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We introduce ChebLieNet, a group-equivariant method on (anisotropic)manifolds. Surfing on the success of graph- and group-based neural networks, wetake advantage of the recent developments in the geometric deep learning fieldto derive a new approach to exploit any anisotropies in data. Via discreteapproximations of Lie groups, we develop a graph neural network made ofanisotropic convolutional layers (Chebyshev convolutions), spatial pooling andunpooling layers, and global pooling layers. Group equivariance is achieved viaequivariant and invariant operators on graphs with anisotropic left-invariantRiemannian distance-based affinities encoded on the edges. Thanks to its simpleform, the Riemannian metric can model any anisotropies, both in the spatial andorientation domains. This control on anisotropies of the Riemannian metricsallows to balance equivariance (anisotropic metric) against invariance(isotropic metric) of the graph convolution layers. Hence we open the doors toa better understanding of anisotropic properties. Furthermore, we empiricallyprove the existence of (data-dependent) sweet spots for anisotropic parameterson CIFAR10. This crucial result is evidence of the benefice we could get byexploiting anisotropic properties in data. We also evaluate the scalability ofthis approach on STL10 (image data) and ClimateNet (spherical data), showingits remarkable adaptability to diverse tasks.

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