LieTorch: Tangent Space Backpropagation

Introduction

The LieTorch library generalizes PyTorch to 3D transformation groups. Just as torch.Tensor is a multi-dimensional matrix of scalar elements, lietorch.SE3 is a multi-dimensional matrix of SE3 elements. We support common tensor manipulations such as indexing, reshaping, and broadcasting. Group operations can be composed into computation graphs and backpropagation is automatically peformed in the tangent space of each element. For more details, please see our paper:

Tangent Space Backpropagation for 3D Transformation Groups
Zachary Teed and Jia Deng, CVPR 2021

@inproceedings{teed2021tangent,
  title={Tangent Space Backpropagation for 3D Transformation Groups},
  author={Teed, Zachary and Deng, Jia},
  booktitle={Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)},
  year={2021},
}

Installation

Installing (from source):

Requires torch >= 2 and CUDA >= 11. Tested up to torch==2.7 and CUDA 12. Make sure PyTorch and CUDA major versions match.

git clone --recursive https://github.com/princeton-vl/lietorch.git
cd lietorch

python3 -m venv .venv
source .venv/bin/activate

# install requirements
pip install torch torchvision torchaudio wheel

# optional: specify GPU architectures
export TORCH_CUDA_ARCH_LIST="7.5;8.6;8.9;9.0"

# install lietorch
pip install --no-build-isolation .

Installing (with pip)

# optional: specify GPU architectures
export TORCH_CUDA_ARCH_LIST="7.5;8.6;8.9;9.0"

pip install --no-build-isolation git+https://github.com/princeton-vl/lietorch.git

To run the examples, you will need these additional libraries

pip install opencv-python open3d scipy pyyaml

Running Tests

After building, you can run the tests

./run_tests.sh

Overview

LieTorch currently supports the 3D transformation groups.

Group	Dimension	Action
SO3	3	rotation
RxSO3	4	rotation + scaling
SE3	6	rotation + translation
Sim3	7	rotation + translation + scaling

Each group supports the following differentiable operations:

Operation	Map	Description
exp	g -> G	exponential map
log	G -> g	logarithm map
inv	G -> G	group inverse
mul	G x G -> G	group multiplication
adj	G x g -> g	adjoint
adjT	G x g*-> g*	dual adjoint
act	G x R^3 -> R^3	action on point (set)
act4	G x P^3 -> P^3	action on homogeneous point (set)
matrix	G -> R^{4x4}	convert to 4x4 matrix
vec	G -> R^D	map to Euclidean embedding vector
InitFromVec	R^D -> G	initialize group from Euclidean embedding

 

Simple Example:

Compute the angles between all pairs of rotation matrices

import torch
from lietorch import SO3

phi = torch.randn(8000, 3, device='cuda', requires_grad=True)
R = SO3.exp(phi)

# relative rotation matrix, SO3 ^ {8000 x 8000}
dR = R[:,None].inv() * R[None,:]

# 8000x8000 matrix of angles
ang = dR.log().norm(dim=-1)

# backpropogation in tangent space
loss = ang.sum()
loss.backward()

Converting between Groups Elements and Euclidean Embeddings

We provide differentiable FromVec and ToVec functions which can be used to convert between LieGroup elements and their vector embeddings. Additional, the .matrix function returns a 4x4 transformation matrix.

# random quaternion
q = torch.randn(1, 4, requires_grad=True)
q = q / q.norm(dim=-1, keepdim=True)

# create SO3 object from quaternion (differentiable w.r.t q)
R = SO3.InitFromVec(q)

# 4x4 transformation matrix (differentiable w.r.t R)
T = R.matrix()

# map back to quaterion (differentiable w.r.t R)
q = R.vec()

Examples

We provide real use cases in the examples directory

1. Pose Graph Optimization
2. Deep SE3/Sim3 Registrtion
3. RGB-D SLAM / VO

Acknowledgements

Many of the Lie Group implementations are adapted from Sophus.