E(n)-equivariant Steerable CNNs (escnn)

escnn is a PyTorch extension for equivariant deep learning. escnn is the successor of the e2cnn library, which only supported planar isometries. Instead, escnn supports steerable CNNs equivariant to both 2D and 3D isometries, as well as equivariant MLPs.

Documentation	Paper ICLR 22	MSc Thesis Gabriele	e2cnn library
	Paper NeurIPS 19	PhD Thesis Maurice	e2cnn experiments

If you prefer using Jax, check our this fork escnn_jax of our library!

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Equivariant neural networks guarantee a specified transformation behavior of their feature spaces under transformations of their input. For instance, classical convolutional neural networks (CNNs) are by design equivariant to translations of their input. This means that a translation of an image leads to a corresponding translation of the network's feature maps. This package provides implementations of neural network modules which are equivariant under all isometries $\mathrm{E}(2)$ of the image plane $\mathbb{R}^2$ and all isometries $\mathrm{E}(3)$ of the 3D space $\mathbb{R}^3$, that is, under translations, rotations and reflections (and can, potentially, be extended to all isometries $\mathrm{E}(n)$ of $\mathbb{R}^n$). In contrast to conventional CNNs, $\mathrm{E}(n)$-equivariant models are guaranteed to generalize over such transformations, and are therefore more data efficient.

The feature spaces of $\mathrm{E}(n)$-equivariant Steerable CNNs are defined as spaces of feature fields, being characterized by their transformation law under rotations and reflections. Typical examples are scalar fields (e.g. gray-scale images or temperature fields) or vector fields (e.g. optical flow or electromagnetic fields).

Instead of a number of channels, the user has to specify the field types and their multiplicities in order to define a feature space. Given a specified input- and output feature space, our R2conv and R3conv modules instantiate the most general convolutional mapping between them. Our library provides many other equivariant operations to process feature fields, including nonlinearities, mappings to produce invariant features, batch normalization and dropout.

In theory, feature fields are defined on continuous space $\mathbb{R}^n$. In practice, they are either sampled on a pixel grid or given as a point cloud. escnn represents feature fields by GeometricTensor objects, which wrap a torch.Tensor with the corresponding transformation law. All equivariant operations perform a dynamic type-checking in order to guarantee a geometrically sound processing of the feature fields.

To parameterize steerable kernel spaces, equivariant to an arbitrary compact group $G$, in our paper, we generalize the Wigner-Eckart theorem in A Wigner-Eckart Theorem for Group Equivariant Convolution Kernels from $G$-homogeneous spaces to more general spaces $X$ carrying a $G$-action. In short, our method leverages a $G$-steerable basis for unconstrained scalar filters over the whole Euclidean space $\mathbb{R}^n$ to generate steerable kernel spaces with arbitrary input and output field types. For example, the left side of the next image shows two elements of a $\mathrm{SO}(2)$-steerable basis for functions on $\mathbb{R}^2$ which are used to generate two basis elements for $\mathrm{SO}(2)$-equivariant steerable kernels on the right. In particular, the steerable kernels considered map a frequency $l=1$ vector field (2 channels) to a frequency $J=2$ vector field (2 channels).

$\mathrm{E}(n)$-Equivariant Steerable CNNs unify and generalize a wide range of isometry equivariant CNNs in one single framework. Examples include:

• Group Equivariant Convolutional Networks
• Harmonic Networks: Deep Translation and Rotation Equivariance
• Steerable CNNs
• Rotation equivariant vector field networks
• Learning Steerable Filters for Rotation Equivariant CNNs
• HexaConv
• Roto-Translation Covariant Convolutional Networks for Medical Image Analysis
• 3D Steerable CNNs
• Tensor Field Networks
• Cormorant: Covariant Molecular Neural Networks
• 3D GCNNs for Pulmonary Nodule Detection

For more details, we refer to our ICLR 2022 paper A Program to Build E(N)-Equivariant Steerable CNNs and our NeurIPS 2019 paper General E(2)-Equivariant Steerable CNNs.

