G-RepsNet: A Fast and General Construction of Equivariant Networks for
  Arbitrary Matrix Groups
Sourya Basu Suhas Lohit Matthew Brand
Group equivariance is a strong inductive bias useful in a wide range of deep
learning tasks. However, constructing efficient equivariant networks for
general groups and domains is difficult. Recent work by Finzi et al. (2021)
directly solves the equivariance constraint for arbitrary matrix groups to
obtain equivariant MLPs (EMLPs). But this method does not scale well and
scaling is crucial in deep learning. Here, we introduce Group Representation
Networks (G-RepsNets), a lightweight equivariant network for arbitrary matrix
groups with features represented using tensor polynomials. The key intuition
for our design is that using tensor representations in the hidden layers of a
neural network along with simple inexpensive tensor operations can lead to
expressive universal equivariant networks. We find G-RepsNet to be competitive
to EMLP on several tasks with group symmetries such as O(5), O(1, 3), and O(3)
with scalars, vectors, and second-order tensors as data types. On image
classification tasks, we find that G-RepsNet using second-order representations
is competitive and often even outperforms sophisticated state-of-the-art
equivariant models such as GCNNs (Cohen & Welling, 2016a) and E(2)-CNNs (Weiler
& Cesa, 2019). To further illustrate the generality of our approach, we show
that G-RepsNet is competitive to G-FNO (Helwig et al., 2023) and EGNN (Satorras
et al., 2021) on N-body predictions and solving PDEs, respectively, while being
efficient.