Minimal Frame Averaging

Overview

Official code repository of paper Equivariance via Minimal Frame Averaging for More Symmetries and Efficiency. In this repository, we have provided decorators to convert any non-equivariant/invariant neural network functions into group equivariant/invariant ones. This enables neural networks to handle transformations from various groups such as $O(d), SO(d), O(1,d-1)$, etc.

In this code base, we provide the equivariance error test for all the groups included in the paper, which suffices to show our idea and corresponding algorithms.

Installation

Note: It is recommended to use a conda environment. First, you can install torch with the following command (if GPU is available):

conda install pytorch pytorch-cuda=11.8 -c pytorch -c nvidia

Then, install our package using:

pip install -e .

Usage

Example: $O(d)$-Equivariant/Invariant Decorator

The od_equivariant_decorator can be used to wrap any forward function to make it $O(d)$​-equivariant. An example model can be

import torch.nn as nn

class MyModel(nn.Module):
    def __init__(self):
        super(MyModel, self).__init__()
        # Define your layers here

    def forward(self, x):
        # Your forward pass implementation
        return x

To apply this decorator to this neural network class’s forward function:

import torch.nn as nn
from minimal_frame.group_decorator import od_equivariant_decorator

class MyModel(nn.Module):
    def __init__(self):
        super(MyModel, self).__init__()
        # Define your layers here
        
    @od_equivariant_decorator
    def forward(self, x):
        # Your forward pass implementation
        return x

To apply this decorator to a neural network instance's forward function:

# Instantiate your model
model = MyModel()

# Apply the O(d)-equivariant decorator
model.forward = od_equivariant_decorator(model.forward)

Similarly, the od_invariant_decorator can be used to make the network $O(d)$-invariant.

Supported Groups

This repository provides a collection of decorators to enforce equivariance or invariance properties in neural network models. Equivariance ensures that the output of the model transforms in the same way as the input under a given group of transformations. Invariance ensures that the output remains unchanged under such transformations.

These decorators in group_decorator.py are model-agnostic and can be applied to any model with input and output shapes of $n\times d$ for equivariance or with input shape of $n\times d$ and output shape of $1$ for invariance. A comprehensive equivariance analysis over various groups can be found here. The groups and MFA decorators (if not specified) include

• Orthogonal Group $O(d)$:
  • od_equivariant_decorator/od_invariant_decorator: Our MFA method for $O(d)$
  • od_equivariant_puny_decorator/od_invariant_puny_decorator: Puny's FA method [1] for $O(d)$
  • od_equivariant_puny_improve_decorator/od_invariant_puny_improve_decorator: Improved [1]'s FA for degenerate eigenvalues, see our Appendix H.3
  • od_equivariant_sfa_decorator/od_invariant_sfa_decorator: Stochastic FA method [2]
• Special Orthogonal Group $SO(d)$: sod_equivariant_decorator/sod_invariant_decorator
• Euclidean Group $E(d)$: ed_equivariant_decorator/ed_invariant_decorator
• Special Euclidean Group $SE(d)$: sed_equivariant_decorator/sed_invariant_decorator
• Unitary Group $U(d)$: ud_equivariant_decorator/ud_invariant_decorator
• Special Unitary Group $SU(d)$: sud_equivariant_decorator/sud_invariant_decorator
• Lorentz Group $O(1,d-1)$: o1d_equivariant_decorator/o1d_invariant_decorator
• Proper Lorentz Group $SO(1,d-1)$: so1d_equivariant_decorator/so1d_invariant_decorator
• General Linear Group $GL(d,\mathbb{R})$ (requiring full-column-rank input): gld_equivariant_decorator/gld_invariant_decorator
• Special Linear Group $SL(d,\mathbb{R})$ (requiring full-column-rank input): sld_equivariant_decorator/sld_invariant_decorator
• Affine Group $Aff(d,\mathbb{R})$ (requiring full-column-rank input): affd_equivariant_decorator/affd_invariant_decorator
• Permutation Group $S_n$: sn_equivariant_decorator/sn_invariant_decorator
• Direct product between Permutation Group $S_n$ and other groups
  • $S_n \times O(d)$: sn_od_equivariant_decorator/sn_od_invariant_decorator
  • $S_n \times SO(d)$: sn_sod_equivariant_decorator/sn_sod_invariant_decorator
  • $S_n \times O(1,d-1)$: sn_o1d_equivariant_decorator/sn_o1d_invariant_decorator

Additionally, the $S_n$-equivariant/invariant decorators in graph_group_decorator.py for undirected graphs are provided for any model with undirected adjacency matrices as input. The corresponding equivariance analysis can be found here, including decorators for

• Undirected unweighted adjacency matrix: undirected_unweighted_sn_equivariant_decorator/undirected_unweighted_sn_invariant_decorator
• Undirected weighted adjacency matrix: undirected_weighted_sn_equivariant_decorator/undirected_weighted_sn_invariant_decorator

References

[1] Puny, Omri, et al. "Frame averaging for invariant and equivariant network design." arXiv preprint arXiv:2110.03336 (2021).

[2] Duval, Alexandre Agm, et al. "Faenet: Frame averaging equivariant gnn for materials modeling." International Conference on Machine Learning. PMLR, 2023.

[3] McKay, Brendan D. "Practical graph isomorphism." (1981): 45-87.

Licence

This project is licensed under the MIT License. Please note that the nauty software is covered by its own licensing terms. All rights to the files mentioned in nauty.py are reserved by Brendan McKay under the Apache License 2.0.

Acknowledgements

We gratefully acknowledge the insightful discussions with Derek Lim and Hannah Lawrence. This work was supported in part by National Science Foundation grant IIS-2006861 and National Institutes of Health grant U01AG070112.

Citation

If you find this work useful, please consider citing:

@inproceedings{
lin2024equivariance,
title={Equivariance via Minimal Frame Averaging for More Symmetries and Efficiency},
author={Yuchao Lin and Jacob Helwig and Shurui Gui and Shuiwang Ji},
booktitle={Forty-first International Conference on Machine Learning},
year={2024},
url={https://openreview.net/forum?id=guFsTBXsov}
}