Tangent Space Backpropagation for 3D Transformation Groups (2103.12032v2)

Abstract: We address the problem of performing backpropagation for computation graphs involving 3D transformation groups SO(3), SE(3), and Sim(3). 3D transformation groups are widely used in 3D vision and robotics, but they do not form vector spaces and instead lie on smooth manifolds. The standard backpropagation approach, which embeds 3D transformations in Euclidean spaces, suffers from numerical difficulties. We introduce a new library, which exploits the group structure of 3D transformations and performs backpropagation in the tangent spaces of manifolds. We show that our approach is numerically more stable, easier to implement, and beneficial to a diverse set of tasks. Our plug-and-play PyTorch library is available at https://github.com/princeton-vl/lietorch.

• The paper introduces a backpropagation method operating directly in the tangent spaces of 3D transformation groups to address numerical instabilities.
• It leverages Lie algebra and the adjoint representation to compute gradients directly, reducing computation graph size and enhancing efficiency.
• Experiments in inverse kinematics, pose graph optimization, and RGB-D registration demonstrate improved convergence and stability over traditional techniques.

Tangent Space Backpropagation for 3D Transformation Groups

The paper "Tangent Space Backpropagation for 3D Transformation Groups" by Zachary Teed and Jia Deng introduces a novel approach to executing backpropagation within computation graphs that involve 3D transformation groups, specifically $SO(3)$, $SE(3)$, and $Sim(3)$. The primary motivation is to address numerical instabilities encountered when 3D transformations, which inherently reside on smooth manifolds, are embedded into Euclidean spaces for computational tasks in deep learning frameworks.

Approach and Methodology

The proposed methodology capitalizes on the group structure of 3D transformations, enabling differentiation directly within tangent spaces rather than embedding these transformations in Euclidean spaces. This approach provides a numerically stable and implementation-friendly strategy for a variety of tasks that require handling 3D transformations, such as SLAM, pose estimation, and scene reconstruction.

The technique involves performing backpropagation through a composition of mappings within Lie groups, applying chain rules within the tangent space to circumvent singularities typically encountered in standard methods. By leveraging the adjoint representation and utilizing Lie algebra, the authors develop a differential framework that computes gradients directly in the tangent spaces. This results in more stable numerical computations and reduced computational overhead, due to smaller computation graphs.

Results and Implications

The authors implement their approach in LieTorch, a PyTorch library designed to handle 3D transformations naturally within deep learning systems. This library serves as a plug-and-play tool, allowing researchers to integrate 3D transformations into computation graphs sans the common pitfalls associated with existing methods.

Several experiments highlight the utility of the proposed methods:

1. Inverse Kinematics: The authors demonstrate improved convergence in solving inverse kinematics problems for robot arms, showcasing the stability and effectiveness of tangent space differentiation over traditional Euclidean methods.
2. Pose Graph Optimization: The paper proposes using Riemannian gradient descent for pose graph optimization, achieving superior convergence compared to existing methods like g2o and gtsam. The ability to efficiently run on GPUs provided a significant computational speedup.
3. RGB-D Registration: The authors extend their method to $Sim(3)$ transformations, successfully solving the RGB-D registration problem. This experiment underscores the flexibility of the approach, effectively tackling tasks previously unresolved within deep learning frameworks due to instability issues.
4. RGB-D SLAM: By integrating with a modified version of DeepV2D, the authors train a SLAM system on geodesic loss instead of an indirect proxy loss, achieving superior tracking accuracy.

Theoretical and Practical Implications

The theoretical implications of this work are significant, as it extends automatic differentiation to handle manifold structures more elegantly. Practically, this paves the way for more robust and scalable implementations in robotics and computer vision applications, where 3D transformations are central. The library's open-source nature invites further exploration and adaptation in other areas.

Future Prospects

Looking ahead, the adoption of tangent space backpropagation can foster advancements in fields requiring dynamic interactions with 3D environments. It encourages future explorations in optimization techniques and neural architectures tailored to smoothly manifold-structured data, pushing boundaries in AI-driven 3D modeling and robotic perception tasks.