lietorch 0.8.0

PDE-G-CNN package for PyTorch

LieTorch

This project provides a python package that expands the functionality of the PyTorch framework with PDE-based group equivariant CNN operators [1].

The name

The name LieTorch is a reference to the Norwegian mathematician Sophus Lie, whose contributions to geometry are extensively used in this project. The name is pronounced /li:/ (as in Lee) and not /ˈlī/ (as in lie).

Installation

The version of LieTorch you install must match the installed PyTorch version per the following table.

PyTorch version	LieTorch version
2.6+cu126	0.8
2.5	0.7
2.0	0.6
1.13	0.5
1.12	0.4

The following commands will install PyTorch and the lietorch package in a fresh virtual environment. Note that installing PyTorch via the Anaconda packages is deprecated since 2.6.

  conda create -n <environment name>
  conda activate <environment name>

  # install python and pip
  conda install python=3.13 pip

  # install PyTorch with pip
  pip install torch torchvision torchaudio --index-url https://download.pytorch.org/whl/cu126

  # and install lietorch with pip
  pip install lietorch

Make sure your Nvidia drivers are up to date!

Neural network modules

Modules are grouped according to the manifold they operate on. Most modules have a functional equivalant in the lietorch.nn.functional namespace.

Euclidean Space ℝ²

Basic operators:

Module	Functional	C++/CUDA backend
MorphologicalConvolutionR2	morphological_convolution_r2	✓
FractionalDilationR2	fractional_dilation_r2	✓
FractionalErosionR2	fractional_erosion_r2	✓
ConvectionR2	convection_r2	⏳
DiffusionR2	diffusion_r2	⏳
LinearR2	linear_r2	✓

Position and Orientation Space 𝕄₂

Basic operators:

Module	Functional	C++/CUDA backend
LiftM2Cartesian	lift_m2_cartesian	-
ReflectionPadM2	reflection_pad_m2	-
ConvM2Cartesian	conv_m2_cartesian	-
MaxProjectM2	max_project_m2	-
AnisotropicDilatedProjectM2	anisotropic_dilated_project_m2	✓
MorphologicalConvolutionM2	morphological_convolution_m2	✓
LinearConvolutionM2	linear_convolution_m2	✓
ConvectionM2	convection_m2	✓
DiffusionM2	diffusion_m2	⏳
FractionalDilationM2	fractional_dilation_m2	✓
FractionalErosionM2	fractional_erosion_m2	✓
LinearM2	linear_m2	✓

High-level modules for implementing PDE-based networks:

Module	Description/PDE
ConvectionDilationPdeM2	$u_t=-\mathbf{c}u + \lVert \nabla u \rVert^{2 \alpha}_{\mathcal{G}}$
ConvectionErosionPdeM2	$u_t=-\mathbf{c}u - \lVert \nabla u \rVert^{2 \alpha}_{\mathcal{G}}$
CDEPdeLayerM2	$u_t=-\mathbf{c}u + \lVert \nabla u \rVert^{2 \alpha}_{\mathcal{G}_1} - \lVert \nabla u \rVert^{2 \alpha}_{\mathcal{G}_2}$ with batch normalization and linear combinations

Loss functions

Additional loss functions.

Module	Functional	Description
lietorch.nn.loss.DiceLoss	lietorch.nn.functional.dice_loss	Binary DICE loss

Generic

The modules in the generic category do not fit into any previous category and include operators that serve as C++/CUDA implementation examples.

Module	Functional	C++/CUDA backend
GrayscaleDilation2D	grayscale_dilation_2d	✓
GrayscaleErosion2D	grayscale_erosion_2d	✓

Extra Dependencies

The included experiments additionally depend on the following packages.

• scikit-learn
• tqdm
• numpy
• sty
• mlflow
• libtiff

Structure

• /lietorch contains the main python package.
• /experiments contains various experiments, including those used in publications.
• /tests contains unit tests.
• /backend contains the source code of the C++/CUDA backend,
  • see ./backend/README.md if you wish to compile the extension yourself.
• /assets various files used in tests and documentation.

Cite

If you use this code in your own work please cite our paper:

[1] Smets, B.M.N., Portegies, J., Bekkers, E.J. et al. PDE-Based Group Equivariant Convolutional Neural Networks. J Math Imaging Vis (2022). https://doi.org/10.1007/s10851-022-01114-x

@article{smets2022pde,
  title={PDE-based Group Equivariant Convolutional Neural Networks},
  author={Smets, Bart and Portegies, Jim and Bekkers, Erik and Duits, Remco},
  journal={Journal of Mathematical Imaging and Vision},
  year={2022},
  doi={10.1007/s10851-022-01114-x},
  url={https://doi.org/10.1007/s10851-022-01114-x}
}