验知识来细化图像特征提取，从而实现图像分类和理解的目标。