Lie Algebra Convolutional Neural Networks with Automatic Symmetry Extraction

Existing methods for incorporating symmetries into neural network architectures require prior knowledge of the symmetry group. We propose to learn the symmetries during the training of the group equivariant architectures. Our model, the Lie algebra convolutional network (L-conv), is based on infinitesimal generators of continuous groups and does not require discretization or integration over the group. We show that L-conv can approximate any group convolutional layer by composition of layers. We demonstrate how CNNs, Graph Convolutional Networks and fully-connected networks can all be expressed as an L-conv with appropriate groups. By allowing the infinitesimal generators to be learnable, L-conv can learn potential symmetries. We also show how the symmetries are related to the statistics of the dataset in linear settings. We find an analytical relationship between the symmetry group and a subgroup of an orthogonal group preserving the covariance of the input. Our experiments show that L-conv with trainable generators performs well on problems with hidden symmetries. Due to parameter sharing, L-conv also uses far fewer parameters than fully-connected layers.