ChebLieNet：不变谱图NN通过李群上的黎曼几何变成了等值的

我们介绍ChebLieNet，这是一种在（各向异性的）流形上的群变量方法。在基于图和群的神经网络的成功基础上，我们利用最近在几何深度学习领域的发展，得出了一种利用数据中任何各向异性的新方法。通过对李群的离散近似，我们开发了一个由各向异性卷积层（切比雪夫卷积）、空间汇集和非汇集层以及全局汇集层组成的图神经网络。群体等值是通过图上的等值和不变算子实现的，图上有各向异性的左不变黎曼尼距离的亲缘关系被编码在边缘。由于其简单的形式，黎曼公制可以模拟任何各向异性，包括在空间和方向领域。这种对各向异性的控制允许平衡图卷积层的等值性（各向异性度量）和不变性（各向同性度量）。因此，我们为更好地理解各向异性的特性打开了大门。此外，我们从经验上证明了CIFAR10上各向异性参数的（与数据相关）甜蜜点的存在。这一关键结果证明了我们可以通过利用数据中的各向异性来获得好处。我们还评估了这种方法在STL10（图像数据）和ClimateNet（球形数据）上的可扩展性，显示其对不同任务的显著适应性。

原文题目：ChebLieNet: Invariant Spectral Graph NNs Turned Equivariant by Riemannian Geometry on Lie Groups

原文：We introduce ChebLieNet, a group-equivariant method on (anisotropic) manifolds. Surfing on the success of graph- and group-based neural networks, we take advantage of the recent developments in the geometric deep learning field to derive a new approach to exploit any anisotropies in data. Via discrete approximations of Lie groups, we develop a graph neural network made of anisotropic convolutional layers (Chebyshev convolutions), spatial pooling and unpooling layers, and global pooling layers. Group equivariance is achieved via equivariant and invariant operators on graphs with anisotropic left-invariant Riemannian distance-based affinities encoded on the edges. Thanks to its simple form, the Riemannian metric can model any anisotropies, both in the spatial and orientation domains. This control on anisotropies of the Riemannian metrics allows to balance equivariance (anisotropic metric) against invariance (isotropic metric) of the graph convolution layers. Hence we open the doors to a better understanding of anisotropic properties. Furthermore, we empirically prove the existence of (data-dependent) sweet spots for anisotropic parameters on CIFAR10. This crucial result is evidence of the benefice we could get by exploiting anisotropic properties in data. We also evaluate the scalability of this approach on STL10 (image data) and ClimateNet (spherical data), showing its remarkable adaptability to diverse tasks.