General E(2)-Equivariant Steerable CNNs

Documentation | Experiments | Paper | Thesis Gabriele | Thesis Maurice | new escnn library

---

Check out our new escnn library which extends e2cnn to a wider class of equivariance groups.

While we will still provide some support for this older version, this library is deprecated and we plan to slowly abandon it in favour of the newer version escnn. Note that escnn already includes all features of e2cnn and many more. You can find a short summary of the main differences in the new version here.

---

---

e2cnn is a PyTorch extension for equivariant deep learning.

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 E(2) of the image plane $\mathbb{R}^2$, that is, under translations, rotations and reflections. In contrast to conventional CNNs, E(2)-equivariant models are guaranteed to generalize over such transformations, and are therefore more data efficient.

The feature spaces of E(2)-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 module instantiates 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. Feature fields are represented 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.

E(2)-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

For more details we refer to our NeurIPS 2019 paper General E(2)-Equivariant Steerable CNNs.

The library also supports equivariant Steerable partial differential operators as described in Steerable Partial Differential Operators for Equivariant Neural Networks.

---

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

Component	Description
e2cnn.group	implements basic concepts of group and representation theory
e2cnn.kernels	solves for spaces of equivariant convolution kernels
e2cnn.diffops	solves for spaces of equivariant differential operators
e2cnn.gspaces	defines the image plane and its symmetries
e2cnn.nn	contains equivariant modules to build deep neural networks

---

Demo

Since 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 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 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

E(2)-steerable convolutions can be used as a drop in replacement for the conventional convolutions used in CNNs. Keeping the same training setup and without performing hyperparameter tuning, this leads to significant performance boosts compared to CNN baselines (values are test errors in percent):

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

The models without * are for a fair comparison 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 paper.

Getting Started

e2cnn 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 e2cnn import gspaces                                          #  1
from e2cnn 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 = nn.GeometricTensor(x, feat_type_in)                            # 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 C₈, which describes discrete rotations by multiples of 2π/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 C₈. Similarly, the output feature space is in line 7 specified to consist of 10 feature fields which transform under the regular representation of C₈. The C₈-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. 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.

A hands-on tutorial, introducing the basic functionality of e2cnn, is provided in introduction.ipynb. Code for training and evaluating a simple model on the rotated MNIST dataset is given in model.ipynb.

More complex 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

Dependencies

The library is based on Python3.7

torch>=1.1
numpy
scipy

Optional:

pymanopt
autograd

The following packages

sympy
rbf

are required to use the steerable differential operators.

Check the branch legacy_py3.6 for a Python 3.6 compatible version of the library.

Installation

You can install the latest release as

pip install e2cnn

or you can clone this repository and manually install it with

pip install git+https://github.com/QUVA-Lab/e2cnn

NOTE: the rbf dependency (necessary to use differential operators) can not be installed automatically since PyPI does not support direct dependencies (see here). You can install rbf manually by following these instructions.

Cite

The development of this library was part of the work done for our paper General E(2)-Equivariant Steerable CNNs. Please cite this work if you use our code:

@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},
}

The implementation of steerable PDOs are part of the work done in the paper Steerable Partial Differential Operators for Equivariant Neural Networks. Please, cite it if you use this code in your own work:

@inproceedings{jenner2022steerable,
    title={Steerable Partial Differential Operators for Equivariant Neural Networks},
    author={Erik Jenner and Maurice Weiler},
    booktitle={International Conference on Learning Representations},
    year={2022},
    url={https://openreview.net/forum?id=N9W24a4zU}
}

Feel free to contact us.

License

e2cnn is distributed under BSD Clear license. See LICENSE file.