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The library is structured into four subpackages with different high-level features:

Component	Description
escnn.group	implements basic concepts of group and representation theory
escnn.kernels	solves for spaces of equivariant convolution kernels
escnn.gspaces	defines the Euclidean spaces and their symmetries
escnn.nn	contains equivariant modules to build deep neural networks

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WARNING: escnn.kernels received major refactoring in version 1.0.0 and it is not compatible with previous versions of the library. These changes do not affect the interface provided in the rest of the library but, sometimes, the weights of a network trained with a previous version might not load correctly in a newly instantiated model. We recommend using version v0.1.9 for backward compatibility.

Demo

Since $\mathrm{E}(2)$-steerable CNNs are equivariant under rotations and reflections, their inference is independent from the choice of image orientation. The visualization below demonstrates this claim by feeding rotated images into a randomly initialized $\mathrm{E}(2)$-steerable CNN (left). The middle plot shows the equivariant transformation of a feature space, consisting of one scalar field (color-coded) and one vector field (arrows), after a few layers. In the right plot we transform the feature space into a comoving reference frame by rotating the response fields back (stabilized view).

The invariance of the features in the comoving frame validates the rotational equivariance of $\mathrm{E}(2)$-steerable CNNs empirically. Note that the fluctuations of responses are discretization artifacts due to the sampling of the image on a pixel grid, which does not allow for exact continuous rotations.

For comparison, we show a feature map response of a conventional CNN for different image orientations below.

Since conventional CNNs are not equivariant under rotations, the response varies randomly with the image orientation. This prevents CNNs from automatically generalizing learned patterns between different reference frames.

Experimental results

$\mathrm{E}(n)$-steerable convolutions can be used as a drop in replacement for the conventional convolutions used in CNNs. While using the same base architecture (with similar memory and computational cost), this leads to significant performance boosts compared to CNN baselines (values are test accuracies in percent).

model	Rotated ModelNet10
CNN baseline	82.5 ± 1.4
SO(2)-CNN	86.9 ± 1.9
Octa-CNN	89.7 ± 0.6
Ico-CNN	90.0 ± 0.6
SO(3)-CNN	89.5 ± 1.0

All models share approximately the same architecture and width. For more details we refer to our paper.

This library supports $\mathrm{E}(2)$-steerable CNNs implemented in our previous e2cnn library as a special case; we include some representative results in the 2D setting from there:

model	CIFAR-10	CIFAR-100	STL-10
CNN baseline	2.6   ± 0.1	17.1   ± 0.3	12.74 ± 0.23
E(2)-CNN *	2.39 ± 0.11	15.55 ± 0.13	10.57 ± 0.70
E(2)-CNN	2.05 ± 0.03	14.30 ± 0.09	9.80 ± 0.40

While using the same training setup (no further hyperparameter tuning) used for the CNN baselines, the equivariant models achieve significantly better results (values are test errors in percent). For a fair comparison, the models without * are designed such that the number of parameters of the baseline is approximately preserved while models with * preserve the number of channels, and hence compute. For more details we refer to our previous e2cnn paper.

Getting Started

escnn is easy to use since it provides a high level user interface which abstracts most intricacies of group and representation theory away. The following code snippet shows how to perform an equivariant convolution from an RGB-image to 10 regular feature fields (corresponding to a group convolution).

from escnn import gspaces                                          #  1
from escnn import nn                                               #  2
import torch                                                       #  3
                                                                   #  4
r2_act = gspaces.rot2dOnR2(N=8)                                    #  5
feat_type_in  = nn.FieldType(r2_act,  3*[r2_act.trivial_repr])     #  6
feat_type_out = nn.FieldType(r2_act, 10*[r2_act.regular_repr])     #  7
                                                                   #  8
conv = nn.R2Conv(feat_type_in, feat_type_out, kernel_size=5)       #  9
relu = nn.ReLU(feat_type_out)                                      # 10
                                                                   # 11
x = torch.randn(16, 3, 32, 32)                                     # 12
x = feat_type_in(x)                                                # 13
                                                                   # 14
y = relu(conv(x))                                                  # 15

Line 5 specifies the symmetry group action on the image plane $\mathbb{R}^2$ under which the network should be equivariant. We choose the cyclic group $\mathrm{C}_8$, which describes discrete rotations by multiples of $2\pi/8$. Line 6 specifies the input feature field types. The three color channels of an RGB image are thereby to be identified as three independent scalar fields, which transform under the trivial representation of $\mathrm{C}_8$ (when the input image is rotated, the RGB values do not change; compare the scalar and vector fields in the first image above). Similarly, the output feature space in line 7 is specified to consist of 10 feature fields which transform under the regular representation of $\mathrm{C}_8$. The $\mathrm{C}_8$-equivariant convolution is then instantiated by passing the input and output type as well as the kernel size to the constructor (line 9). Line 10 instantiates an equivariant ReLU nonlinearity which will operate on the output field and is therefore passed the output field type.

