ChebLieNet: Invariant spectral graph NNs turned equivariant by Riemannian geometry on Lie groups

Hugo Aguettaz, Erik J. Bekkers, Michaël Defferrard

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.

Paper: arXiv:2111.12139, OpenReview:WsfXFxqZXRO.

Installation

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1. Optionally, create and activate a virtual environment.

   python -m venv cheblienet
   source cheblienet/bin/activate
   python -m pip install --upgrade pip setuptools wheel

2. Clone this repository.

   git clone https://github.com/haguettaz/ChebLieNet.git

3. Install the ChebLieNet package and its dependencies.

   python -m pip install -e ChebLieNet

Notebooks

• graph_manifold.ipynb: building graphs from sampled Lie groups.
• eigen_space.ipynb: visualizing the Fourier modes (eigenspaces) of Lie groups.
• graph_diffusion.ipynb: heat diffusion on Lie groups.
• nn_layers.ipynb: convolution, pooling and unpooling layers.

Reproducing our results

Train a WideResNet on MNIST with anisotropic kernels.

python -m train_mnist --path_to_graph ./saved_graphs --path_to_data ./data \
    --res_depth 2 --widen_factor 2 --anisotropic --coupled_sym --cuda

Train a WideResNet on CIFAR10 with spatial random pooling and anisotropic kernels.

python -m train_cifar10 --path_to_graph ./saved_graphs --path_to_data ./data \
    --res_depth 2 --widen_factor 4 --anisotropic --pool --reduction rand --cuda

Train a WideResNet on STL10 with spatial random pooling and anisotropic kernels.

python -m train_stl10 --path_to_graph ./saved_graphs --path_to_data ./data \
    --res_depth 3 --widen_factor 4 --anisotropic --reduction rand --cuda

Train a U-Net on ClimateNet with spatial max pooling, average unpooling, and anisotropic kernels.

python -m train_artc --path_to_graph ./saved_graphs --path_to_data ./data \
    --anisotropic --reduction max --expansion avg --cuda

License & citation

The content of this repository is released under the terms of the MIT license. Please cite our paper if you use it.

@inproceedings{cheblienet,
  title = {{ChebLieNet}: Invariant spectral graph {NN}s turned equivariant by Riemannian geometry on Lie groups},
  author = {Aguettaz, Hugo and Bekkers, Erik J. and Defferrard, Michaël},
  year = {2021},
  archivePrefix={arXiv},
  eprint={2111.12139},
  url = {https://arxiv.org/abs/2111.12139},
}