Lines 12 and 13 generate a random minibatch of RGB images and wrap them into a nn.GeometricTensor to associate them with their correct field type feat_type_in. The equivariant modules process the geometric tensor in line 15. Each module is thereby checking whether the geometric tensor passed to them satisfies the expected transformation law.

Because the parameters do not need to be updated anymore at test time, after training, any equivariant network can be converted into a pure PyTorch model with no additional computational overhead in comparison to conventional CNNs. The code currently supports the automatic conversion of a few commonly used modules through the .export() method; check the documentation for more details.

To get started, we provide some examples and tutorials:

• The introductory tutorial introduces the basic functionality of the library.
• A second tutorial goes through building and training an equivariant model on the rotated MNIST dataset.
• Note that escnn also supports equivariant MLPs; see these examples.
• Check also the tutorial on Steerable CNNs using our library in the Deep Learning 2 course at the University of Amsterdam.

More complex 2D equivariant Wide Resnet models are implemented in e2wrn.py. To try a model which is equivariant under reflections call:

cd examples
python e2wrn.py

A version of the same model which is simultaneously equivariant under reflections and rotations of angles multiple of 90 degrees can be run via:

python e2wrn.py --rot90

You can find more examples in the example folder. For instance, se3_3Dcnn.py implements a 3D CNN equivariant to rotations and translations in 3D. You can try it with

cd examples
python se3_3Dcnn.py

Useful material to learn about Equivariance and Steerable CNNs

If you want to better understand the theory behind equivariant and steerable neural networks, you can check these references:

• Erik Bekkers' lectures on Geometric Deep Learning at in the Deep Learning 2 course at the University of Amsterdam
• The course material also includes a tutorial on group convolution and another about Steerable CNNs, using this library.
• Gabriele's MSc thesis provides a brief overview of the essential mathematical ingredients needed to understand Steerable CNNs.
• Maurice's PhD thesis develops the representation theory of steerable CNNs, deriving the most prominent layers and explaining the gauge theoretic viewpoint.

Dependencies

The library is based on Python3.7

torch>=1.3
numpy
scipy
lie_learn
joblib
py3nj

Optional:

torch-geometric
pymanopt>=1.0.0
autograd

WARNING: py3nj enables a fast computation of Clebsh Gordan coefficients. If this package is not installed, our library relies on a numerical method to estimate them. This numerical method is not guaranteed to return the same coefficients computed by py3nj (they can differ by a sign). For this reason, models built with and without py3nj might not be compatible.

To successfully install py3nj you may need a Fortran compiler installed in you environment.

Installation

You can install the latest release as

pip install escnn

or you can clone this repository and manually install it with

pip install git+https://github.com/QUVA-Lab/escnn

Contributing

Would you like to contribute to escnn? That's great!

Then, check the instructions in CONTRIBUTING.md and help us to improve the library!

Do you have any doubts? Do you have some idea you would like to discuss? Feel free to open a new thread under in Discussions!

Cite

The development of this library was part of the work done for our papers A Program to Build E(N)-Equivariant Steerable CNNs and General E(2)-Equivariant Steerable CNNs. Please cite these works if you use our code:

   @inproceedings{cesa2022a,
        title={A Program to Build {E(N)}-Equivariant Steerable {CNN}s },
        author={Gabriele Cesa and Leon Lang and Maurice Weiler},
        booktitle={International Conference on Learning Representations},
        year={2022},
        url={https://openreview.net/forum?id=WE4qe9xlnQw}
    }
    
   @inproceedings{e2cnn,
       title={{General E(2)-Equivariant Steerable CNNs}},
       author={Weiler, Maurice and Cesa, Gabriele},
       booktitle={Conference on Neural Information Processing Systems (NeurIPS)},
       year={2019},
       url={https://arxiv.org/abs/1911.08251}
   }

Feel free to contact us.

License

escnn is distributed under BSD Clear license. See LICENSE file.